2ND Quarter Modules Gen Math

SDO MALABON CITY 11
GENERAL MATHEMATICS
Second Quarter
GENERAL MATHEMATICS SHS SECOND QUARTER
G11 SLEM # 1 – WEEK 1 – 2nd QUARTER
ILLUSTRATING SIMPLE AND COMPOUND INTEREST
OVERVIEW
This module was designed and written with you in mind. It is here to
help you understand how to illustrate and distinguish simple and
compound interest. The scope of this module permits it to be used in many
different learning situations. The language used recognizes the diverse
vocabulary level of students. The lessons are arranged to follow the
standard sequence of the course. But the order in which you read them
can be changed to correspond with the textbook you are now using. The
module focuses on achieving this learning competency:
Illustrate simple and compound interest

Distinguish between simple and compound interest.
After going through this module, you are expected to:

• compare simple interest and compound interest; and,
• differentiate simple and compound interest.
PRETEST
Choose the letter of the best answer. Write the chosen letter on a
separate sheet of paper.
1. What do you call to a person or institution who owes the money

or avails of the funds?
A. Borrower C. Lender
B. Co- maker D. Investor
2. What is the amount of time in years the money is borrowed or

invested?
A. Interest C. Rate
B. Principal D. Term
3. What do we call the amount of money or goods which are usually

offered by the banks to individuals and businessmen for short-
term periods?
A. Compound Interest C. Interest

B. Simple Interest D. Loan
LOOKING BACK
Let us recall the linear and exponential function.
➢ Linear Function – is a function whose graph is a line in the plane
with unique values, when the input variable is changed, the change
in the output is proportional to the change in the input.
This is a function in the form of 𝑓 (𝑥) = 𝑚𝑥 + 𝑏, where 𝑚 is the slope

and 𝑏 represents in 𝑦-intercept.
➢ Exponential Function – is a one-to-one function whose exponent

is a variable.
This function is in the form of 𝑓 (𝑥) = 𝑎 𝑥 , where 𝑎 > 0, 𝑎 ≠ 1 and

whose 𝑥 ∈ ℝ.
BRIEF INTRODUCTION OF THE LESSON
There are basic terms that we need to know about this lesson.
❖ Simple Interest (Is) – is the interest charged on the principal alone
for the entire length of the loan.
❖ Compound Interest (Ic) – is the interest on the principal and on the
accumulated past interest.
❖ Lender or creditor – a person (or institution) who invests the money
or makes the funds available.
❖ Borrower or debtor – person (or institution) who owes the money
or avails of the funds from the lender.
❖ Origin or loan date – date on which money is received by the
borrower.
❖ Repayment date or maturity date – date on which the money

borrowed, or loan is to be completely repaid.
❖ Time or term (t) – amount of time in years the money is borrowed
or invested; length of time between the origin and maturity dates.
❖ Principal (P) – amount of money borrowed or invested on the origin
date.
❖ Rate (r) – annual rate, usually in percent, charged by the lender, or
rate of increase of the investment.
❖ Interest (I) – amount paid or earned for the use of money.
❖ Maturity value or Future value (F) – amount after t years; that the
lender receives from the borrower on the maturity date.
Illustration of Simple and Compound Interest
Example: Suppose you won P5,000 and you plan to invest it for 6 years.
Bank A offer a 3% simple interest rate per year while Bank B offers 3%
compounded annually.
Solution:
Investment 1 Bank A: Simple interest, with annual rate 𝒓.
Principal Simple Interest (𝑰𝒔 ) Amount after t years

Time (𝒕)
(𝑷) Solution Answer (Maturity Value) (𝑭)
1 (5,000) (0.03) (1) 150 5,000 + 150 = 5,150
2 (5,000) (0.03) (2) 300 5,000 + 300 = 5,300
3 (5,000) (0.03) (3) 450 5,000 + 450 = 5,450
5,000
4 (5,000) (0.03) (4) 600 5,000 + 600 = 5,600
5 (5,000) (0.03) (5) 750 5,000 + 750 = 5,750
6 (5,000) (0.03) (6) 900 5,000 + 900 = 5,900
Investment 1 Bank B: Compound interest, with annual rate 𝒓.

Amount at Compound Interest (Ic)
Time Amount at the end of year 𝒕
the start of
(𝒕) Solution Answer (Maturity Value)
year 𝒕
1 5,000 (5,000) (0.03) (1) 150 5,000 + 150 = 5,150
2 5,150 (5,150) (0.03) (1) 154.50 5,150 + 154.50 = 5,304.50
3 5,304.50 (5,304.50) (0.03) (1) 159.14 5,304.50 + 159.14 = 5,463.64
4 5,463.64 (5,463.64) (0.03) (1) 163.91 5,463.64 + 163.99 = 5,627.55
5 5,627.55 (5,627.55) (0.03) (1) 168.83 5,627.55 + 168.83 = 5,796.38
6 5,796.38 (5,796.38) (0.03) (1) 173.89 5,796.38 + 173.89 = 5,970.27
Comparing the two investments:
Simple Interest (in pesos): 5,900 – 5,000 = P900

Compound Interest (in pesos): 5,970.35 – 5,000 = P970.35
It can be seen from the illustration above that as investment period

becomes greater than 1, the investment in compound interest
becomes greater than the investment in simple interest. This can be
graphically interpreted in the linear and exponential graphs above.
ACTIVITY
Use an interest table (like what was presented above) to illustrate

the following situations below. Write your answer on a separate sheet.
1. Aling Nette borrowed P108,500 at an annual interest rate of

1
15 4 %. How much will she pay at the end of 1.5 years? Illustrate
your answer.
2. Thess deposited P30,000 in a bank that pays 0.6% interest

compounded annually. How much will her money be after 3
years?
REMEMBER
• Simple interest remains constant throughout the investment term.
• This is an example of a linear function.
• Compound interest, on the other hand, shows that the interest from
the previous year also earns interest. Thus, the interest grows every
year.
• Compound interest is an example of an exponential function.
CHECKING YOUR UNDERSTANDING
Choose the letter of the correct answer. Write it on a separate sheet

of paper.
1. If Juana invested money in a bank and after a period, she

collected the whole amount. What she has in hand is called?
A. Principal C. Future Value

B. Borrower D. Repayment Date
2. Due to COVID-19 pandemic, Mark affected his financial status,

so he decided to apply online for an emergency loan offered by
the Social Security System. Mark is what we call?
A. Lender C. Principal
B. Borrower D. Future Value
3. After 5 working days, Mark received his emergency loan

amounted of P20,000 with an interest of 2.5% and he must pay
this loan for 3 years. The period of time he must pay his
emergency load is called what?
A. Loan Date C. Term

B. Interest D. Rate
4. Mark computed the amount of his loan to be paid in 3 years at

2.5% interest which amounted to P1,500. This value of the
interest is based on P20,000 is called?
A. Principal C. Future Value

B. Borrower D. Repayment Date
5. Which of the following is not mentioned in the given situation

above?
A. Loan Data C. Term

B. Interest D. Rate
POSTTEST
1. Which of the following are used by banks in calculating interest

for long term investments and loans?
A. Compound Interest C. Interest

B. Simple Interest D. Loan
2. Which of the following are NOT considered exponential?
A. Bumbay System C. Cooperative Loan

B. Bank Investment D. Car Loan
3. What should be added to the principal in order to get the future

value?
A. Maturity Value C. Present Value

B. Downpayment D. Interest
COMPUTE SIMPLE INTEREST
OVERVIEW
help you understand how to compute interest, maturity value, future value,
and present value in simple interest and compound interest environment.
The scope of this module permits it to be used in many different learning
situations. The language used recognizes the diverse vocabulary level of
students. The lessons are arranged to follow the standard sequence of
the course. But the order in which you read them can be changed to
correspond with the textbook you are now using. The module focuses on
achieving this learning competency:
Compute interest, maturity value, future value, and present value in

simple interest and compound interest environment.

• finding simple interest, principal, rate, time, maturity value, and
• solving real-life problems involving simple interest.
PRETEST
1. What should be added to the principal in order to get the future

value?
A. Maturity Value C. Downpayment

B. Present Value D. Interest
2. How much should be paid if P12 000.00 is borrowed from a bank

that offers a simple interest rate of 4% in 6 years?
A. P 2 880.00 C. P 14 880.00
B. P 12 880.00 D. P 24 880.00
3. Which of the following is the correct simple interest yield of P

100.00 charged at 40% per annum?
A. P 140.00 C. P 400.00
B. P 40.00 D. P 104.00
LOOKING BACK
Let us recall how to change percent to decimal.
To change percent to decimal, remove the percent sign and move the
decimal point 2 places to the left. If the given percent has fraction,
change the fraction form to decimal then follow the steps in changing
percent to decimal.
Examples: Change the following percent in decimal.

1. 18% → 18. → 𝟎. 𝟏𝟖
2. 5.25% → 2.25 → 𝟎. 𝟎𝟐𝟐𝟓
3
3. 9 4 % → 9.75% → 9.75 → 𝟎. 𝟎𝟗𝟕𝟓
Simple interest is usually used for short borrowing. An annual interest is

based on the 3 factors:
➢ Principal which is the amount invested or borrowed.

➢ Simple interest rate, usually expressed in percent
➢ Time or term of loan, in years
The formula for annual simple interest, 𝐼𝑠 , is given by:
𝑰𝒔 = 𝑷𝒓𝒕
where: 𝐼𝑠 is the simple interest

𝑃 is the principal
𝑟 is the rate
𝑡 is the term or time, in years
The maturity value or the future value, 𝐹, is given by:
𝑭 = 𝑷 + 𝑰𝒔
where: 𝐹 is the maturity or future value

𝐼𝑠 is the simple interest
If 𝐼𝑠 = 𝑃𝑟𝑡 will be substituted in the previous formula, 𝐹 = 𝑃 + 𝐼𝑠 , another

alternative formula for the maturity value can be derived, which is:
𝑭 = 𝑷 + 𝑷𝒓𝒕 → 𝑭 = 𝑷(𝟏 + 𝒓𝒕)
Example: Joel borrowed P 20,000 from a bank to charge 12% simple

interest rate. The loan plus the interest paid in one lump sum at the end
of 3 years. How much will he pay the bank when the loan is due?
1
Given: 𝑃 = 20,000 𝑟 = 12% or 0.12 𝑡 = 3 4 years
Find 𝐹:
Solution:
Change 12% to 0.12 and 3 ¼ years to 3.25 years. Substitute the values
in the formula.
𝐼𝑠 = 𝑃𝑟𝑡
𝐼𝑠 = 20000 (0.12)(3.25)
𝑰𝒔 = 𝑷 𝟕𝟖𝟎𝟎
This means:
𝐹 = 𝑃 + 𝐼𝑠
𝐹 = 20000 + 7800
𝑭 = 𝟐𝟕𝟖𝟎𝟎
Therefore, the future value after 𝟑 years is P 𝟐𝟕𝟖𝟎𝟎.
Alternatively, we can just use the formula 𝐹 = 𝑃(1 + 𝑟𝑡 ) to directly solve

the future value. Thus:
𝐹 = 𝑃(1 + 𝑟𝑡 )
𝐹 = 20000[1 + (0.12)(3.25)]
𝐹 = 20000(1.39)
𝑭 = 𝟐𝟕𝟖𝟎𝟎
Example: Mario invested at an annual interest rate of 7%, an amount

earned P11200 of simple interest in two years. How much money was
originally invested, and the amount paid at the end of 2 years?
Given: 𝑟 = 7% 𝐼𝑠 = 11,200 𝑡=2

Find 𝑃 and 𝐹:
Solution:
To compute for the original amount (𝑃) invested, use the equation
𝑰
𝑷=
𝒓𝒕
11,200
𝑃=
(0.07)( 2 )
𝑃 = 80000
Thus, the amount invested is P 80,000.
In computing the amount paid for 2 years, use the formula:

𝐹 = 𝑃(1 + 𝑟𝑡)
𝐹 = 80000 + 11200
𝑭 = 𝟗𝟏𝟐𝟎𝟎
Therefore, the future or maturity value after 2 years is P 91,200.
Example: For how long must P 45000 be invested at 8% simple interest

rate in order to have P 46800?
Given: 𝐹 = 46800 𝑃 = 45000 𝑟 = 8% or 0.08

Find 𝑡:
Solution:
𝐼
𝑡 =
𝑃𝑟
But 𝐼 = 𝐹 − 𝑃, so:
𝐹−𝑃
𝑡 =
𝑃𝑟
46800 − 45000
𝑡=
(45000)(0.08)
1800
𝑡=
3600
𝑡 = 0.5
1
Note that the time is expressed in years, so 0.5 years is equal to 2 of a
year or 6 months. Therefore, it takes 6 months to have 𝑷 𝟒𝟔𝟖𝟎𝟎.
Example: A man deposits P 90000 in a savings bank for 5 months. If the

interest amounts to P 1875, find the rate of interest.
Given: 𝑃 = 90000 𝐼𝑠 = 𝑃 1875 𝑡 = 5 months

Find 𝑟.
Solution:
Solve 𝑟 as follows:
𝐼𝑠
𝑟 =
𝑃𝑡
Since 𝑡 is 5 months, and 𝑡 must be expressed in years, we immediately
5
take 𝑡 = 12. Hence,
1875
𝑟=
5
(90,000) ( )
12
1875
𝑟=
37500
𝑟 = 0.05 → 𝑟 = 5%
Therefore, the rate of interest is 5%.
ACTIVITY
A. Find the unknown principal 𝑃, rate 𝑟, time 𝑡, and interest 𝐼𝑠 , by

completing the table.
Principal (𝑷) Rate (𝒓) Time (𝒕) Interest (𝑰𝒔 )

P10,000 8.5% 15 (1)
(2) 2% 5 P10,000
P260,000 (3) 2 P2,600
P500,000 10.5% (4) P175,500
P880,000 91/4% 2.5 (5)
B. Complete the table by finding the unknown.
Principal Rate Time (𝒕) in Future Value

Interest (𝑰𝒔 )
(𝑷) (𝒓) years (𝑭)
(1) 8% 2 1/4 P4,500 (2)
P200,000 (3) 1.5 P37,000 (4)
P50,000 9.75% (5) (6) P59,500
(7) 10.5% (8) P157,000 P457,000
P60,000 (9) 18 months (10) P88,200
C. Solve the following problems. (Round amounts to the nearest

centavo.)
1. Mrs. Delos Santos deposited P 105000 in a savings bank. If the
3
bank pays 8 % simple interest, determine the following:
5
a. amount of interest earned per year
b. total interest earned after 3 years and 5 months.
2. Marife got a loan of P 75000 from a credit union payable at the end
of 5 years and 8 months to put up a new business. If the credit union
charges 10.25% simple interest, how much will she repay the credit
union?
3. Perla borrowed a certain amount from the bank that charges 92/5%
simple interest. If she received P 49750, how much was the amount
borrowed?
4. If a bank gives 11.5% interest per annum, for how long will an
investment of P 25000 accumulate to P 45000?
REMEMBER
Simple interest is a simple way of investing the money by multiplying the
principal, rate, and time. To find the simple interest and the future value,
use the formulas:
Simple Interest Formula: Marurity (Future) Value Formula:
𝑰𝒔 = 𝑷𝒓𝒕 𝑭 = 𝑷 + 𝑰𝒔 𝑭 = 𝑷 (𝟏 + 𝒓𝒕)
In doing problems involving interest:
a. Analyze the problem.
b. Determine the unknown.
c. List down all the given in the problem.
d. Substitute the given in the formula.
e. Solve for the unknown.

of paper.
1
1. A man borrowed P 25000 for 2 4 years at 8% per year. Find the
amount of simple interest.
A. P 5500 C. P 3500
B. P 4500 D. P 2500
2. Mr. Paul Lopez is a health center nurse. Due to COVID-19

pandemic the means of transportation was affected. He decided
to borrow P 200000 from his company to buy a motorcycle as his
means of transportation going to his work. If he will pay an
interest of P 37000 for one and a half years, what is the rate of
interest of his loan?
A. 11.5% C. 12.5%
B. 12% D. 13%
3. If Miss Lorna Fuentabella paid an interest o P 3000 for a loan

obtained at 9% per annum for eight months, what is the original
loan?
A. P 40000 C. P 50000
B. P 45000 D. P 55000
4. PAGIBIG Fund offered a Calamity Loan due to COVID-19

pandemic with an interest rate of 5.95% per year. Mrs. Santos
applied for P 50000 emergency loan. If the total interest of the
loan was 14875, how long will it take to pay her loan?
A. 5 years C. 7 years
B. 6 years D. 8 years
5. Peter deposited an amount of P 12800 in a savings bank that

gives 6.5% interest for 8 years. How much would he have in his
account at the end of 8 years assuming that no withdrawals were
made?
A. P 19556 C. P 19536
B. P 19546 D. P 19526
POSTTEST
1
1. Andrew deposited P 80000 in a savings account that pays 4 2 %
simple interest and earned P 10800 in interest. How long in years
did it take Andrew to earn the interest?
A. 2 years C. 4 years
B. 3 years D. 5 years
2. How much should be paid if P 10000.00 is borrowed from a bank

that offers a simple interest rate of 5.25% in 5.5 years?
A. P 2887.50 C. P 4887.50
B. P 12887.50 D. P 24887.50
3. Matti plans to buy a new laptop and printer using her credit card,
it will cost her P 56800. How much rate of interest will be charged
if the interest is P 2156 for 18 months based on the banker’s rule?
A. 2.56% C. 2.36%
B. 2.46% D. 2.26%
COMPUTE SIMPLE INTEREST
OVERVIEW
help you understand how to compute interest, maturity value, future value,
and present value in simple interest and compound interest environment.
Computes interest, maturity value, future value, and present value

in simple interest and compound interest environment.

• finding compound interest and maturity value, and
• solving real-life problems involving simple interest.
PRETEST
1. Pamela invested P 130000.00 in a fund that motions 2.5%

interest compounded annually. How much will be her money after
24 months?
A. P 163851.87 C. P 149740.23
B. P 157342.11 D. P 136581.25
2. In relation to QUESTION 1, what will be the new interest when

Pamela decides to extend its investment for 6 months more?
A. P 10223.43 C. P 9001.34
B. P 9653.98 D. P 8277.78
3. Luisa borrowed an amount of money from a bank that offers 8%

interest compounded yearly. How much did she borrowed (in
nearest peso) if she paid an amount of Php 514 264.83 for 5
years?
A. P 410000.00 C. P 370000.00
B. P 390000.00 D. P 350000.00
LOOKING BACK
Let us recall the law of exponents for power and the negative exponent.
➢ Power Rule (Powers to Powers): (𝑎 𝑚)𝑛 = 𝑎 𝑚𝑛 , this rule states that to

raise a power to a power, you need to multiply the exponents. There
are several other rules that go along with the power rule, such as the
product-to-powers rule and the quotient-to-powers rule.
Example: (2𝑥 3 )4 = (24 ) (𝑥 3 )4 = (2 ∗ 2 ∗ 2 ∗ 2)(𝑥 12 ) = 16𝑥12
1
➢ Negative Exponent Rule: 𝑎−𝑛 = 𝑎𝑛, this rules states that negative
exponents in the numerator get moved to the denominator and
become positive exponents. Negative exponents in the denominator
get moved to the numerator and become positive exponents. Only
move the negative exponents.
1 1 1
Example: 5−3 = = =
53 5∗5∗5 125
Many savings accounts pay compound interest. This is usually used by

banks in calculating interest for long term investments and loans such as
saving account and the time deposits.
The compound amount or future value is the final amount of the

investment or loan at the end of the term or last period.
The formula for the maturity (future) value, 𝐹 is given by:
𝑭 = 𝑷(𝟏 + 𝒓)𝒕
where: 𝐹 is the maturity value at the end of the term

𝑟 is the rate
Similarly, the present value can be obtained by:
𝑭
𝑷 = 𝑭(𝟏 + 𝒓)−𝒕 → 𝑷 =
(𝟏 + 𝒓)𝒕
The compound interest, 𝐼𝑐 , can be found by subtracting the future value

to the present value, that is:
𝑰𝒄 = 𝑭 − 𝑷
Example: Find the future value and compound interest if P 50000 is

invested for 5 years at 6% compounded annually.
Given: 𝑃 = 50000 𝑟 = 6% or 0.06 𝑡 = 5 years

Find 𝐹 and 𝐼𝑐 :
Solution:
Change 6% to 0.06. Substitute the values in the formula.
𝐹 = 𝑃(1 + 𝑟)𝑡
𝐹 = (50000)(1 + 0.06)5
𝐹 = 50000(1.3382226)
𝑭 = 𝟔𝟔𝟗𝟏𝟏. 𝟑𝟎
This means that the compound interest is:
𝐼𝑐 = 𝐹 − 𝑃
𝐼𝑐 = 66911.30 − 50000.00
𝑰𝒄 = 𝟏𝟔𝟗𝟏𝟏. 𝟑𝟎
Therefore, the future value after 𝟓 years is P 𝟔𝟔𝟗𝟏𝟏. 𝟐𝟖 and the

compound interest is P 𝟏𝟔𝟗𝟏𝟏. 𝟐𝟖.
Example: Supposed your father deposited in your bank account P 10000

at an annual interest of 0.5% compounded yearly when you graduate from
kindergarten and did not get the amount until you finish Grade 12. How
much will you have in your bank account after 12 years?
Given: 𝑃 = 10000 𝑟 = 0.5% = 0.005 𝑡 = 12

Find 𝐹:
Solution:
The future value can be obtained by:
𝐹 = 𝑃(1 + 𝑟)𝑡
𝐹 = 10000(1 + 0.005)12
𝐹 = 10000(1.005)12
𝑭 = 𝟏𝟎𝟔𝟏𝟔. 𝟕𝟖
Thus, the amount will become P 𝟏𝟎𝟔𝟏𝟔. 𝟕𝟖 after 12 years.
Example: How much must be invested today in a savings account in

order to have Php300,000 in 6 years if money earns 8.5% compounded
annually?
Given: 𝐹 = 300000 𝑡=6 𝑟 = 8.5% or 0.085

Find 𝑃:
Solution:
The present value can be solved by using the formula:
𝐹
𝑃=
(1 + 𝑟)𝑡
300000
𝑃=
(1 + 0.085)6
300000
𝑃=
1.631468
𝑷 = 𝟏𝟖𝟑𝟖𝟖𝟑. 𝟒𝟕
Therefore, the present value is P 𝟏𝟖𝟑𝟖𝟖𝟑. 𝟒𝟕.
Example: How much money should Mr. Reyes place in a time deposit If
a bank that pays 1.1% compounded annually so that he will have Php
200,000.00 after 6 years?
Given: 𝐹 = 200000 𝑟 = 1.1% = 0.11 𝑡 = 6 years

Find 𝑃.
Solution:
The present value can be solved by using the formula:
𝐹
𝑃=
(1 + 𝑟)𝑡
200000
𝑃=
(1 + 0.11)6
200000
𝑃=
1.067842
𝑷 = 𝟏𝟖𝟕𝟐𝟗𝟑. 𝟔𝟑
Therefore, Mr. Reyes should deposit P 𝟏𝟖𝟕𝟐𝟗𝟑. 𝟔𝟓 so that he will

have P 𝟐𝟎𝟎𝟎𝟎𝟎. 𝟎𝟎.
ACTIVITY
Track the right path to finish the maze. Write your answer on a
separate sheet of paper. Show your complete solution.
REMEMBER
Steps in Finding the Maturity value at compound interest:

1. Analyze the given problem
2. Determine what values are given and values you need to find.
3. Apply the formula 𝑭 = (𝟏 + 𝒓)𝒕
4. Substitute all the known values in the formula.
To find the present value at compound interest, use the formula

𝑭
𝑷= 𝒕
𝒐𝒓 𝑷 = (𝟏 + 𝒓)−𝒕
(𝟏+𝒓)

of paper.
1. A man borrowed Php25,000 for 2.5 years at 8% compounded

yearly. How much was the amount after the end of the term?
A. Php30,303.96 B. Php30,330.96 C. Php30,333.96
2. Mr. Paul Lopez is a health center nurse, due to COVID-19 pandemic
the means of transportation was affected. He decided to borrow
Php100,000 from his company to buy a motorcycle as his means of
transportation going to work. If the interest rate of his loan is 11.5%
compounded annually for 3 years, what was the total amount of his
loan after the end of the term?
A. Php138,619.79 B. Php138,619.69 C. Php148,619.59
3. Find the present value of Php300,000 due in 6 years at 7.½%
A. Php174,388.46 B. Php184,388.46 C. Php194,388.46
4. On the birth of his grandson, a grandmother wished to invest money
to accumulate Php150,000 by the time his grandson turned 7 years
old. If the interest rate of her investment is 6.6% compounded
yearly, how much should she invest now?
A. Php95,793.87 B. Php95,893.87 C. Php95,983.87
5. A man will need Php800,000 to buy a vehicle at the end of 5 years.
How much should he place in a savings account now that pays
7.25% interest compounded annually to be able to buy the vehicle?
A. Php563,770.97 B. Php563,771.97 C. Php563,717.97
POSTTEST
1. How much should be paid if Php. 10 000.00 is borrowed from a

bank that offers a 5.25% compounded interest in 5.5 years?
A. Php 2 887.50 C. Php 14 887.50
B. Php 12 887.50 D. Php 24 887.50
2. Matti plans to buy a new laptop and printer using her credit card,
it will cost her P56,800. How much is the present value if the
interest rate is 2.75% compounded for 1.5 years based on the
banker’s rule?
A. Php 55,545.03 C. Php 44,530.03

B. Php 54,535.03 D. Php 40,555.03
3. Find the present value of Php235,700 due in 4 years at 6½%

A. Php393, 325.50 C. Php 183,215.05

B. Php 273,415.10 D. Php 93,875.25

COMPOUND INTEREST MORE THAN ONCE A YEAR
OVERVIEW
help you understand how to solve problems involving simple and
Solve problems involving simple and compound interest.
After going through this module, you are expected to solve for maturity
value, interest and present value more than once a year.
PRETEST
1. Sean invested Php 100 000.00 in a bank that offers 7% interest

compounded monthly. How much will be his money in 4 years?
A. Php 176 251.78 C. Php 147 234.97

B. Php 163 615.10 D. Php 132 205.39
2. In relation to QUESTION 1, what if the bank decided to change

the interest rate to 12.5% compounded quarterly?
A. Php 176 851.72 C. Php 169 534.22

B. Php 173 251.78 D. Php 163 615.10
3. What is the present value of Php 25 000.00 due in 2 years and 6

months if money is worth 10% compounded quarterly?
A. Php 24 577.12 C. Php 20 445.29

B. Php 22 129.01 D. Php 19 529.00
LOOKING BACK
Whenever a simple interest is added to the principal at regular intervals,
and the sum becomes the new principal, the interest is said to be
compounded.
It means that in compound interest, the interest earned at the end of the
term is automatically reinvested to earn more interest. How about if this
interest is compounding more than once a year?
Before we calculate compound amount and interest, let us first

familiarize ourselves with the following terms:
Interest compounded Number of Interest/
Compounding Periods in
1Year
1. Annually or once a year 1
2. Semi-annually or every 6 months 2
3. Quarterly or every 3 months 4
4. Monthly or once a month 12
5. Daily or once a day 365/366(leap year)
The formula for the maturity (future) value, 𝐹 is given by:
𝑭 = 𝑷(𝟏 + 𝒓)𝒕
where: 𝐹 is the maturity value at the end of the term

𝑟 is the rate
Bear in mind the following terminologies:

• Conversion or Interest Period. This is the time between two
successive conversions of interest.
• Frequency of Conversion (𝒎). This is the number of conversion
periods of the investment or loan in one year. It is denoted by 𝒎.
Example. Monthly (𝑚 = 12) and quarterly (𝑚 = 4)
• Nominal Rate (𝒊(𝒎) ). This is the started rate of interest per year,
denoted 𝒊(𝒎) .
Example: 𝑖 (𝑚) = 8% means 8% interest per year.
• Rate per Conversion Period (𝒋). This is the rate of Interest for
each conversion period. Therefore,
𝒊(𝒎) 𝒂𝒏𝒏𝒖𝒂𝒍 𝒐𝒓 𝒏𝒐𝒎𝒊𝒏𝒂𝒍 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕
𝒋= =
𝒎 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒔𝒊𝒐𝒏
Examples: For 8% compounded quarterly:

0.08
𝑗= → 𝑗 = 2% 𝑜𝑟 0.02
4
For 9% compounded monthly:
0.09
𝑗= → 𝑗 = 0.75% 𝑜𝑟 0.075
12
• Number of conversion periods (𝒏). This is the total number of
times interest is calculated for the entire term of the investments or
loan. It is denoted by 𝒏 and obtained by multiplying the time in years
by the frequency of conversion per year.
𝒏 = 𝒕𝒎
Examples: If 𝑡 = 2 years compounded semi-annually, then
𝑛 = 𝑡𝑚 → 𝑛 = (2)(2) → 𝑛 = 4
If 𝑡 = 3 years & 9 months compounded every month,
then
9
𝑛 = 𝑡𝑚 → 𝑛 = (3 ) (4) → 𝑛 = 15
12
Recall that the formula for compound interest is:

𝐹 = 𝑃(1 + 𝑟)𝑡
𝑖 (𝑚)
We modify this formula by letting 𝑟 = . So:
𝑚
(𝒎) 𝒎𝒕
𝒊
𝑭 = 𝑷 (𝟏 + )
𝒎
where:
𝐹 is the maturity (future) value
𝑖 (𝑚) is the nominal rate of interest (annual rate)
𝑚 is the frequency of conversion
𝑡 is the term/time of investment/loan in years
Example: Dolor’s Kakanin wants to open a new account of Php 500000,

in 5 years’ time they will open a new branch in Quezon City and wants to
invest this amount yielding 3.75% interest compounded quarterly. How
much will be the amount after 5 years, assuming that there was no
withdrawal made? How much is the interest?
Given: 𝑃 = 500000 𝑡 = 5 years 𝑗 = 3.75% or

0.0375
Find: (a) 𝐹 (b) 𝐼𝑐
Solution:
Compute the interest rate in a conversion:
𝑖 (𝑚) 0.0375
𝑗= = = 0.009375
𝑚 4
Compute for the total number of conversion periods:

𝑛 = 𝑚𝑡 → 𝑛 = (4)(5) → 𝑛 = 20
Compute for the maturity value using 𝑭 = 𝑷(𝟏 + 𝒊)𝒏 :

𝐹 = 𝑃(1 + 𝑖)𝑛
𝐹 = 500000(1 + 0.009375)20
𝑭 = 𝟔𝟎𝟐𝟓𝟓𝟓. 𝟓𝟎
Compute for the interest using:
𝐼𝑐 = 𝐹 − 𝑃
𝐼𝑐 = 602588.57 − 500000
𝑰𝒄 = 𝟏𝟎𝟐𝟓𝟖𝟖. 𝟓𝟕
Therefore, Dolor’s Kakanin will have Php 602,588.57 after 5 years

with an of interest of Php102,588.57
Example: Suppose that Arny Dading’s Peachy-Peachy needs P125,000

in forty-eight months to start a new product. An online business offers
4.5% interest, compounded semi-annually for partnership and a
cooperative offers 3.5% interest compounded monthly. How much should
he invest now, and which company will he choose?
Given: 𝐹 = 125,000 𝑡 = 48 months

( ) ( )
𝑖1𝑚 = 4.5% 𝑖2𝑚 = 3.5%
𝑚1 = 2 𝑚2 = 12
Find 𝑃:
Solution:
The online business offers 4.5% compounded semi- annually. Compute
the interest rate in a conversion period by:
𝑖 (𝑚) 0.045
𝑗= →𝑗= → 𝑗 = 0.0225
𝑚 2
Compute for the total number of conversion periods. Convert 48 months
to years. Hence:
48
𝑡= →𝑡=4
12
𝑛 = 𝑚𝑡 → 𝑛 = (4)(4) → 𝑛 = 16
Compute for the present value using:
𝐹
𝑃=
(1 + 𝑖)𝑛
125000
𝑃=
(1 + 0.0225)16
125000
𝑃=
1.42762146
𝑷 = 𝟖𝟕𝟓𝟓𝟖. 𝟐𝟐
Arny Dading needs P 87558.22 to invest in Online Business.
The computation of present value for cooperative is left as an exercise.

ACTIVITY
Write your answer on a separate sheet of paper. Show your

complete solution.
REMEMBER
To solve for maturity value and present value compounding more than
once a year, follow the steps below:
3. Compute the interest rate in a conversion period and the total
number of conversion period
4. Apply the formula
Maturity Value Present Value
𝒎𝒕
𝒊(𝒎) 𝑭
𝑭 = 𝑷 (𝟏 + ) or 𝑭 = 𝑷(𝟏 + 𝒋)𝒏 𝑷= or 𝑷 = 𝑭 (𝟏 + 𝒋)−𝒏
𝒎 (𝟏+𝒋)𝒏

of paper.
1. Mrs. San Jose borrowed Php 30,750 and promised to pay the
3
principal and interest at 7 5 % compounded monthly. How much did
she pay after 3 years and 6 months?
A. P 39,340.93 B. P 39,340.95 C. P 39,340.97

5
2. Find the interest earned if Php 22,550 is invested at 5 8 %
9
compounded quarterly for 4 12 years.
A. P 6,851.95 B. P 6,851.96 C. P 6,851.97
3. How much must a parent deposit in a bank now that gives 7.5%
interest compounded semi-annually so that after 10 years, he will
have Php 750,000?
A. P 359,369.26 B. P 359,269.26 C. P 359,169.26
4. San Fabian Beach Resort wants to renovate their cottages. The

owner borrowed P850,000 and promised to pay the principal and
interest at 7.35% compounded monthly for 5 years and 4 months.
How much will he pay at the end of the term?
A. P 1,652,344.93 B. P 1,265,434.93 C. P1,256,443.93
5. How much should a Grade 11 working student of Senior High

School set aside from his salary and invest in a fund earning 12.75%
compounded semi-annually if he needs P10,000 in 60 months?
A. P 5,930.19 B. P 5,390.19 C. P 5,309.19

POSTTEST
1. How much must Jerico deposit in a bank that pays 0.75%

compounded quarterly so that she will have P200,000 after 15
years?
A. 𝑃 179,739.30 C. 𝑃 177,737.30
B. 𝑃 178,738.30 D. 𝑃 176.736.30
2. Find the accumulated value of Php15,000 in 6 years if it is

invested at 12% compounded quarterly.
A. 𝑃 40,696.96 C. 𝑃 30,491.91
B. 𝑃 35,584.90 D. 𝑃 25,050.81
3. How many times does interest compound if it compounds 4%
bimonthly?
A. Twice a year C. Six times a year
B. Thrice a year D. Twelve times a year
FINDING THE RATES OF COMPOUND INTEREST MORE

THAN ONCE A YEAR
OVERVIEW
help you understand how to solve problems involving simple and
Solves problems involving simple and compound interest.
After going through this module, you are expected to find the rate of
interest compounding more than once a year.
PRETEST
1. The “Bombay’s System” of Loan is also known as 5 – 6. What is

its interest rate?
A. 6% B. 10% C. 20% D. 60%

2. What is the nominal rate of interest if P24,000 accumulates to
P30,000 in 4 years with interest compounded quarterly?
A. 7.86% B. 6.72% C. 5.62% D. 4.59%

3. Find the nominal rate of interest compounded quarterly at which
Php50,000 accumulates to P70,000 in 8 years.
A. 6.76% B. 5.54% C. 4.23% D. 3. 16%

LOOKING BACK
Let us recall the meaning and the formula used in compound interest.
Compound Interest – the interest earned at the end of the term is
automatically reinvested to earn more interest.
Maturity Value Formula: Rate per Conversion Period (𝑗):

𝒊(𝒎) nominal rate
𝑭 = 𝑷(𝟏 + 𝒋)𝒏 𝒋= = frequence of conversion
𝑚
Number of conversion periods 𝑛:

𝒏 = 𝒕𝒎 → 𝒏 = (𝐭𝐢𝐦𝐞 𝐢𝐧 𝐲𝐞𝐚𝐫𝐬)(𝐟𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲 𝐨𝐟 𝐜𝐨𝐧𝐯𝐞𝐫𝐬𝐢𝐨𝐧)
From the maturity value formula 𝑭 = 𝑷 (𝟏 + 𝒋)𝒏 ; the interest rate (𝒊(𝒎) )
can be determine if 𝐹, 𝑃, and 𝑛 are given.
𝐹 = 𝑃 (1 + 𝑗)𝑛 Given
𝐹
= (1 + 𝑗)𝑛 Division Property of Equality
𝑃
𝐹
(1 + 𝑗)𝑛 = Symmetric Property of Equality
𝑃
1
𝐹 𝑛 Applying the Law of Exponent
1+𝑗 =( )
𝑃
1
𝐹 𝑛 Subtraction Property of Equality
𝑗 =( ) −1
𝑃
1
(𝑚 ) 𝑖 (𝑚)
𝑖 𝐹 𝑛 Let 𝑗 = .
= ( ) −1 𝑚
𝑚 𝑃
1
𝐹 𝑛
𝑖 (𝑚 ) = 𝑚 [( ) − 1] Multiplication Propery of Equalityj
𝑃
Examples: At what nominal rate compounded quarterly will

P50,000 accumulate P65,000 in 5 years?
Given: 𝑃 = 50,0000 𝑡 = 5 years 𝑚=4 𝐹=

65,000
Find 𝑖 (𝑚) :
Solution:
We solve first for the total number of compounding periods.
𝑛 = 𝑚𝑡 → 𝑛 = (4)(5) → 𝑛 = 20
Substitute all the known values in the formula, we have:
𝑭 = 𝑷 (𝟏 + 𝒋)𝒏
65,000 = 50,000(1 + 𝑗)20
65,000
= (1 + 𝑗)10
50,000
1.3 = (1 + 𝑗)20
1 1
(1.3)20 = [(1 + 𝑗)20 ]20
1
(1.3)20 = 1 + 𝑗
1
(1.3)20 − 1 = 𝑗
1
Using scientific calculator to find (1.3)20 − 1, thus, 𝑗 = 0.013205. Thus,
the interest per conversion is 1.32%
The nominal rate (annual rate of interest) can be computed using the
formula
𝑖 (𝑚)
𝑗=
𝑚
With 𝑗 = 0.013205 and 𝑚 = 4, we can obtain 𝑖 (𝑚) :
𝑖 (𝑚)
0.013205 =
4
(𝑚 ) (𝑚 )
𝑖 = 0.05282 → 𝑖 = 5.28%
Therefore, the nominal rate is 5.28%.

Alternatively, we can just solve for 𝑖 (𝑚) using the formula:

𝟏
𝑭 𝒏
𝒊(𝒎) = 𝒎 [( ) − 𝟏]
𝑷
Using the same information as above, we have:
1
𝐹 𝑛
𝑖 (𝑚) = 𝑚 [( ) − 1]
𝑃
1
65000 20
𝑖 (𝑚) =4 [( ) − 1]
50000
Using a scientific calculator to simplify the above expression, we obtain:
𝒋 = 𝟓. 𝟐𝟖%
Example: At what interest rate compounded semi-annually will

money double itself in 12 years?
Given: 𝐹 = 2𝑃 𝑡 = 12 years 𝑚 = 2
(𝑚 )
Find 𝑖
Solution:
Solve first for the total number of compounding periods, 𝑛:

𝑛 = 𝑚𝑡 → 𝑛 = (2)(12) → 𝑛 = 24
We can just solve for 𝑖 (𝑚) using the formula:

𝟏
𝑭 𝒏
𝒊(𝒎) = 𝒎 [(𝑷) − 𝟏]
By substitution:
1
2𝑃 24
𝑖 (𝑚) = 2 [( ) − 1]
𝑃
1
𝑖 (𝑚) = 2 [224 − 1] → 𝑖 (𝑚) = 0.05860447
𝑖 (𝑚) = 5.29%

ACTIVITY
A. Complete the table by computing for unknown values.
Nominal Interest Frequency of the Interest Rate

Rate Compounded Conversion Period per Period
12% quarterly (1) (2)
8% Semi-annually 2 (3)
16% monthly 12 (4)
B. Find the unknown.

1. F = P105,000 P = P12,000 t = 8.5 years
(𝑚)
Money is compounded monthly, 𝑗 =? 𝑖 =?
2. Shane borrowed an amount of P65,000 which she paid with

an interest of P3,400 at the of 4 years. At what nominal rate
compounded semi-annually was it invested?
REMEMBER
A. From the maturity value formula 𝑭 = 𝑷 (𝟏 + 𝒋)𝒏 ; the interest rate

(𝒋) can be determine if F, P, and n are given.
B. Use the alternative formula to find the interest rate:
𝟏
𝑭 𝒏
𝒊(𝒎) = 𝒎 [( ) − 𝟏]
𝑷
C. Steps in finding the interest rate
number of conversion period. Apply the formula

of paper.
1. Jose borrowed P60,825 accumulates to P87,104.09 at the end of 6

years. Find the nominal rate of interest compounded monthly.
A. 𝟖% B. 𝟕% C. 𝟔%
2. Mrs. Cruz needs P800,000 to purchase a lot and she plans to invest
P250,000 in a savings account for 15 years. At what interest rate
compounded quarterly should she invest her money?
A. 7.82% B. 7.83% C. 7.84%
3. At what rate compounded monthly is Mr. Santos paying the interest
if he borrows P25,000 and agrees to pay P33,000 for the debt 2
years and 3 months from now?
A. 12.41% B. 12.40% C. 12.39%
4. Hagiozune needs to determine the nominal rate compounded semi-
annually so that her Php 10 000.00 accumulate P 15 000.00 in 10
years. Which of the following should she choose?
A. 3.90% B. 4.05% C. 4.10%
5. At what rate of interest compounded annually will P60,000

accumulate to P70,000 in 2 years?
A. 8.01% B. 9.05% C. 10.08%
POST TEST
1. If P55,846 becomes P77,930 .85 in 5 years, how much is the rate

of interest compounded semi-annually?
A. 6.76% B. 5.54% C. 4.23% D. 3.16%

2. At what rate of interest compounded annually will P80,000

accumulate to P90,000 in 4 years?
A. 2.89% B. 2.98% C. 2.99% D. 3.00%
3. Jose borrowed P70,500 accumulates to P97,800.50 at the end of 5

years. Find the nominal rate of interest compounded quarterly.
A. 6.60% B. 7.70% C. 8.80% D. 9.90%

FINDING TIME INVOLVING COMPOUND INTEREST
OVERVIEW
help you understand how to find time in compound interest. The scope of
this module permits it to be used in many different learning situations. The
language used recognizes the diverse vocabulary level of students. The
lessons are arranged to follow the standard sequence of the course. But the
order in which you read them can be changed to correspond with the
textbook you are now using. The module focuses on achieving this learning
competency:

a. find time involving compound interest.
PRETEST
1. At the end of investment period, Php 750 000.00 earned an interest of

Php 270 000.00 at 4% simple interest rate. How long is the investment
period?
A. 9 years B. 12 years C. 15 years D. 18 years
2. Kit borrowed an amount of Php 150 000.00 in a cooperative that offers

an interest of 5% compounded annually. If he paid a total of Php 173
643.75 as payment of his debt, how long did it take Kit to pay the debt?
A. 3 years B. 4 years C. 5 years D. 6 years
3. Arthur received a loan of Php20,200 from Alex with interest at 10%

converted monthly. He promised to pay Alex in full on the day when
Php36,600 will be due. When should Arthur pay?
A. 5.97 years B. 5.54 years C. 4.23 years D. 3.16 years

LOOKING BACK
Let us laws of logarithm and the maturity value formula:
Logarithm of a Product log b 𝑥𝑦 = log 𝑏 𝑥 + log 𝑏 𝑦

𝑥
Logarithm of a Quotient log 𝑏 = log 𝑏 𝑥 − log 𝑏 𝑦
𝑦
Logarithm of a Power log 𝑏 𝑥 𝑛 = 𝑛 log 𝑏 𝑥
Also , the maturity value, 𝐹, of any compound interest is given by the formula:
𝐹 = 𝑃(1 + 𝑗)𝑛
From the maturity value formula 𝑭 = 𝑷 (𝟏 + 𝒋)𝒏 ; the time or length of the term
can be determined using logarithms.
𝐹 = 𝑃(1 + 𝑗)𝑛 Given

𝐹
= (1 + 𝑗)𝑛 Division Property of Equality
𝑃
𝐹
(1 + 𝑗)𝑛 = Symmetric Property of Equality
𝑃
𝐹
𝑛 log(1 + 𝑗) = log Take the common logarithm of both sides.
𝑃
𝐹
log 𝑃
𝑛= Division Property of Equality
log(1 + 𝑗)
𝐹
log 𝑃
𝑚𝑡 = Let 𝑛 = 𝑚𝑡.
log(1 + 𝑗)
𝐹
log 𝑃
𝑡= Division Propery of Equality
𝑚 log(1 + 𝑗)
Hence, to find the time under compound interest, use the formula:
𝑭
𝐥𝐨𝐠 𝑷
𝒕=
𝒎 𝐥𝐨𝐠(𝟏 + 𝒋)
Example: How long will it take P13,500 become P20,000 if invested at 8%

compounded quarterly?
Given: 𝐹 = P20000 𝑃 = P13,500 𝑖 (𝑚) = 0.08 𝑚 = 4

Find: 𝑡
Solution:
𝑖 (𝑚)
Since 𝑗 = , we can immediately solve for 𝑗.
𝑚
𝑖 (𝑚) 0.08
𝑗= →𝑗= → 𝑗 = 0.02
𝑚 4
To find 𝑡, we have:
𝒏
𝑛 = 𝑚𝑡 ↔ 𝒕 =
𝒎
𝑛
𝑡=
4
To find the total number of compounding periods, substituting the values in

the formula:
𝑭 = 𝑷 (𝟏 + 𝒋)𝒏
20000 = 13500 (1 + 0.02)𝑛
20,000
= (1 + 0.02)𝑛
13,500
(1.02)𝑛 = 1.481481
Taking the logarithm of both sides,
log(1.02)𝑛 = log 1.481481

𝑛 log 1.02 = log 1.481481
𝑛 (0.008600) = 0.170696
0.187087
𝑛=
0.008600
𝑛 = 19.8483
𝒏 = 𝟐𝟎
𝑛
But since 𝑡 = 4,
𝑛 20
𝑡= →𝑡= →𝒕=𝟓
4 4
Alternatively, we can immediately solve for 𝑡 using the derived formula

above. Using the given information above, we can have:
𝑭
𝐥𝐨𝐠 𝑷
𝒕=
20000
log
𝑡= 13500
4 log(1 + 0.02)
𝑡 = 4.962001
𝒕=𝟓
Thus, it will take 5 years for P13,500 to become P20,000.
Example: How long in years will it take for P25,000 to earn an interest
of P10,000 if it is invested at 6% compounded monthly?
Given: 𝑃 = P25000 𝐼𝑐 = P10000 𝑗 = 6% 𝑚 = 12

Find 𝑡
Solution:
𝑖 (𝑚)
Since 𝑗 = , we can immediately solve for 𝑗.
𝑚
𝑖 (𝑚) 0.06
𝑗= →𝑗= → 𝑗 = 0.005
𝑚 12
The future value is missing. To find the future value 𝐹, recall that 𝐹 = 𝑃 + 𝐼𝑐
𝐹 = 𝑃 + 𝐼𝑐
𝐹 = 25000 + 10000
𝐹 = 35000
To find the total number of compounding periods, substituting the values in

the formula:
𝐹 = 𝑃 (1 + 𝑗)𝑛
35000 = 25,000(1 + 0.005)𝑛
35,000
= (1 + 0.005)𝑛
25,000
1.005𝑛 = 1.4
Taking the logarithm of both sides:

log(1.005)𝑛 = log 1.4
𝑛 log 1.005 = log 1.4
log 1.4
𝑛=
log 1.005
𝑛 = 67.4625436
𝑛 = 68
NOTE: Interest is earned only at the end of the period, then 68-
month periods are needed so that the interest can reach Php10,000.
(ROUND UP).
𝑛
But since 𝑡 = 12,
68 𝟏𝟕
𝑡= →𝒕=
12 𝟑
Therefore, the time required is 5 years and 8 months.
NOTE: Compute the previous problem using the above formula to find time.
Check if it will come up of the same answer. Discuss your observations and
identify the part in the solutions which resulted their indifference.
ACTIVITY
A. Complete the table by computing the unknown values (time and rate).
(Round answers to the hundredths place.)
B. Find the unknown
𝐹 = P40,000 𝑃 = P15000
(𝑚)
𝑖 = 16.25% compounded quarterly
𝑗 =? ; 𝑛 =? ; 𝑡 =?
C. In how many years will it take P48,000 accumulate to P60,000 when

deposited in a savings account that earns 6% compounded monthly?
REMEMBER
A. From the maturity value formula 𝑭 = 𝑷 (𝟏 + 𝒋)𝒏 ; the time (𝒕) can
be determine if 𝐹, 𝑃, and 𝑖 are given.
B. Use the alternative formula to find the interest rate:
𝑭
𝐥𝐨𝐠 𝑷
𝒕=
C. Steps in finding the interest rate
number of conversion period
Choose the letter of the correct answer. Write it on a separate sheet of

paper.
1
1. Ricael invested Php50,000 in a bank that gives 12 8 % interest
converted semi-annually. If he decides to withdraw his money once it
accumulates to Php89,500 when will that be?
A. 4.96 years B. 4.95 years C. 4.94 years
2. How many years will it take for Php125,000 to earn an interest of

1
Php60,000 if it is invested at 7 4 % compounded quarterly?
3. How long will it take Php6,500 to amount to Php8,500, if the interest

rate is 6% compounded quarterly?
A. 5. 5 years B. 4.5 years C. 3.5 years
4. Lowell borrowed Php9,250 from Nestor with the agreement that

interest is charged at 8% compounded monthly. If the maturity value of
his loan is Php11,500, when it is due?
5. Delia deposits Php100,000 in savings account that pays 13% interest

converted semi-annually. If she decides to withdraw her money when
it grows to Php150,000, when should she withdraw her money?
POSTTEST
2
1. Sarah invested Php225,000 in a savings account that pays 6 5 %
interest compounded quarterly. If she decides to withdraw her money
once it accumulates to Php350,000, after how many years will that be?
A. 3.42 years B. 4.54 years C. 5.83 years D. 6. 96 years

2. How long will it take Php 10,000 to earn P3,000 if the interest is 12%
compounded semi-annually?
A. 5.5 years B. 4. 3 years C. 2.5 years D. 1.3 years
3. Ana invested Php30,000 in her time deposit account. How long will it
take to accumulate to Php35,000 at 2.5% compounded monthly?
A. 6.17 years B. 7.16 years C. 8.15 years D. 9.14 years

FINDING NOMINAL RATE, EQUIVALENT RATES AND

EFFECTIVE RATE MORE THAN ONCE A YEAR
OVERVIEW
help you understand how to find nominal rate, equivalent rates and
effective rates. The scope of this module permits it to be used in many
Solves problems involving simple and compound interest.
After going through this module, you are expected to find nominal rate,
equivalent rates and effective rate.
PRETEST
1. What effective rate is equivalent to 10% compounded quarterly?
A. 10.38% B. 10.21% C. 10.15% D. 10.03%
2. The rate compounded annually that will give the same compound
amount as a given nominal rate, denoted by i(1) .
A. Effective Rate C. Equivalent Rate

B. Nominal Rate D. Interest Rate
3. What effective rate is equivalent to 10% compounded quarterly?
A. 10.38% B. 10.21% C. 10.15% D. 10.03%

LOOKING BACK
Recall the formula of compound interest.
𝑭 = 𝑷(𝟏 + 𝒋)𝒏
𝒎𝒕
𝒊(𝒎)
𝑭 = 𝑷 (𝟏 + )
𝒎
Nominal Rate (𝒊(𝒎) ) - is a rate of interest that is compounded more often

than once a year such as semi-annually, quarterly or monthly; denoted
by 𝑖 (𝑚) .
Effective Rate (𝒊(𝟏) ) – the rate compounded annually that will give the
same compound amount as a given nominal rate 𝑖 (𝑚) ; denoted by 𝑖 (1) .
Equivalent Rates – two annual rates of interest with different conversion

periods that earn the same compound amount at the end of a given
number of years.
To determine the effective rate 𝑖 (1) equivalent to a nominal rate

𝑖 (𝑚) compounded 𝑚 times per year, we shall use the following formula,
𝒎
(𝟏)
𝒊(𝒎)
𝒊 = (𝟏 + ) −𝟏
𝒎
where: 𝑖 (1) is the effective rate
𝑖 (𝑚) is the nominal rate
𝑚 is the conversion period per year
The nominal interest rate 𝑖 (𝑚) formula is derived from the effective interest
rate 𝑖 (1) formula, as follows:
𝟏
(𝒎) ( 𝟏) 𝒎
𝒊 = 𝒎 [(𝟏 + 𝒊 ) − 𝟏]
Example: What nominal rate compounded monthly, will yield effective

rate 4%?
Given: 𝑖 (1) = 4% 𝑚 = 12
(𝑚)
Find 𝑖
Solution:
Since we are looking for the nominal rate, we shall use the formula,
1
(𝑚) ( 1) 𝑚
𝑖 = 𝑚 [(1 + 𝑖 ) − 1]
1
𝑖 (𝑚) = 12 [(1 + 0.04)12 − 1]
Using a scientific calculator to calculate for the above expression, we

obtain:
𝑖 (𝑚) = 0.39288
𝒊(𝒎) = 𝟑. 𝟗𝟑%
Example: What is the effective rate that is equivalent to 12%

Given: 𝑖 (𝑚) = 0.12 𝑚 = 4

(1)
Find 𝑖
Solution:
This time we are looking for the effective rate, we shall use the formula:
𝑚
(1)
𝑖 (𝑚)
𝑖 = (1 + ) −1
𝑚
(1)
0.12 4
𝑖 = (1 + ) −1
4

obtain:
𝑖 (1) = 0.125509
𝒊(𝒎) = 𝟏𝟐. 𝟓𝟓%
Example: Which is better investment: 9.8% compounded quarterly

or 9% compounded monthly?
To solve this problem, we need to compare the two investments by finding

their corresponding equivalent effective rates.
Given: 𝑖 (𝑚) = 0.098 𝑚 = 4

Find 𝑖 (1)
Solution:
We should use the formula:
𝑚
(1)
𝑖 (𝑚)
𝑖 = (1 + ) −1
𝑚
(1)
0.098 4
𝑖 = (1 + ) −1
4

obtain:
𝑖 (1) = 0.101661
𝒊(𝒎) = 𝟏𝟎. 𝟏𝟕%
Given: 𝑖 (𝑚) = 0.09 𝑚 = 12

(1)
Find 𝑖
Solution:
We should use the formula:
𝑚
(1)
𝑖 (𝑚)
𝑖 = (1 + ) −1
𝑚
(1)
0.09 12
𝑖 = (1 + ) −1
12

obtain:
𝑖 (1) = 0.093807
𝒊(𝒎) = 𝟗. 𝟑𝟖%
Thus, the 10.17% effective rate equivalent of 9.8% (m = 4) is better

investment than 9.38% effective rate equivalent of 9% (m = 12).
Example: If P1,000 is invested, what rate compounded quarterly

is equivalent to 18% compounded semi-annually?
Given: 𝑃 = 1000 𝑖 (𝑚) = 0.18 𝑚 = 2

(𝑚)
Find 𝑖 when 𝑚 = 4
Solution:
Investing P1,000 at 18% compounded semi-annually in one year will yield
the amount of:
𝑚𝑡
𝑖 (𝑚 )
𝐹 = 𝑃 (1 + )
𝑚
0.18 (2)(1)
𝐹 = 1000 (1 + )
2
𝑭 = 𝟏𝟎𝟎𝟎(𝟏. 𝟎𝟗)𝟐
Investing P1,000 at the rate of 𝑖 (𝑚) compounded quarterly in one year will
yield the amount of:
𝑚𝑡
𝑖 (𝑚 )
𝐹 = 𝑃 (1 + )
𝑚
(4)(1)
𝑖 (𝑚)
𝐹 = 1000 (1 + )
4
𝟒
𝒊(𝒎)
𝑭 = 𝟏𝟎𝟎𝟎 (𝟏 + )
𝟒
Two investments must be equal, and hence by transitivity:

4
2
𝑖 (𝑚 )
1000(1.09) = 1000 (1 + )
4
4
2
𝑖 (𝑚 )
1.09 = (1 + )
4
𝑖 (𝑚)
1+ = 1.044030651
4
𝑖 (𝑚)
= 0.044030651
4
𝑖 (𝑚) = 0.176122 ≈ 𝟏𝟕. 𝟔𝟏%
ACTIVITY
Solve the following problems on compound interest. Show your

complete solution.
3
1. What nominal rate compounded monthly is equivalent to 8 4 %
3
2. Find the effective rate equivalent to 12 8 % b compounded monthly.
3
3. Lowell earns on his account 9 5 % compounded quarterly. At what
rate compounded monthly could he just as well invest his money?
4. Darren wants to open a new account. In which bank would he prefer

to put his money, if Bank B offers 8.5% compounded semi-annually
on savings account while Bank C offers 8% compounded quarterly?
5
5. Find the rate compounded semi-annually that is equivalent to 6 6 %
compounded monthly.
REMEMBER
A. To differentiate nominal rate and effective rate

➢ In nominal rate, periodic rate times the number of periods per
year while effective rate caters the compounding periods during
a payment plan.
B. To convert one interest rate to an equivalent interest rate
➢ Use the compound interest formula 𝐹 = 𝑃 (1 + 𝑗)𝑛
C. To determine the effective rate 𝒊(𝟏) equivalent to a nominal rate
𝒊(𝒎) compounded 𝒎 times per year, use the following formula:
𝒎
(𝟏)
𝒊(𝒎)
𝒊 = (𝟏 + ) −𝟏
𝒎
D. To convert the effective interest rate to nominal rate, use the
formula:
𝟏
(𝒎) ( 𝟏) 𝒎
𝒊 = 𝒎 [(𝟏 + 𝒊 ) − 𝟏]

of paper.
1. If Maro charges 12% interest, compounded quarterly, what effective

annual interest rate is Maro charging?
A. 12.68% B. 12.55% C. 12.45%
2. What nominal rate compounded quarterly is equivalent to 6% effective

rate?
A. 5.80% B. 5.86% C. 5.87%
3. What is the effective rate compounded annually that is equivalent to

12% compounded annually?
A. 12. 64% B. 12.66% C. 12.68%
4. Which is the better investment: 8% compounded semi-annually or

7.8% compounded quarterly?
A. 7.8% B. 8% C. both A & B
5. Delia borrowed an amount of Php40,000 which she paid with interest

of Php2,000 at the end of 3 years. At what nominal rat compounded
semi-annually was it invested?
A. 3.30% B. 2. 68% C. 1.63%
POSTTEST
1. What effective rate is equivalent to 10% compounded monthly?
A. 10.47% B. 10.38% C. 10.25% D. 10.13%

2. What is the nominal rate of interest if Php4,000 accumulates to

Php10,000 in 8 years with interest compounded quarterly?
A. 11. 47% B. 11. 58% C. 11.62% D. 11.76%
3. At what annual interest rate compounded semi-annually will a

certain amount triple itself in 20 years?
A. 8.57% B. 7.57% C. 6.57% D. 5.57%

ANNUITIES
OVERVIEW
help you understand how to illustrate and distinguish between simple and
general annuities. The scope of this module permits it to be used in many
After going through this module, you are expected to distinguish

between simple and general annuities.
PRETEST
1. It is a method of accumulating a lump sum of money through a

series of regular and equal payments and the reverse, being the
liquidation of a lump sum through a series of regular an equal
payment.
A. Compound Interest C. Annuity

B. Amortization D. Loan
2. In the context of computation of interests, annuities and

amortizations, the time 𝑡 should always be expressed in terms of:
A. Days B. Months C. Years D. Decades

3. The following are considered exponential EXCEPT:
A. Simple Interest C. Simple Annuity

B. Compound Interest D. General Annuity
LOOKING BACK
We learned in the previous modules that an individual or business owner
can invest a sum of money in a bank or financial institution over a period
of time that will earn simple or compound interest.
The formula used to find the future value in compound interest was
𝒎𝒕
𝒊(𝒎)
𝑭 = 𝑷 (𝟏 + )
𝒎
An individual or business owner can still invest even they don’t have large
amount of money. This can be done by making series of equal deposits
or payments at a regular interval at specific interest rate that will earn
compound interest called annuity.
➢ Annuity – is a series of periodic payments made at equal(fixed)

regular intervals of time. This payment scheme by installment are
done periodically, and in equal amounts.
Examples: major purchases in installment basis, premiums on life

insurance, loan payment, etc.
Annuities can be classified in different ways, as follows:
Simple Annuity General Annuity

According to Annuity whose interest An annuity whose
payment conversion period (𝑚) is interest conversion
interval and equal or the same as the period (m) is unequal or
interest payment interval (𝑝𝑖). NOT the same as the
period. [𝑚 = 𝑝𝑖] payment interval (𝑝𝑖).
[𝑚 ≠ 𝑝𝑖]
Example: Installment Example: Installment

payment for an payment for an
appliance at the end of appliance at the end of
each month with interest each month with
compounded monthly. interest compounded
annually (to be
discussed in the next
lesson)
Ordinary Annuity
(or Annuity Immediate)
A type of annuity in
which the payments are
made at the end of each Annuity Due
payment interval. A type of annuity in
According to
Examples: Interest which the periodic
time of
payments from bonds payments are made at
payment
generally made semi- beginning of each
annually and from stock payment interval.
that has maintained
stable payout level of
quarterly dividends for
years.
Annuity Uncertain or
Annuity Certain
Contingent Annuity
An annuity in which
An annuity payable for
payments begin and end
an indefinite duration in
at definite/fixed date.
which the beginning or
Example: Monthly
the termination is
payment on a car form
According to dependent on some
an annuity certain
Duration certain event.
because the payment
Example: Pension, life
starts on a fixed date
insurance coverage
and continue until the
and mortgage
required number of
insurance redemption
payments has been
plans are examples of
made.
contingent annuity.
• Payment Interval - time between successive payments
• Term of an Annuity (𝒕) - a time between the first payment interval

and last payment interval.
• Regular or Periodic Payment (𝑹) - Amount of each payment

• Amount (Future Value) of an Annuity (𝑭) – sum of future values

of all the payments to be made during the entire term of the annuity.
• Present Value of an Annuity (𝑷) – sum of present value of all the

payments to be made during the entire term of the annuity.
Let us illustrate simple and general annuities:
Both simple and general annuities have a time diagram for its cash
flow shown below.
P F
R R R R R R … R
0 1 2 3 4 5 6 … n
The main difference is that:

Simple Annuity – the payment interval is the same as the interest
period.
General Annuity – the payment interval is not the same as the
interest period.
Example of a simple annuity – Installment payment for an appliance at

the end of each month with interest compounded monthly.
Example of a general annuity – Installment payment for an appliance at

the end of each month with interest compounded annually (to be
discussed in the next lesson)
ACTIVITY
Fill in the blanks.

1. A sequence of payments made an equal time period is a/an _____.
2. A simple annuity in which payment are made at the end of each period
is a/an _______.
3. An annuity where the payment intervals is not the same as the interest
period is a/an _______.
4. An annuity where the payment interval is the same as the interest

period is a/an _______.
5. An annuity in which payments begin and end at a definite time is a/an

_______.
REMEMBER
1. To illustrate simple and general annuities you need to show the cash
flows through time diagram.
2. The main difference is that in a simple annuity the payment interval

is the same as the interest period while in general annuity the
payment interval is not the same as the interest period.

of paper.
1. An installment payment of Php 2,578 every month for 6 months is a

representation of ________.
A. General Annuity C. Simple Interest

B. Simple Annuity D. Compound Interest
2. If Miel started to deposit a fixed amount monthly in her savings

account that pays an interest rate compounded quarterly and she
plans to withdraw her savings after 3 years. This refers to ________.
A. General Annuity C. Simple Interest

B. Simple Annuity D. Compound Interest
3. A series of payment made at fixed time period is_______.
A. Term of Annuity C. Payment

B. Amount of Annuity D. Annuity
4. This refers to time between the first payment interval and last
payment interval.
A. Present Value of Annuity C. Periodic Payment Annuity

B. Future Value of Annuity D. Term of Annuity
5. According to time of payment, this annuity payments are made at

the beginning of each payment interval.
A. Ordinary Annuity C. Annuity Certain

B. Contingent Annuity D. Annuity Due
POSTTEST
1. Teacher Lowell purchase a laptop using her credit card. He

will pay this in an installment scheme every end of each month
with an interest compounded monthly. This kind of annuity is
______.
A. Future Value of Annuity C. General Annuity

B. Present Value of Annuity D. Simple Annuity
2. A pension and life insurance policies are examples of______.
A. Ordinary Annuity C. Contingent Annuity

B. Annuity Certain D. Annuity Due
3. Delia had a car loan which her monthly payment starts on a

fixed date and will continue until the required number of
payments has been made. This car loan form a/an________.
A. Ordinary Annuity C. Contingent Annuity

B. Annuity Certain D. Annuity Due
FINDING THE FUTURE, PRESENT VALUE AND PERIODIC

PAYMENT OF SIMPLE OR ORDINARY ANNUITY
OVERVIEW
help you understand in finding the future and present value of simple or
ordinary annuity. The scope of this module permits it to be used in many
vocabulary level of students. The lessons are arranged to follow the standard
sequence of the course. But the order in which you read them can be
changed to correspond with the textbook you are now using. The module
focuses on achieving this learning competency:
Find the future and present value of both simple and ordinary
annuities.

• compute the future value of a simple annuity, and
• find the periodic payment of a simple annuity.
PRETEST
1. Find the future value of Php1,500 payable every three months for
six years and 6 months. If money worth 6% compounded quarterly.
A. Php 57,370.95 C. Php 37,170.95

B. Php 47,270.95 D. Php 27,070.95
2. In relation to QUESTION 1, what is the present value?
A. Php 52,487.95 C. Php 32,097.95

B. Php 43,157.95 D. Php 23,067.95
3. The formula below is used to find the future value of an annuity with
periodic payment 𝑅 and with 𝑗 as nominal rate of interest over the
number of compounding periods. Why is it that 𝐹 cannot be zero if
𝑅 is not zero?
( 1 + j )n − 1
F=R [ ]
j
A. Because (1 + j)n is always non – integral.
B. Because (1 + j)n is always nonzero.
C. Because (1 + j)n is always greater than 1.
D. Because j cannot be zero.
LOOKING BACK
We learned in the previous module the basic terms of simple annuity such
as:
• Term of an Annuity (𝒕) - a time between the first payment interval
and last payment interval.
• Regular or Periodic Payment (𝑹) - Amount of each payment
• Amount (Future Value) of an annuity (𝑭) – sum of future values of

all the payments to be made during the entire term of the annuity.
• Present Value of an annuity (𝑷) – sum of present value of all the

payments to be made during the entire term of the annuity.
The following notations will be used extensively throughout the discussion of

annuities.
𝐹 = sum of amount of an annuity

𝑃 = present value of an annuity
𝑅𝑠 = periodic payment of the sum
𝑅𝑎 = periodic payment of the present value
𝑡 = term of an annuity
𝑖 = rate of an annuity
𝑛 = number of conversion periods for the whole term (𝑡 × 𝑚)
𝑚 = number of conversion period per year
𝑗 = interest per conversion period (𝑖 ÷ 𝑚)
𝑝𝑖 = payment interval
To compute the future and present value of simple annuity we will use the
formula below:
a. Future Value of an Ordinary Annuity (Annuity – Immediate)

( 𝟏 + 𝒋 )𝒏 – 𝟏
𝑭=𝑹 [ ]
𝒋
b. Present Value of an Ordinary Annuity (Annuity –Immediate)
𝟏 − ( 𝟏 + 𝒋 )−𝒏
𝑷=𝑹 [ ]
𝒋
c. Periodic Payment of an Ordinary Annuity
𝑭
𝑹=
𝟏 − ( 𝟏 + 𝒊 )−𝒏
𝒊
Example: Suppose you are planning to have your own business and
you want to franchise Master Siomai Food Cart (as of 2020 the Franchise
amount is P280,000). In order to do that, you plan to deposit P50,000 each
year for the next 6 years in an ordinary annuity account that pays 8.25%
interest compounded annually. Find the future amount of the annuity at the
end of 6 years. Is the amount enough to franchise the said food cart?
Given: 𝑅 = P50000 is the periodic payment of the sum

𝑡 = 6 years is the term of annuity
𝑚 = 1 is the number of conversions per year
𝑖 = 8.5% = 0.085 is the rate of annuity
Find 𝐹:
Solution:
Solve for 𝑛 (number of payment periods) and 𝑗 (interest per conversion
period:
𝑛 = 𝑚𝑡 → 𝑛 = (1)( 6) → 𝒏 = 𝟔 number of payment period
𝑖 0.085
𝑗 = 𝑚 → 𝑗 = 1 → 𝒋 = 𝟎. 𝟎𝟖𝟓 interest rate per conversion period
We can now solve for 𝐹. Substitute the given values in the formula, we have:
( 𝟏 + 𝒋 )𝒏 – 𝟏
𝑭=𝑹 [ ]
𝒋
( 1 + 0.085)6 – 1
𝐹 = 50000 [ ]
0.085
( 1.085)6 – 1
𝐹 = 50000 [ ]
0.085
1.631468 – 1
𝐹 = 50000 [ ]
0.085
0.631468
𝐹 = 50000 [ ]
0.085
𝐹 = 50000 (7.429030)
𝑭 = 𝟑𝟕𝟏, 𝟒𝟓𝟏. 𝟒𝟖
Therefore, you will be able to save P371,451.48 at the end of 6 years,
which is enough to franchise Master Siomai Food Cart.
Example: Mrs. Martinez paid P500,000 as down payment for a house

and lot. The remaining amount is to be settled by paying P22,710 at the end
of each month for 10 years. If interest is 6.5% compounded monthly, what is
the cash price of the house and lot?
Given: Down payment = P550,000

𝑅𝑎 = 22,710
𝑡 = 10
𝑚 = 12
𝑖 = 6.5% = 0.065
Find 𝐶𝑉 (Cash Value):
Solution:
Solve for 𝑛 (number of payment periods) and 𝑗 (interest per conversion
period:
𝑛 = 𝑚𝑡 → 𝑛 = (12)(10) → 𝑛 = 120
𝑖 0.065
𝑗 = →𝑗= → 𝑗 = 0.0054167
𝑚 12
Get the present value, 𝑃.

1 − ( 1 + 𝑗 )−𝑛
𝑃=𝑅 [ ]
𝑗
1 − (1 + 0.0054167)−120
𝑃 = 22710 [ ]
0.0054167
𝑃 = 22710(88.06838)
𝑃 = 2000032.94
Find the cash value by adding the downpayment to the present value.
Hence:
𝐶𝑉 = 𝐷𝑃 + 𝑃
𝐶𝑉 = 550000 + 2000032.94
𝑪𝑽 = 𝟐𝟓𝟓𝟎𝟎𝟑𝟐. 𝟗𝟒
Therefore, the cash price of the house and lot is P 2,550,032.94.
Example: Paolo borrowed P200,000 from a lending company and

agrees to pay the principal plus interest by paying an equal amount of money
each for 4 years. What should be his quarterly payment if interest is 7%
Given: 𝑃 = 200,000
𝑡=4
𝑚=4
𝑖 = 7% = 0.07
Find 𝑅 (Periodic Payment of the Present Value):
Solution:
To find the periodic payment of the present value, we shall use the formula:
𝐹
𝑅=
1 − ( 1 + 𝑖 )−𝑛
𝑖
Substitute the given values:
𝐹
𝑅=
1 − ( 1 + 𝑖 )−𝑛
𝑖
200000
𝑅=
1 − (1 + 0.0175)−16
0.0175
200000
𝑅=
13.850497
𝑹 = 𝐏𝟏𝟒𝟒𝟑𝟗. 𝟗𝟐
Thus, Paolo should pay P 14,439.92 every quarter for 4 years.
ACTIVITY
Identify which of the following problems is Future Value Annuity, Present

Value Annuity or Periodic Payment.
1. Jenny plans to travel in Cagayan De Oro 3 years from now. How much
money must Jenny deposit on her account now if the bank offers 7%
interest compounded monthly in order to be able to withdraw
P160,000 at the end of 3 years?
2. Mr. De Jesus invested P20,000 every quarter over a 12year period. If

4
his money earns an annual rate of 8 5% compounded quarterly, how
much would be the amount at the end of the time period. How much
is the interest earned?
3. Suppose your family wants to celebrate the 55th birthday of your

mother in Cebu City for 5 days. To make it possible your family begin
a monthly deposit of P10,000 in your account that pays 11%
compounded monthly. If your mother is 52 years old now, how much
money will be in your account when your mother reaches her 55th
birthday? How much is the interest earned?
4. If money is invested at 6% compounded quarterly, find the quarterly

rent of an annuity for 15 years if its present value is 120,000.
Solve all the problems above. (Round your answer to the nearest
centavo.)
REMEMBER
To compute the future and present value of simple annuity and to find
the periodic payment of simple annuity we need the following formulas:
a. Future Value of an Ordinary Annuity (Annuity – Immediate)
( 𝟏 + 𝒋 )𝒏 – 𝟏
𝑭=𝑹 [ ]
𝒋
b. Present Value of an Ordinary Annuity (Annuity –Immediate)
𝟏 − ( 𝟏 + 𝒋 )−𝒏
𝑷=𝑹 [ ]
𝒋
c. Periodic Payment of an Ordinary Annuity
𝑭
𝑹=
𝟏 − ( 𝟏 + 𝒊 )−𝒏
𝒊
A. Steps in finding the Future Value, Present Value and Periodic
payment.
5. Solve.

paper.
1. Find the amount (future value) of an ordinary annuity P5,000 payable

semi-annually for 10 years if money is worth 6% compounded semi-
annually.
A. P59,964.50 B. B. P74,387.37 C. P134,351.87
2. To pay for his debt at 12% compounded, Ruben committed for 8

quarterly payments of P28,491.28 each. How much did he borrow?
A. P200,000 B. P300,000 C. P400,000

3. Ronalyn is a high school student. She would like to save P50,000 for
her college graduation. How much should she deposit in a savings
1
account every month for 5 2 years if interest is at 0.25% compounded
monthly?
A. P750.56 B. P752. 46 C. P754.36
4. Mr. Londonio buy a second-hand car and pays P169,000 cash and
P12,000 every month for 5 years. If money is 10% compounded
monthly, how much is the cash price of his car?
A. P569,784.43 B. P664,784.43 C. P733,784.43
5. Due to COVID-19 pandemic SM Appliance Center is on sale in all their

appliances, cash or installment. The automatic washing machine price
on sale at P23,499 in cash, for on installment terms, it cost P2,500
monthly for the next 9 months at 8.5% compounded monthly. How
much would be the cost of the washing machine if the buyer will choose
for an installment term?
A. P21,723.39 B. B. P21,473.28 C. P21,183.49
POSTTEST
1. Theresa wants to save money for her college graduation, she

decides to save P350 at the end of each month. If the bank pays
1.25% compounded monthly, how much will her money be at the
end of 4 years?
A. P 20,340.50 C. P 18,140.30
B. P 19,560.40 D. P 17,220.20
2. Sam selected a new car to purchase for P1,014,000. If the car can
be financed over a period of 5 years at an annual rate of 6.4%
compounded monthly, how much will be his monthly payment?
A. P25,000.21 C. P23,000.21
B. P24,000.21 D. P13,000.21
3. If money is invested at 6% compounded quarterly, find the quarterly

rent of an annuity for 15 years if its present value is P120,000.
A. P5,138.33 C. P3,047.21
B. P4,067.28 D. P2,432.16
GENERAL ANNUITY
OVERVIEW
help you understand in finding the future and present value of general
annuity. The scope of this module permits it to be used in many different
learning situations. The language used recognizes the diverse vocabulary
level of students. The lessons are arranged to follow the standard
Find the future and present value of both simple and general
annuities.
After going through this module, you are expected to compute the
future value and present value of general annuity.
PRETEST
1. If the interest rate is 6% compounded quarterly, what is the

present value of P7,700 payable at the end of each 6 months for
8 years?
A. P 87,370.95 C. P 107,170.67
B. P 96,544.33 D. P 136,070.86
2. Find the cash equivalent of a sala set with a down payment of P

12,000 and P 1,450 for the balance payable every end of the
month for 30 months with interest rate of 8% compounded
quarterly.
A. P 19,879.17 C. P 17,685.34
B. P 18,789.71 D. P 16,956.21
3. Which of the following is true for any annuity as the term of

compounding becomes longer?
A. The interest increases as periodic payment also increases.

B. The interest decreases as periodic payment also decreases.
C. The interest increases as periodic payment also decreases.
D. The interest decreases as periodic payment also increases.
LOOKING BACK
We learned in the previous module the basic terms of general annuity
such as:
• General Annuity – an annuity where the payments interval is not
the same as the interest compounding period.
• General Annuity Ordinary – a general annuity in which the

periodic payment is made at the end of the payment interval.
• Equivalent Rates – two annual rates of interest with different

conversion periods that earn the same compound amount at the
end of a given number of years.
To solve problems under general annuity;
• we should get the payment interval, 𝒑𝒊, and the compounded

period, 𝒄, to coincide in order to facilitate out computations.
The formula for the future value is just the same as that for
simple ordinary annuity.
• the extra step occurs in finding 𝒋, the given interest rate period
must be converted to an equivalent rate per payment interval.
• the amount or the final value of a general annuity is the sum

of all payments and the accumulated interest. We shall use
𝑭(𝒈𝒐) to denote the amount of a general annuity.
The following are examples of annuity:
1. Monthly installment payment of a car

2. Lot or house and lot with an interest that is compounded annually
3. Paying a debt semi-annually when the interest is compounded
quarterly.
The amount of a general ordinary annuity is calculated using the following

formula:
(𝟏 + 𝒋)𝒏 − 𝟏
𝑭(𝒈𝒐) = 𝑹(𝒈𝒐) [ ]
(𝟏 + 𝒋)𝒌 − 𝟏
The present value of a general ordinary annuity is the sum of money today
which if invested at a specified rate will amount to all the payments and
the compound interests at the end of the term of the annuity. We shall use
𝑷(𝒈𝒐) to denote the present value of annuity.
𝟏 − (𝟏 + 𝒋)−𝒏
𝑷(𝒈𝒐) = 𝑹(𝒈𝒐) [ ]
(𝟏 + 𝒋)𝒌 − 𝟏
where:
𝐹(𝑔𝑜) = amount of general ordinary annuity
𝑅(𝑔𝑜) = amount of periodic payment
𝑛 = total number of compounding periods
𝑟 = rate of interest per year
𝑗 = interest rate per compounding period
𝑚 = number of compounding periods per one year
𝑝
𝑘=𝑐=
number of compounding periods per payment interval
𝑐 = number of months per compounding period
𝑝 = number of months per payment interval
Example: Micah started to deposit P1,000 monthly in a fund that

pays 6% compounded quarterly. How much will be in the fund after 15
years?
Given: 𝑅(𝑔𝑜) = P1000 𝑡 = 15 years 𝑝 = 1 month

𝑖 = 6% 𝑚=4 𝑐 = 3 months (quarterly)
Find 𝐹(𝑔𝑜) :
Solution:
Solve for 𝑗, 𝑘 and 𝑛.
𝑖 0.06
𝑗= →𝑗= → 𝒋 = 𝟎. 𝟎𝟏𝟓 → 𝒋 = 𝟏. 𝟓%
𝑚 4
𝑝 𝟏
𝑘= →𝒌=
𝑐 𝟑
𝑛 = 𝑚𝑡 → 𝑛 = (4)(15) → 𝒏 = 𝟔𝟎
Substituting the given values into the formula, we have:

(1 + 𝑗)𝑛 − 1
𝐹(𝑔𝑜) = 𝑅(𝑔𝑜) [ ]
(1 + 𝑗)𝑘 − 1
(1 + 0.015)60 − 1
𝐹(𝑔𝑜) = 1000 [ 1 ]
(1 + 0.015)3 − 1
2.44321977 − 1
𝐹(𝑔𝑜) = 1000 [ ]
1.00497521 − 1
𝑭(𝒈𝒐) = 𝟐𝟗𝟎𝟎𝟖𝟐. 𝟕𝟓
Therefore, Micah will have P290,082.75 in the fund after 15 years.
Example: If money worth 6.5% compounded semi-annually, find

the present value of a general annuity where P12,000 is payable at the
end of every 3 months for 8 years.
Given: 𝑅(𝑔𝑜) = P12000 𝑡 = 8 years 𝑝 = 3 months

𝑖 = 6.5% 𝑚 = 2 𝑐 = 6 months (semi-annually)
Find 𝑃(𝑔𝑜) :
Solution:
Solve for 𝑗, 𝑘 and 𝑛.
𝑖 0.065
𝑗= →𝑗= → 𝒋 = 𝟎. 𝟎𝟑𝟐𝟓 → 𝒋 = 𝟑. 𝟐𝟓%
𝑚 2
𝑝 3 𝟏
𝑘= →𝑘= →𝒌=
𝑐 6 𝟐
𝑛 = 𝑚𝑡 → 𝑛 = (2)(8) → 𝒏 = 𝟏𝟔
Substituting the given values into the formula, we obtain:

1 − (1 + 𝑗)−𝑛
𝑃(𝑔𝑜) = 𝑅(𝑔𝑜) [ ]
(1 + 𝑗)𝑘 − 1
1 − (1 + 0.0325)−16
𝑃(𝑔𝑜) = 12000 [ 1 ]
(1 + 0.0325)2 − 1
𝑃(𝑔𝑜) = 12000(24.847384)
𝑃(𝑔𝑜) = 298168.61
Hence, the present value of the general annuity is P 298,168.61.
ACTIVITY
Answer the following problems. (Use 6 or more decimal places.

Final answer rounded to nearest centavo.)
1. A travel company offers a 5-day group tour to Kalanggaman Island

Polompon, Leyte with side trip in Cebu City for only P36,599 with at
least 5 people. The offer can be avail in cash or on an installment at
the end of every month for 2 years at 5% interest compounded
quarterly. How much is the amount to be shared by a person joining
the tour at the end of 2 years?
2. In the previous problem, how much will Nestor, Ivan, Paolo, Lowell and
Darren pay for joining the tour?
3. Joanne deposits 25,000 at the end of every quarter for the next 8 years
that pays 6% compounded semi-annually. Find the amount of the
general annuity at the end of 8 years?
4. Ana is an OFW and plan to take her vacation this year. How much is
the cash value that she has whose periodic payment of P2,500 was
deposit at the end of each month for the past 10 years at 8%

compounded annually?
5. Mrs. Pingol would like to buy a new laptop and printer payable monthly
1
for 2 2 years starting at the end of the month. How much is the cost of
the laptop and printer if her monthly payment is P2,300 and interest is
9% compounded semi-annually?
REMEMBER
To compute the future and present value of general annuity, we need

the following formulas:
The amount of a general ordinary annuity is calculated using the
following formula:
(𝟏 + 𝒋)𝒏 − 𝟏
𝑭(𝒈𝒐) = 𝑹(𝒈𝒐) [ ]
(𝟏 + 𝒋)𝒌 − 𝟏
The present value of a general ordinary annuity is the sum of money

today which if invested at a specified rate will amount to all the payments
and the compound interests at the end of the term of the annuity. We
shall use 𝑷(𝒈𝒐) to denote the present value of annuity.
𝟏 − (𝟏 + 𝒋)−𝒏
𝑷(𝒈𝒐) = 𝑹(𝒈𝒐) [ ]
(𝟏 + 𝒋)𝒌 − 𝟏
Steps in finding the Future and Present Value of General Annuity

5. Solve.

of paper.
1. On the 1st birthday of my grandson I started to deposit P5,000 quarterly

at the end of each term in a fund that pays 1% compounded monthly.
How much will be in the fund on his 7th birthday?
A. P 144,833.93 B. P 144,832.94 C. P 144,832.95
2. Find the cash equivalent of a dining set with a down payment of

P12,000 and P1,450 for the balance payable every end of the month for
30 months with interest rate at 8% compounded quarterly.
A. P 19,879.17 B. P 19,897.17 C. P 19,987.17
3. Darren deposit P5,000 every end of the year for 10 years in a savings
account paying 5% compounded semi-annually. How much is the
amount at the end of the term?
A. P 65,075.25 B. P 64,074.24 C. P 63,063.23
4. Mr. Sanchez buys a lot near Nanay’s Pansit Malabon agreeing to pay
P250,000 at the end of each year for 10 years. Find the equivalent cash
price, if money is worth 12% compounded quarterly.
A. P 1,831,263.91 B. P 1,681,632.91 C. P 1,381,263.91
5. If the money that you save in a savings account gives 6% compounded

quarterly, find the present value of a general annuity where you deposit
P12,000 every end of 6 months for 15 years.
A. P 234,422.69 B. P 234,522.69 C. P 234,422.69

POSTTEST
1. Sarah will deposit 12,000 at the end of every 6 months for the
next 10 years in a general ordinary annuity that pays 6% interest
compounded quarterly. Find the amount of annuity at the end of
10 years.
A. P123,283.64 C. P323,183.48
B. P223,383.56 D. P423,438.32
2. Hannah borrowed an amount of money from San Jose Multi-

purpose Cooperative which she will use in her clothing business.
She agrees to pay the principal plus interest by paying P78,849
each year for 5 years. How much money did she borrow if interest
is 8% compounded quarterly?
A. P412,428.27 C. P212,313.27
B. P312,813.27 D. P112,973.27
3. Robert deposit P 5,000 every end of the year for 10 years in a

savings account paying 5% compounded semi-annually. What
amount is in the account at the end of the term?
A. P65,138.33 C. P61,047.21
B. P63,073.23 D. P60,432.16
GENERAL ANNUITY – CASH FLOW
OVERVIEW
help you understand calculating the fair market value of a cash flow
stream that includes an annuity. The scope of this module permits it to be
used in many different learning situations. The language used recognizes
the diverse vocabulary level of students. The lessons are arranged to
follow the standard sequence of the course. But the order in which you
read them can be changed to correspond with the textbook you are now
using. The module focuses on achieving this learning competency:
Calculate the fair market of a cash flow stream that includes an

annuity.

• find the future value and present value of annuity;
• solve the equivalent interest rate; and
• calculate the market value of a cash flow.
PRETEST
1. If the interest rate is 6% compounded quarterly, what is the

present value of P7,700 payable at the end of each 6 months for
8 years?
A. P 87,370.95 C. P 107,170.67
B. P 96,544.33 D. P 136,070.86
2. Find the cash equivalent of a sala set with a down payment of P

12,000 and P 1,450 for the balance payable every end of the
month for 30 months with interest rate of 8% compounded
quarterly.
A. P 19,879.17 C. P 17,685.34
B. P 18,789.71 D. P 16,956.21
3. Which of the following is true for any annuity as the term of

compounding becomes longer?
A. The interest increases as periodic payment also increases.

B. The interest decreases as periodic payment also decreases.
C. The interest increases as periodic payment also decreases.
D. The interest decreases as periodic payment also increases
LOOKING BACK
To solve problems about cash flows we will apply the concepts of present
value and future values. Recall the formula:
Present Value Future Value

𝑃 = 𝐹 (1 + 𝑟)−𝑛 𝐹 = 𝑃(1 + 𝑟)𝑛
Compound −𝑚𝑡 𝑚𝑡
Interest 𝑖 (𝑚 ) 𝑖 (𝑚 )
𝑃 = 𝐹 (1 + ) 𝐹 = 𝑃 (1 + )
𝑚 𝑚
1 − (1 + 𝑗)−𝑛 (1 + 𝑗)𝑛 − 1
𝑃 =𝑅[ ] 𝐹 =𝑅[ ]
𝑗 𝑗
Ordinary
Annuity
1 − (1 + 𝑗)−𝑛 (1 + 𝑗)𝑛 − 1
𝑃(𝑔𝑜) = 𝑅(𝑔𝑜) [ ] 𝐹(𝑔𝑜) = 𝑅(𝑔𝑜) [ ]
(1 + 𝑗)𝑘 − 1 (1 + 𝑗)𝑘 − 1
• Cash flow – is a term that refers to payments received (cash flows)

or payments or deposits made (cash outflows).
• Cash inflows – can be represented by positive numbers.
• Cash outflows – can be represented by negative numbers.
• The Fair Market Value or Economic Value - a cash flow (payment
stream) on a particular date refers to a single amount that is
equivalent to the value of the payment stream at that date.
Example: Mr. Dela Cruz received two offers on a lot that he wants
to sell. Mr. Alfonso has offered P50,000 and P1,000,000 lump sum
payment for 5 years from now. Mr. Co has offered P50,000 plus P40,000
every quarter for five years. Compare the fair market values of the two
offers if money can earn 5% compounded annually. Which offer has a
higher market value?
Given:
Mr. Alfonso’s offer Mr. Co’s Offers
P50,000 down payment P50,000 down payment
P1,000,000 after 5 years P40,000 every quarter for 5 years
Find: the fair market values of each offer.
Solution:
Illustrate the cash flows of the two offers using time diagrams. It is
convenient to choose focal dates to either be at the start or at the end of
the term.
Mr. Alfonso’s offer:
50000 1000000
0 1 2 3 4 …. 20
Mr. Co’s offer:
50000 40000 40000 40000 40000 40000

0 1 2 3 4 …. 20
Solution 1: Choose the focal date to be the start of the term. Since the
focal date is at 𝑡 = 0, compute for the present value of each offer.
Since P50,000 is offered today, then its present value is still P50,000. The
resent value of P1,000,000 offered five years from now is, we shall use
the present value compound interest formula. Using:
𝐹 = 1000000 𝑗 = 0.05 𝑛=5
𝑃 = 𝐹 (1 + 𝑗)−𝑛
𝑃 = 1000000(1 + 0.05)−5
𝑃 = 783526.17
Hence, the present value is P 783,526.17. To know the fair market value
(FMV), we must add the amount of down payment (DP) to the present
value (P). Hence:
𝐹𝑀𝑉 = 𝐷𝑃 + 𝑃
𝐹𝑀𝑉 = 50000 + 783526.17
𝑭𝑴𝑽 = 𝟖𝟑𝟑𝟓𝟐𝟔. 𝟏𝟕
Mr. Co’s offer:
Compute the present value of a general annuity with quarterly payments

but with annual compounding at 5%.
First, find the equivalent rate of 5% compounded annually.

𝑚𝑡
𝑖 (𝑚 )
𝐹 = 𝑃 (1 + )
𝑚
For 𝑚 = 4 (quarterly) and 𝑡 = 1 (in years), we have:
4(1)
𝑖 (𝑚 )
𝐹 = 𝑃 (1 + )
4
Similarly, for 𝑖 (𝑚) = 0.05, 𝑚 = 1 (annually) and 𝑡 = 1 (in years), we have:
0.05 1(1)
𝐹 = 𝑃 (1 + )
1
The two 𝐹 values must be equal, so:

4( 1) ( )
𝑖 (𝑚 ) 0.05 1 1
𝑃 (1 + ) = 𝑃 (1 + ) Transitivity
4 1
4(1)
𝑖 (𝑚 ) 0.05 1(1)
(1 + ) = (1 + ) Dividing both sides by 𝑃.
4 1
4(1)
𝑖 (𝑚 )
(1 + ) = 1.05
4 Evaluating the expression on the
(𝑚 ) 4 right.
𝑖
(1 + ) = 1.05
4
𝑖 (𝑚 ) 4 Taking the fourth root of both
1+ = √1.05 sides of the equation.
4
𝑖 (𝑚 ) 4
= √1.05 − 1 Subtraction Property of Equality
4
𝑖 (𝑚 )
= 0.0122722 Simplifying the right side.
4
𝑖 (𝑚) = 0.0490888 Multiplication Property of Equality
Since 𝑖 (𝑚) is found, we can immediately solve for 𝑗 with 𝑚 = 4.

𝑖 (𝑚) 0.0490888
𝑗= →𝑗= → 𝑗 = 0.0122722
4 4
Substitute all the known value using the ordinary annuity formula for the
present value.
1 − (1 + 𝑗)−𝑛
𝑃 = 𝑅[ ]
𝑗
1 − (1 + 0.0122722)−20
𝑃 = 40000 [ ]
0.0122722
𝑷 = 𝟕𝟎𝟓𝟓𝟕𝟐. 𝟗𝟐
Hence, the fair market value is:

𝐹𝑀𝑉 = 𝐷𝑃 + 𝑃
𝐹𝑀𝑉 = 500000 + 705572.92
𝐹𝑀𝑉 = 755572.92
Comparing the two fair market values, Mr. Alfonso’s offer has a higher
market value.
The difference between the market values, Δ, of the two offers at the start
of the term is:
Δ = |𝐹𝑀𝑉1 − 𝐹𝑀𝑉2 |
Δ = 833526.17 − 755572.92
𝚫 = 𝟕𝟕𝟗𝟓𝟑. 𝟐𝟓
Solution 2: Choose the focal date to be the end of the term

At the end of the term, P1,000,000 is valued as such (because this is the
value at 𝑡 = 5). The future value of P50,000 at the end of the term at 5%
compounded annually is given by:
𝐹 = 𝑃 (1 + 𝑖)𝑛
𝐹 = 50,000 (1 + 0.05)5
𝐹 = 50,000 (1.05)5
𝐹 = 50,000 (1.2762816)
𝑭 = 𝟔𝟑, 𝟖𝟏𝟒. 𝟎𝟖
Fair Market Value = Future Value + Lump Sum Payment

𝐹𝑀𝑉 = 63,814.08 + 1,000,000
𝑭𝑴𝑽 = 𝑷𝟏, 𝟎𝟔𝟑, 𝟖𝟏𝟒. 𝟎𝟖
Mr. Co’s offer:
The future value of ordinary annuity is given by:

( 𝟏 + 𝒋 )𝒏 − 𝟏
𝑭=𝑹 [ ]
𝒋
( 1 + 0.0122722 )20 − 1
𝐹 = 40,000 [ ]
0.0122722
( 1.0122722 )20 − 1
𝐹 = 40,000 [ ]
0.0122722
( 1.2762807 − 1
𝐹 = 40,000 [ ]
0.0122722
( 0.2762807
𝐹 = 40,000 [ ]
0.0122722
𝐹 = 40,000 (22.5127275)
𝑭 = 𝐏𝟗𝟎𝟎, 𝟓𝟎𝟗. 𝟏𝟎
The future value of P50,000 at the end of the term is P63,814.08, which
was already determined earlier. Thus,
Fair Market Value = Future Value + Future Ordinary Annuity

𝐹𝑀𝑉 = 63,814.08 + 900,509.10
𝑭𝑴𝑽 = 𝑷𝟗𝟔𝟒, 𝟑𝟐𝟑. 𝟒𝟖
Thus, Mr. Alfonso’s offer still has a higher market value, even if we solve
the focal date to be the end of the term. The difference between the market
values of the two offers at the end of the term is:
Δ = |𝐹𝑀𝑉1 − 𝐹𝑀𝑉2 |
Δ = 1,063,814.08 − 964,323.48
𝚫 = 𝑷𝟗𝟗, 𝟒𝟗𝟎. 𝟔𝟎
Check the present value using the difference between the market values
of the two FMV, P99,490.60.
𝐹 = 𝑃99,490.60 𝑛 = 5 𝑖 = 5% or 0.05
−𝒏
𝑷 = 𝑭 (𝟏 + 𝒊)
𝑃 = 99,490.60 (1 + 0.05)−5
𝑃 = 99,490.60 (1.05)−5
𝑃 = 99,490.60 ( 0.7835262)
𝑷 = 𝐏𝟕𝟕, 𝟗𝟓𝟑. 𝟒𝟗
What did you notice about the present value when we use the difference
between the fair market values of the two FMV?
ACTIVITY
Answer the following problems. (Use 6 or more decimal places. Final

answer rounded to nearest centavo.)
1. Mrs. Jarina wishes to sell her house and lot, Mrs. Cruz is offering
her P1,550,000 in cash while Mrs. Dalmacio is offering a down
payment of P150,000 and monthly periodic payments of P30,000 at
the end of each month for 5 years. Which of the offers should Mrs.
Jarina accept if money can be invested at 8% compounded quarterly
and how much is the difference between the offers in terms of their
equivalent cash values?
Note: a. Illustrate the cash flows of the two offers using time
diagrams.
b. Which has a better market value?
2. Due to COVID-19 pandemic, Darren wants to sell his used vehicle.

Mrs. Ebio is offering a down payment of P50,000 and monthly
payment of P6,000 payable at the end of each month for 4 years.
Mrs. Pingol is offering P300,000 cash. Which of the offers should
1
Darren accept if money can be invested at 8 2% compounded semi-
annually? Find the difference between the two offers.
REMEMBER
To find the fair market value, you will need the following:
a. Illustrate the cash flows of the two offers using time diagrams.
b. Choose a focal date.
c. Determine the values of the two offers at the chosen focal date.
d. Compute the Future and Present Value
Using the Compound Interest
𝑭 = 𝑷 (𝟏 + 𝒋)𝒏 → 𝑷 = 𝑭 (𝟏 + 𝒋)−𝒏
OR
Using the formula of Ordinary Annuity:
( 𝟏 + 𝒋 )𝒏 − 𝟏 𝟏 − ( 𝟏 + 𝒋 )−𝒏
𝑭=𝑹 [ ]→𝑷= 𝑹 [ ]
𝒋 𝒋
e. Compute the Fair Market Value start of the term
Fair Market Value = Down payment (DP) + Present Value (PV)
f. Compute for the difference between the market values of the two
offers at the end of the term
𝜟 = |𝐹𝑀𝑉1 – 𝐹𝑀𝑉2 |
Read and analyze the given problem. Write your answer on a

Performance Task:
Find situations involving annuities in your home. For example:

• Look for an appliance store, find a certain appliance that you wish
to have. How much is the cost if it is (a) paid in full, or (b) paid by
installment.
• Look for a bank that offers loan such as; car, house and lot or
cash loan. Know their terms and conditions for the loans that they
offer.
• If you have known someone borrowing from a five-six
moneylender, you can ask how much will be the charged if you
want to loan 5,000 payable in 1 year.
1. For the situation you choose, determine the interest rate for the
period and the annual interest rate.
2. Illustrate using time diagram. Compute for the Fair Market Value.
3. Based on the interest rates you computed, do you think it is good
idea to loan? Discuss.
NOTE: Since face to face is not allowed, you can search through internet
to perform this task. (ONLINE). You can search for appliance store, bank
or message someone you know who works in a bank or appliance store,
someone who are moneylender. Make sure that you will write the site that
you used for your research. Below Rubric for this Performance Task.
POSTTEST
Bank A offers P150,000 at the end of 3 years plus P300,000

at the end of 5 years. Company offers P25,000 at the end of each
quarter for the next 5 years. Assume that money is worth 8%
compounded semi-annually.
1. What is the fair market value of Bank A?
A. P118,547.18 C. P321,216.43
B. P202,669.25 D. P409, 560.47
2. What is the fair market value of Bank B?
A. P 118,547.18 C. P 321,216.43
B. P 202,669.25 D. P 409, 560.47
3. Which bank has a better market value?
A. Bank A C. Both banks

B. Bank B D. None
DEFERRED ANNUITY
OVERVIEW
help you understand calculating the present value and period of deferral of
a deferred annuity. The scope of this module permits it to be used in many
vocabulary level of students. The lessons are arranged to follow the standard
Calculate the present value and period of deferral of a deferred

annuity

• give examples of deferred annuity in real-life situations, and
• find the present value and period of deferred annuity.
PRETEST
1. Find the quarterly payment for 21 quarters to discharge an

1
obligation of P120,000, if money worth 4 2 % compounded quarterly
and the first payment is due at the end of 3 years and 9 months.
A. P8,451.00
B. P7,541.01
C. P 6,154.02
D. P5,571.03
2. At 10% converted semi-annually, find the present value of 10 semi-

annually payments of P 8,500 each, the first due is due in 5½ years.
A. P60,942.04
B. P50,492.04
C. P40,294.04
D. P30,249.04
3. In a series of monthly payments of P7500 each, the first payment is

due at the end of 6 years and the last at the end of 11 years and 9
months. If money is worth 3.5% compounded monthly, find the
present value of the deferred annuity.
A. P349,976.08
B. P349,873.08
C. P349,766.08
D. P349,673.08
LOOKING BACK
Let us recall the following terms in annuity:
• Annuity - is a sequence of payments made at equal (fixed) intervals
• Annuity Immediate or Ordinary Annuity - a type of annuity in which
the payments are made at the end of each period.
• Deferred Annuity – an annuity wherein the first payment interval does
not coincide with the first interest period. The first payment off to some
later date.
• Period of Deferral – length of time from the present to the beginning
of the first payment.
• Amount of a deferred annuity of 𝒏 payment is equivalent to amount
of 𝑭 of an ordinary annuity of 𝒏 payments.
Examples of this deferred annuity in real life:
1. A credit card company offering its clients to purchase today to start

paying monthly whether 3 months, 6 months, …, 24 months
depends on the choice of the clients.
2. A real estate agent is pleading the buyer to purchase a
condominium unit now and start paying after 2 years when the
condominium is ready for occupancy.
3. A worker who has gained extra income now and wants to save his
money so that he can withdraw his monthly starting on the day of
his retirement from work.
The present value 𝑃𝑑𝑒𝑓 of an annuity deferred for d periods is the value of
the annuity or the value of the 𝑛 payments in lump sum amount at the
beginning of the term.
Since the amount of a deferred annuity n payment is equivalent to amount

of F of an ordinary annuity of n payments. Let us illustrate the deferred
annuity using time diagram.
Ordinary annuity of n payments

Value of same annuity of n Translated for d periods
payments at one period
before 1st payment date
which is the end of d period
Deferred annuity of n payments
Pdef Discount P for d periods P F
________ _ _ _ R R_ _ _ _ R R R
0 1 2 … d-2 d-1 d d+1 d+2 … d+(n-2 ) d+(n-1) d+n
No payment for d periods
𝒅 = period of deferment 1st payment starts on the (𝑑 + 1)nth period
Therefore, from the time diagram we can now derived the deferred annuity
formula using the present value formula:
𝟏 − ( 𝟏 + 𝒋 )−𝒏
𝑷𝒅𝒆𝒇 = 𝑹 [ ] (𝟏 + 𝒋)−𝒅
𝒋
Or equivalently,
𝟏 − (𝟏 + 𝒋)−(𝒌+𝒏) 𝟏 − (𝟏 + 𝒋)−𝒌
𝑷=𝑹 −𝑹
𝒋 𝒋
where
𝑅 = is the regular payment,
𝑗 = is the interest rate per period,
𝑛 = is the number of payments,
𝑑 or 𝑘 = is the number period of deferment
(or number of artificial payments)
Examples: Find the present value of a P5,000 annuity payable

1
quarterly for 7 years but deferred for 52 year. Money is worth 9.6%
compounded quarterly.
Given: 𝑅 = 5,000 𝑡 = 7 years 𝑚=4 𝑖 (𝑚) = 9.6% = 0.096

Find 𝑃𝑑𝑒𝑓 :
Solution:
1
The period of deferral is also given in the problem which is 5 2 years. We can
solve for 𝑑.
𝑑 = period of deferral × 𝑚
11
𝑑 = ( ) (4) → 𝒅 = 𝟐𝟐
2
We also find the number of compounding periods, 𝑛 and the rate of interest
per period, 𝑗, using the formulas we had in the earlier modules:
𝑛 = 𝑚𝑡 → 𝑛 = (4)(7) → 𝒏 = 𝟐𝟖
𝑖 (𝑚) 0.096
𝑗 = →𝑗= → 𝒋 = 𝟎. 𝟎𝟐𝟒
𝑚 4
For convenience, we prepare a time diagram:
Pdef P28
d = 22 n = 28
___ 5,000 5,000_ R 5,000
5,000
0 1 2 … 20 21 22 23 24 … 49 50
1st payment starts on the 23rd period

NOTE: n = (last – first) + 1
28 last – 23+1
29 = last – 22
Solve the Deferred Annuity by substituting all the known values in the formula:
1 − ( 1 + 𝑗 )−𝑛
𝑃𝑑𝑒𝑓 = 𝑅 [ ] (1 + 𝑗)−𝑑
𝑗
1 − ( 1 + 0.024 )−28
𝑃𝑑𝑒𝑓 = 5,000 [ ] (1 + 0.024)−22
0.024
1 − ( 1.024 )−28
𝑃𝑑𝑒𝑓 = 5,000 [ ] (1.024)−22
0.024
1 − 0.514755758
𝑃𝑑𝑒𝑓 = 5,000 [ ] (0.593472984)
0.024
0.485244242
𝑃𝑑𝑒𝑓 = 5,000 ( ) (0.593472984)
0.024
𝑃𝑑𝑒𝑓 = 5,000 (20.21851008)(0.593472984)
𝑷𝒅𝒆𝒇 = 𝑷𝟓𝟗, 𝟗𝟗𝟓. 𝟕𝟎
Thus, the present value of the deferred annuity is P 59,995.70.

Example: Lowell is planning to buy a pension plan for himself on his

th
40 birthday. This plan will allow him to claim P30,000 quarterly for 5 years
starting 3 months after his 60th birthday. What one-time payment should he
make on his 40th birthday to pay off this pension plan, if the interest rate is
8% compounded quarterly?
Given: 𝑅 = 30000 𝑡 = 5 years 𝑚=4 𝑖 (𝑚) = 0.08

Find 𝑃𝑑𝑒𝑓 :
Solution:
The period of deferral is also given in the problem which is 20 years. We can
solve for 𝑑.
𝑑 = period of deferral × 𝑚
𝑑 = (20)(4) → 𝒅 = 𝟖𝟎
Note that the value of 𝑑 is also the same as the value of 𝑘. (We are going to
use the alternative formula.)
We also find the number of compounding periods, 𝑛 and the rate of interest
per period, 𝑗, using the formulas we had in the earlier modules:
𝑛 = 𝑚𝑡 → 𝑛 = (4)(5) → 𝒏 = 𝟐𝟎
𝑖 (𝑚) 0.08
𝑗 = →𝑗= → 𝒋 = 𝟎. 𝟎𝟐
𝑚 4
P ___ 30,000 30,000 _ _ _ _ _30,000

0 1 2 … 80 81 82 83 … 100
1 − (1 + 𝑗)−(𝑘+𝑛) 1 − (1 + 𝑗)−𝑘
𝑃=𝑅 −𝑅
𝑗 𝑗
−(80+20)
1 − (1 + 0.02) 1 − (1 + 0.02)−80
𝑃𝑑𝑒𝑓 = 30,000 − 30,000
0.02 0.02
1 − (1.02)−100 1 − (1.02)−80
𝑃𝑑𝑒𝑓 = 30,000 − 30,000
0,02 0.02
1 − 0.13803297 1 − 0.20510973
𝑃𝑑𝑒𝑓 = 30,000 − 30,000
0,02 0.02
1 − 0.13803297 1 − 0.20510973
𝑃𝑑𝑒𝑓 = 30,000 − 30,000
0,02 0.02
0.86196703 0.79489027
𝑃𝑑𝑒𝑓 = 30,000 − 30,000
0,02 0.02
𝑃𝑑𝑒𝑓 = 30,000( 43.0983516) − 30,000(39.74451)
𝑃𝑑𝑒𝑓 = 1,292,950.50 − 1192335.41
𝑷𝒅𝒆𝒇 = 𝐏𝟏𝟎𝟎, 𝟔𝟏𝟓. 𝟏𝟒
Therefore, the present value of Lowell’s monthly pension is

P100,615.14
ACTIVITY
A. Find the period of deferral in each of the following deferred annuity

problem. (Note: One way to find the period of deferral is to count the
number of artificial payments)
1. If Maro’s monthly payments of P3,000 for 5 years that will start 7

months from now, find the period of deferral.
2. Suppose that Lowell’s annual premium payments in his life insurance
of P8,000 for 15 years will start 5 years from now.
3. Thess quarterly payments for her health insurance of P6,000 for 6
years that will start three years from now.
4. Paolo applied for a housing loan which his semi-annual payments of
P60,000 for 15 years that will start 5 years from now.
5. Susan gained an extra income and she wants to save P30,000 every
2 years for 8 years starting at the end of 4 years.
B. Answer the following problems completely. (Use separate sheet for the
computation
1. Joanne availed of an emergency loan offered by GSIS that gave her an

option to pay P5,550 monthly for 3 years. The first payment is due after
6 months. How much is the present value of the loan if the interest rate
is 9.5% compounded monthly?
2. Mr. and Mrs. Cabuslay decided to buy a car. They made the down
payment of P250,000 and they will pay P26,800 monthly for 5 years.
The first payment is due after 3 months. How much is the present value
of the car if the interest rate is 3.92% convertible monthly?
3. Ivan purchased a laptop through the credit card of his colleague. The
credit card company provides an option for a deferred payment. Lowell
decided to pay after 6 months of purchase. His monthly payment is
computed as P2,800 payable in 18 months. How much is the cash value
of the laptop if the interest rate is 7.5% converted monthly?
REMEMBER
To find the deferred annuity

a. Read and analyze the given problem.
b. Make a time diagram.
c. Determine all the known values.
d. Solve the deferred annuity by substituting all the known values in
the formula:
𝟏− ( 𝟏+𝒋 )−𝒏
𝑷𝒅𝒆𝒇 = 𝑹 [ ] (𝟏 + 𝒋)−𝒅
𝒋
𝟏 − (𝟏 + 𝒋)−(𝒌+𝒏) 𝟏 − (𝟏 + 𝒋)−𝒌
𝑷=𝑹 −𝑹
𝒋 𝒋
where
𝑅 = is the regular payment,
𝑗 = is the interest rate per period,
𝑛 = is the number of payments,
𝑑 or 𝑘 = is the number period of deferment
(or number of artificial payments)
Choose the letter of the correct answer. Then write it on a separate

sheet of paper.
1. Find the present value of a deferred annuity of P900 every 3 months

for five years that is deferred 3 years, if the money is worth 10%
compounded quarterly.
A. P 10,234.27 B. P 10,324.27 C. P 10,432.27
2. Sarah converted her loan to light payments which gives her an option
to pay P4,800 every six months for 7 years, if the first payment is made
in 4 years and money is worth 11% compounded semi-annually. How
much is the amount of the loan?
A. P 32,642.94 B. P 31,642.93 C. P 30,642.92
3. Dexter made a series of quarterly payments of P5,700 each in his loan,

the first payment is due at the end of 5 years and the last at the end of
10 years and 9 months. How much is the amount of the loan if the
interest rate is 6% converted quarterly?
` A. P 86,041.96 B. P 86,041.86 C. P 86,041.76

4. Find the present value of 10 semi-annual payments of P7,500 each,

the first due in 5 years if the interest rate is 11% converted semi-
annually.
A. P 33,075.76 B. P 33,085.70 C. P 33,095.67
5. A certain fund is to be established today in order to pay for the P5,000

worth of monthly rent for a car. If the payment for the car rental will start
next year and the fund must be enough to pay for the monthly rental
for 2 years, how much must be deposited at 2.5% interest compounded
monthly?
A. P 116,930.64 B. P 115,844.60 C. P 114,046.58
POSTTEST
1. What is the period of deferral for payments of P6,000 every 4 months

for 10 years that will start five years from now?
A. 5 periods C. 12 periods
B. 9 periods D. 14 periods
2. If money is worth 9.75% semi-annually, find the present value of 12

semi-annual payments of P10,000 each, the first payment is due in
6½ years?
A. P45,175.59 C. P50,975.59
B. P48,075.59 D. P53,875.59
3. Nestor plans to buy a 55” slim smart television set with monthly
payments of P5,000 for 2 years. A credit card company offers a
deferred payment option for the purchase of this television set and the
payments will start at the end of 3 months. How much is the cash price
of the 52” slim smart TV set if the interest rate is 9.5% compounded
monthly?
A. P107,194.11 C. P113,123.15
B. P109,415.21 D. P116,514.24
BASIC CONCEPTS OF STOCKS AND BONDS
OVERVIEW
help you understand how to illustrate and distinguish between stocks and
bonds. The scope of this module permits it to be used in many different
Illustrate and distinguish between stocks and bonds

• illustrates stocks and bonds, and
• distinguish stocks and bonds.
PRETEST
1. It is a type of security that signifies ownership in a corporation

and represents a claim on part of the corporation’s assets and
earnings.
A. Amortization C. Stock
B. Mortgage D. Bond
2. It is a debt investment in which an investor loans money to an

entity which borrows the funds for a defined period of time at a
variable of fixed interest rate.
A. Amortization C. Stock
B. Mortgage D. Bond
3. The following cases are true for bonds EXCEPT.
A. If P = F, the bond is purchased at par.

B. If P = F, the bond is purchased at coupon.
C. If P > F, the bond is purchased at premium
D. If P < F, the bond is purchased at a discount.
LOOKING BACK
In our previous lesson of simple and compound interest, some
individuals or entities may borrow or lend or deposits their money to
secure funds for their own purposes. In this lesson this individual or
entities can invest and be part owner of a certain company or may apply
for an interest-bearing security which promises to pay on the maturity date
or on regular interest payments.
DEFINITION OF TERMS IN RELATION TO STOCKS
Stocks – share in the ownership of a certain company. Information may

vary about stock transactions; it can be obtained from tables published by
newspapers and online.
Stocks are classified into two types, the common stock and the preferred
stock.
➢ Common stock – is the one mostly owned by investors.

➢ Preferred stock – holds special treatment in relation to distribution
and sharing earnings.
Stockholders/Investors – one of the many owners of a certain company

who receive money called dividend.
Dividend – are part of the company’s profit distributed to the stockholders.

It is based on the number of shares of stock that a stockholder owns.
Dividend Per Share – ratio of the dividends to the number of shares.
Stock Market - Investors can buy or sell shares of stock. These are the
places where an investor can buy or sell stocks such as Philippine Stock
Exchange and Makati Stock Exchange.
Market value – the current price of a stock at which it can be sold.
Stock Yield Ratio – ratio of an annual dividend per share and the market
value per share. It is also called current stock yield.
Par value – the per share amount as stated on the company certificate.
Unlike market value, it is determined by the company and remains stable
over time.
DEFINITION OF TERMS IN RELATION TO BONDS
Bonds – are interest bearing security which promises to pay amount of

money on a certain maturity date as stated in the bond certificate and a
regular interest payment called coupons.
Coupon – the contract to pay a periodic payment on a specified date. The

coupons can be detached and cashed through banks.
Coupon Rate – the rate per coupon payment period.
Price of a Bond – the price of the bond at purchase time.
Par or Face Value (FV) – the amount payable on the maturity date.
Cases:
➢ If 𝑃 = 𝐹, the bond is purchased at par
➢ If 𝑃 < 𝐹, the bond is purchased at a discount.
➢ If 𝑃 > 𝐹, the bond is purchased at premium.
Term of a bond – fixed period of time (in years) at which the bond is
redeemable as stated in the bond certificate; number of years from time
of purchase to maturity date.
Fair Price of a bond – present value of all cash inflows to the bondholder.
Bondholders – are lenders/debtors to the institution which may be a

government or private company. He can sell his bonds to the highest
bidder. Bond issuers are the national government, government agencies,
government owned and controlled corporation, non-bank corporations,
banks, and multilateral agencies.
Bond Rate – the rate at which the bond pays interest on its par or face
value.
Yield Rate – the true overall rate of return that an investor receives on the
invested capital.
Stocks Bonds
A form of equity financing or
A form of debt financing or
raising money by allowing
raising money by borrowing
investors to be part owners of the
from investors.
company.
Stock prices vary every day. Investors are guaranteed
These prices are reported in interest payments and a return
various media (newspaper, TV, of their money at the maturity
internet, etc). date.
Uncertainty comes from the
Investing in stock involves some ability of the bond issuer to pay
uncertainty. Investors can earn if the bond holders. Bonds issued
the stock prices increase, but they by the government pose less
can lose money if the stock prices risk than those by companies
decrease or worse, if the company because the government has
goes bankrupt. guaranteed funding (taxes)
from which it can pay its loan.
Higher risk but with possibility of
Lower risk but lower yield.
higher returns.
Can be appropriate for retirees
Can be appropriate if the
(because of the guaranteed
investment is for the long term (10
fixed income) or for those who
years or more). This can allow
need the money soon (because
investors to wait for stock prices to
they cannot afford to take a
increase if ever they go low.
chance at the stock market).
REMEMBER
Stocks – share in the ownership of a certain company. This can be

bought or sold at its current price.
Stocks are classified into two types, the common stock and the
preferred stock.
➢ Preferred stock – holds special treatment in relation to
distribution and sharing earnings.
Bonds – are interest bearing security which promises to pay amount
of money on a certain maturity date as stated in the bond certificate
and a regular interest payment called coupons.
Cases:
• If 𝑃 = 𝐹, the bond is purchased at par
• If 𝑃 < 𝐹, the bond is purchased at a discount.
• If 𝑃 > 𝐹, the bond is purchased at premium.

of paper.
1. Which of the following cannot be true for stocks?
A. Stocks can be bought or sold

B. Stocks is mostly owned by investors
C. Stocks are published in newspaper and online.
D. Stocks is a written promise to pay a specified sum of money.
2. Which of the following is an example of bond?
A. Investors can earn if the security prices increase but they lose
money if the prices decrease
B. This suits for retirees since it is a guaranteed fixed income.
C. Investors can invest for a long term (10 years or more).
D. Investors can be part owners of the company.
3. Which statement best illustrates for stocks?
A. Lower risk but lower yield.

B. Higher risk but with possibility of higher returns.
C. Investors still need to consider the borrower’s credit rating.
D. Investors are guaranteed interest payments and return of
their money at the maturity date.
4. This refers to share in company’s profit for holders.
A. Par Value
B. Dividend
C. Coupon
D. Rate
POSTTEST
Read and analyze the given problem. Write your answer on a
If you are an investor which is better for you to invest your money,
in stocks or in bonds? Why? Explain in 2 paragraphs not more than
250 words (site an example)
Rubric to be used:
Skills 4 3 2 1
Level of Content Content Content Shows some
Content indicates indicates indicates thinking and
synthesis of original thinking and reasoning but
ideas, in- thinking and reasoning most ideas are
depth develops applied with underdeveloped
analysis and ideas with original and unoriginal.
evidences enough and thought on a
original firm few ideas.
thought and evidence
support for
the topic.
Quality of Use specific Use relevant Use Examples use
information and examples examples has nothing to
convincing clearly but the do with the
examples information lesson.
clearly relates in the has little to

relates in the lesson. do in the
lesson lesson.
Organization Information Information The The information

is veryis organized information is inappropriate
organized but appears to and inaccurate.
with well-paragraphs disorganized
constructed are not well- and
paragraphs, constructed, insufficient.
factual and and
correct information
is factual
Mechanics No Almost no A few Many
grammatical, grammatical, grammatical, grammatical,
spelling or spelling or spelling or spelling or
punctuation punctuation punctuation punctuation
errors; used errors; used errors; used errors; used
250 words less than less than less than 100
250 words 200 words words.
COMPUTATION OF STOCKS AND BONDS
OVERVIEW
help you understand how to illustrate and distinguish between stocks and
Illustrate and distinguish between stocks and bonds

• computes for stocks and bonds, and
• determine the difference between stocks and bonds.
PRETEST
Using the table below, If company JMC has the following shares of
common stocks of DMC Corporation and dividend.
Amount of
Price per Number of
Time Period Quarterly
Share Share
Dividend
st
1 Quarter P105.50 1,000
nd
2 Quarter P112.25 1,000
rd
3 Quarter P98.75 1,000
th
4 Quarter P101.35 1,000
Total Annual Dividend
1. What is the amount of dividend in the 2nd quarter of JMC

company?
A. P105,000
B. P112,250
C. P98,750
D. P101,350
2. How much is the total annual dividend of JMC company?
A. P317,950
B. P417,850
C. P517,750
D. P617,650
3. XYZ Commodities Corporation purchased a bond having a

P150,000 par value at 12%. Find the annual interest earned.
A. P15,000
B. P14,000
C. P13,000
D. P12,000
LOOKING BACK
Stocks – share in the ownership of a certain company. Information may
vary about stock transactions; it can be obtained from tables published by
newspapers and online.
Stocks are classified into two types, the common stock and the preferred
stock.

➢ Preferred stock – holds special treatment in relation to distribution
and sharing earnings.
Bonds – are interest bearing security which promises to pay amount of

money on a certain maturity date as stated in the bond certificate and a
regular interest payment called coupons.
Cases:
➢ If 𝑃 = 𝐹, the bond is purchased at par
➢ If 𝑃 < 𝐹, the bond is purchased at a discount.
➢ If 𝑃 > 𝐹, the bond is purchased at premium.
Example: DMC Company purchased P5,000 shares of stocks at

P195 par value with a preferred dividend rate of 8. 5% quarterly. How
much dividend will DMC receive per quarter?
Given:
1
Par Value = P195 Dividend Rate = 82%
Total Number of Shares = P5,000 T = quarterly (1/4 of a
year)
Solution:
𝑸𝑫 = 𝑷𝑽 × 𝑻𝑵𝑺 × 𝑫𝑹 × 𝑻𝑷
where: 𝑄𝐷 is the quarterly dividend

𝑃𝑉 is the par value
𝑇𝑁𝑆 is the total number of shares
𝐷𝑅 is the dividend rate
𝑇𝑃 is the time period
1
𝑄𝐷 = 𝑃195 𝑥 5,000 𝑥 0.085 𝑥
4
𝑄𝐷 = P20,718.75
Therefore, the quarterly dividend is P20,718.75
Example: ABC Corporation has a current market value of P52,

gave a dividend of P8 per share for its common stock. XYZ Corporation
has a current market value of P95, gave a dividend of P12 pe share. Use
the stock yield ratio to measure how much dividends shareholders are
getting in relation to the amount invested.
Given:
ABC Corporation XYZ Corporation
Given: Given:
Dividend per share = P8 Dividend per share = P12
Market value = P52 Market value = P95
Find: Stock Yield Ratio (SYR)
Solution:
The stock yield ratio can be obtained by dividing the dividend per share
and the maket value.
𝐝𝐢𝐯𝐢𝐝𝐞𝐧𝐝 𝐩𝐞𝐫 𝐬𝐡𝐚𝐫𝐞
𝑺𝒀𝑹 =
𝐦𝐚𝐫𝐤𝐞𝐭 𝐯𝐚𝐥𝐮𝐞
Hence,
8
ABC Corporation: 𝑆𝑌𝑅 = 52 → 𝑆𝑌𝑅 = 0.1538 → 𝑺𝒀𝑹 = 𝟏𝟓. 𝟑𝟖%
12
XYZ Corporation: 𝑆𝑌𝑅 = 95 → 𝑆𝑌𝑅 = 0.1263 → 𝑺𝒀𝑹 = 𝟏𝟐. 𝟔𝟑%
Therefore, ABC Corporation has a higher stock yield ratio than XYZ
Corporation. This means that, each peso would earn you more if you
invest in ABC Corporation than in XYZ Corporation. If all other things
are equal, then it is wiser to invest in ABC Corporation.
Example: Mr. Reyes purchased 900 shares of LBC stock for

P56.49 per share. If the broker’s commission is 1.5%, how much is the
amount of the broker’s commission and the amount the investors paid for
the stock plus the commission?
Given:
Number of shares of stock = 900 shares
Price per share of stock = P56.49
Broker’s commission = 1.5% or 0.015
Find: Amount of Broker’s Commission (BC) and Amount of Payment (AP)

Solution:
The amount of payment (𝐴𝑃) can be obtained by adding the purchase
price of stocks (𝑃𝑃) and the broker’s commission (𝐵𝐶). That is:
𝑨𝑷 = 𝑷𝑷 + 𝑩𝑪
The purchase price (𝑃𝑃) can be solved by multiplying the price per share
of stocks (𝑃𝑠 )and the number of shares (𝑛𝑠 ).
𝑃𝑃 = 𝑃𝑠 𝑛𝑠
𝑃𝑃 = 56.49(900)
𝑷𝑷 = 𝟓𝟎𝟖𝟒𝟏
The broker’s commission (𝐵𝐶) is computed by multiplying the purchase

price (𝑃𝑃) to the broker’s commission rate (𝑟𝑐 ).
𝐵𝐶 = 𝑃𝑃𝑟𝑐
𝐵𝐶 = 50841(0.015)
𝑩𝑪 = 𝟕𝟔𝟐. 𝟔𝟐
Thus, the amount of payment (𝐴𝑃):

𝐴𝑃 = 𝑃𝑃 + 𝐵𝐶
𝐴𝑃 = 50841 + 762.62
𝑨𝑷 = 𝟓𝟏𝟔𝟎𝟑. 𝟔𝟐
Therefore, the investor paid P51,603.62 for the stock transaction.
Example: RQS Corporation purchased a bond having a par value

1
of P350,000 with a price quotation of 95 5%. What is the market value of
the bond?
Given:
Par or Face Value = P350,000
1
Price Quotation = 955 % = 95. 20% = 0.952
Find: Market Value (𝑀𝑉)
Solution:
The market value of a bond can be obtained by multiplying the face value
of bond (𝐹𝑣 ) to its price quotation (𝑄).
𝑀𝑉 = 𝐹𝑣 𝑄
𝑀𝑉 = 350000(0.952)
𝑴𝑽 = 𝟑𝟑𝟑𝟐𝟎𝟎
Therefore, the market value of the bond is P 333,200.00.
Example: Suppose that a bond has a face value of P150,000 and

its maturity date is 5 years from now. The coupon rate is 6% payable semi-
annually. Find the fair price of this bond, assuming that the annual market
is 3.5%
Given:
Face Value (𝐹𝑣 ) = P150,000 Coupon Rate (𝑟) = 6%
Maturity Date = 5 years Market Rate = 3.5%
Frequency of Conversion (𝑚) = 2 (semi-annually)
Number of periods (𝑛) = 𝑚𝑡 = (2)(5) = 10
Find: Fair Price of the Bond

Solution:
First, find the amount of semi-annual coupon which is the product of the
face value and coupon rate semi-annually.
Amount of semi-annual coupon = face value x coupon rate semi-annually

1
= 150,000 𝑥 (0.06 𝑥 2)
= 150,000 𝑥 0.03
𝑨𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒔𝒆𝒎𝒊 − 𝒂𝒏𝒏𝒖𝒂𝒍 𝒄𝒐𝒖𝒑𝒐𝒏 = 𝑷𝟒, 𝟓𝟎𝟎
Bondholder receives 10 payments of P4,500 each, and P150,000 at 𝑡 =

5.
Let us find the present value of P150,000 for 5 years with market rate of
3.5%. We shall use the formula of present value in compound interest
formula which is;
𝐹
𝑃1 =
(1 + 𝑖)𝑛
150,000
𝑃1 =
(1 + 0.035)5
150,000
𝑃1 =
(1.035)5
150,000
𝑃1 =
1.18768631
𝑷𝟏 = 𝑷𝟏𝟐𝟔, 𝟐𝟗𝟓. 𝟗𝟕
Now let us compute the present value of p150,000 with 3.5% annual
market rate for 5 years. We need to convert 3.5% to equivalent semi-
annual rate since the coupon rate is 6% payable semi-annually. We shall
use the formula of present value of an ordinary annuity which is,
𝟏 − (𝟏 + 𝒋)−𝒏
𝑷 = 𝑹[ ]
𝒋
Convert 3.5% to equivalent semi-annual rate:
𝐹1 = 𝐹2
𝑃(1 + 𝑗)𝑛 = 𝑃(1 + 𝑗)𝑛
2
1
𝑖 (𝑚)
𝑃(1 + 0.035) = 𝑃 (1 + )
2
𝑖 (𝑚)
1.017349497 = 1 +
2
𝒊(𝒎)
𝟎. 𝟎𝟏𝟕𝟑𝟒𝟗𝟒𝟗𝟕 =
𝟐
Thus,
1 − (1 + 𝑖)−𝑛
𝑃2 = 𝑅 [ ]
𝑖
1 − (1 + 0.017349497)−10
𝑃2 = 4,500 [ ]
0.017349497
1 − (1.017349497)−10
𝑃2 = 4,500 [ ]
0.017349497
1 − 0.84197317
𝑃2 = 4,500 [ ]
0.017349497
0.15802683
𝑃2 = 4,500 [ ]
0.017349497
𝑃2 = 4,500[9.10843871]
𝑷𝟐 = 𝑷𝟒𝟎, 𝟗𝟖𝟕. 𝟗𝟕
Now, the fair price is the sum of the two present values.
Fair Price = Present Value1 + Present Value2

𝐹𝑃 = 126,295.97 + 40,987.97
𝑭𝑷 = 𝑷𝟏𝟔𝟕, 𝟐𝟖𝟑. 𝟗𝟒
Therefore, a fair price of 𝑷𝟏𝟔𝟕, 𝟐𝟖𝟑. 𝟗𝟒 is equivalent to all future

payments, assuming an annual market rate of 3.5%.
ACTIVITY
A. The table below shows the data on 5 stockholders. Find the dividend
of the 5 stockholders.
(Note:Use Dividend = Dividend Percentage x Par Value x Number

of shares)
Par Value (in Dividend Number of

Stockholder
Peso) (%) Shares
A 55 3.5% 105
B 43 5.25% 125
C 36 1.75% 250
D 68 6% 400
E 52 2.15% 600
B. Answer the following problems completely.
1. Calculate the market value and determine if each of the

following bonds is purchased at a premium or a discount.
a. Par Value of P100,000 with Price Quotation of 9.15%]
b. Par Value of P200,000 with Price Quotation of 1.24%
2. RFO distributor bought 18,800 shares of stocks at P1.43 par

value with a preferred quarterly dividend rate of 8.5%. How
much is the amount of quarterly dividend?
3. Find the amount of a semi-annual coupon for a P350,00 bond

which pays 8% convertible semi-annually for its coupons.
4. A property holding declared a dividend of P10 per share for

the common stock. If the common stock closes at P86, how
large is the stock yield ratio on this investment?
5. RSP Company issued a P200,000 bond paying interest rate

of 7.75%. If the closing price is 110.12%, find the current yield
of the bond. (Note: Current Yield = Annual Interest divided by
Current Price)
REMEMBER
To compute stocks:
𝑸𝑫 = 𝑷𝑽 × 𝑻𝑵𝑺 × 𝑫𝑹 × 𝑻𝑷
Dividend per Share

𝑆𝑡𝑜𝑐𝑘 𝑌𝑖𝑒𝑙𝑑 𝑅𝑎𝑡𝑖𝑜 =
Market Value
To compute bonds:
𝑭
Present Value in Compound Interest formula: 𝑷𝟏 = (𝟏+𝒋)𝒏
𝟏−(𝟏+𝒋)−𝒏
Present value of an ordinary annuity formula: 𝑷 = 𝑹 [ ]
𝒋
Fair Price = Present Value1 + Present Value2

of paper.
1. A food corporation declared a dividend of P25,000,000 for its common

stock. Suppose there are P180,000 shares of common stock, how
much is the dividend per share?
A. 138.79 B. 138.89 C. 138.99
2. A certain financial institution declared P57 dividend per share for its
common stock. The market value of the stock is P198. Determine the
stock yield ratio.
A. 28.79% B. 28.89% C. 28.99%
3. A certain land developer declared a dividend of P28 per share for the
common stock. If the common stock closes at P99, how large is the stock
yield ratio on this investment?
A. 29. 29% B. 28.28% C. 27.27%

4. Calculate the total semi-annual interest earned by two bonds each

having a par value of P250,000 at 8% interest rate.
A. 𝑃21,775 B. 𝑃21,875 C. 𝑃21,975
5. A P450,000 bond is redeemable at P550,000 after 5 years. Coupons

are given at 5% convertible semi-annually. Find the amount of the semi-
annual coupon.
A. 𝑃11, 450 B. 𝑃11,350 C. 𝑃11,250
POSTTEST
(No posttest is intended)
Provide newspaper clippings or search online that show information

about stocks and bonds. Look for the important words that you see
in and think about what these terms might mean. (We will use these
in our next topic)
MARKET INDICES FOR STOCKS AND BONDS
OVERVIEW
help you understand analyzing the different market indices for stocks and
level of students. The lessons are arranged to follow the standard sequence
of the course. But the order in which you read them can be changed to
Analyze the different market indices for stock and bonds.

• explain the terms in stock and market tables.
• differentiate stock indices and bond indices.
PRETEST
1. The market index/indices for stocks represents which of the

following?
A. Collection of different stocks that act as benchmark of

investment in the market.
B. Determine the supply and demand in economic model in the
market
C. Classification of capital comes in selling the shares in the
market.
D. Evaluate the future growth shares of investors in the
market.
2. The acquisition of capital for bonds comes in two forms which are?
A. Stocks and bonds

B. Debt capital and equity capital
C. Public offering and secondary markets.
D. Relationship of Investors and corporation
3. Which of the following the characteristics of a useful indices?
A. Indices function as a status report on the general economy.

B. The inability to duplicate the most successful fund managers'
approaches.
C. The index funds merely followed the stock indexes downward.
D. Calculation of indices adjusted based on stock splits or
other changes in the market.
LOOKING BACK
In the previous modules we studied on how to compute the simple
interest and compound interest whether it is deposit or loan.
Let say your planning to invest your money, so how are you going to
allocate your money?
Let’s study some key points that we need for a well-diversified portfolio
strategy for stocks and bonds.
Stock Market Index – is a measure of a stock market, or a small subset of

the market, that helps investors compare current price levels with past prices
to calculate market.
➢ This is the barometers of the stock market.
➢ The indices give a broad outline of the market movement and
represent the market.
➢ Some of the stock market major indices in the Philippines are:
a. PSE Composite Index or PSE

It is composed of 30 companies carefully selected to represent the
general movement of the market prices.
b. FTSE Philippines
c. PHS All Shares
d. Other indices are sector indices, each representing a particular sector
(e.g. financial institutions, industrial corporations, holding firms,
service corporations, mining/oil, property)
Visit https://www.pse.com.ph/stockMarket/home.html for example of Stocks.
Usefulness of indices
• Indices help to recognize the broad trends in the market.
• Index can be used as a benchmark for evaluating the investor’s
portfolio.
• Indices function as a status report on the general economy. Impact of
various economic policies are reflected on stock market.
• The investor can use the indices to allocate funds rationally among
stocks. To earn returns on par with the market returns, he can choose
the stocks that reflect the market movements.
List of index values:

Val – Value of the index
Chg – change of the index value from the previous trading day.
(Chg = value today – value yesterday)
% Chg – ratio of Chg to Val
(% Chg = Chg divided by Val)
Stock Tables – shows about stocks information.

52-WK Hi/LO – highest/lowest selling price of the stock in the past 52
weeks.
HI/LO – highest/lowest price of the stock in the last trading day.
STOCK – three-letter symbol the company is using for trading.
DIV – dividend per share last year
VOL (100s) – number of shares (in hundreds) traded in the last
trading day.
CLOSE – closing price on the last trading day.
NETCHG – net change between the two last trading days.
Buying or Selling Stocks

To buy or sell stocks, one may go to the PSE personally. Most
transactions now a days are done through phone call to a registered
broker or by logging on to a reputable online trading platform.
Bid Size – the number of individual buy orders and the total numbers
of shares they wish to buy at a given price.
Bid Price – the price these buyers are willing to pay for the stock.
Ask Price – the price the sellers of the stock are willing to sell the
stock.
Ask Size – how many individual sell orders have been placed in the
online platform and the total number of shares these sellers
wish to share at specific price.
Bond Market Indices - is made up of selected bonds and used to measure

the value of a part of the bond market. While it can be useful tool to gauge
the value of specific investments, a bond market index isn’t without its pitfalls,
and should be used judiciously.
• It is a tool used by investors and financial managers to describe
the market, and to compare the return on specific investments.
The main platform for bonds or fixed income securities in the Philippines is
the Philippine Dealing and Exchange Corporation (PDEx)
Types of Bonds
1. Maturity-based bonds
Bonds categorized based on the length of time it will mature.
• Treasury Bills (T-bills) – Bonds that mature in less than 1 year (short
term). The most common tenors (length of maturity) for T-bills are 91
days, 181 days, and 364 days.
➢ Matures in less than a year (shorter investment time frame)
➢ Sold at a discount from their face value but the investor will
get the full amount upon maturity (works like a zero-coupon
bond)
➢ Doesn’t pay income or coupon interest
• Treasury Bonds (T-bonds) – Bonds that have tenors of more than 1

year. The most common maturity lengths for T-bonds are 2-year, 5-
year, 7-year, 10-year, 20-year, and 30-year bonds.
➢ Pays investor coupon interest (fixed income) at fixed intervals for
the duration of the bond
➢ Can present a higher risk due to the longer length of time before
it matures
2. Issuer-based bonds
These are bonds that are classified according to who issued it:
• Treasury Securities – Bonds issued by the Bureau of Treasury
➢ Low(er) risk since investment is backed by the full faith and credit
of the government (vs other fixed income investments)
➢ The lower risk comes with a lower yield potential compared to
other fixed income instruments
• Government Bonds – Bonds that are issued by various government

agencies like HDMF, Government National Mortgage Association
(GNMA), Federal National Mortgage Association, and others.
➢ Low(er) default risk (similar with Treasury Securities)
➢ Favorable tax treatment
➢ Interest rate risk. Gov’t bonds may lose value if market interest
rates rise beyond the bond’s face value
• Municipal Bonds – Bonds issued by the local government units

(LGUs).
➢ Low(er) default risk
➢ Low volatility
➢ Interest rate risk. Gov’t bonds may lose value if market interest
rates rise beyond the bond’s face value
• Corporate Bonds – Bonds issued by public and private companies.

➢ Potentially higher returns vs gov’t-issued banks
➢ Highly liquid
➢ Multiple options
➢ Higher risk compared to gov’t-issued bonds
ACTIVITY
Fill in the blanks.

1. _________ the price these buyers are willing to pay for the stock.
2. It is a tool used by investors and financial managers to describe the
____________, and to compare the return on specific investments.
3. This __________ issued by the Bureau of Treasury.
4. The measure of ___________, or a small subset of the market, that
helps investors compare current price levels with past prices to
calculate market.
5. This __________ shows about stocks information.
REMEMBER
The bond market is where investors go to trade (buy and sell) debt
securities, prominently bonds, which may be issued by corporations or
governments. It is also known as the debt or the credit market. Securities
sold on the bond market are all various forms of debt. By buying a bond,
credit, or debt security, you are lending money for a set period and
charging interest—the same way a bank does to its debtors.
A stock market is a place where investors go to trade equity securities

such as common stocks and derivatives including options and futures.
Stocks are traded on stock exchanges. Buying equity securities, or
stocks, means you are buying a very small ownership stake in a
company. While bondholders lend money with interest, equity holders
purchase small stakes in companies on the belief that the company
performs well and the value of the shares purchased will increase.

paper.
1. What is the most major difference between bond and stock markets?
A. It was the trading and introducing new products and asset classes
B. The stock market has central places or exchanges where stocks are
bought and sold while bonds are default by issuers.
C. The market is facilitated by underwriters who set the initial price for
securities.
D. Bond index can help investors track the performance of bond
portfolios than stock index.
2. Which of this is true for bonds coupon rate?
A. Temporary
B. Fluctuate
C. Auctioned
D. Fixed
3. Which of this need to consider before investing in bonds?
A. Be aware not only of the financial condition of the issuer of the bond
but also the prevailing market conditions.
B. Be aware that investing in bonds is relatively safer investment than
investing in stocks.
C. Be aware that these prices are determined by supply and demand.
D. Be aware that as the price of the bond may increase or decrease.
4. What is the main platform for bonds or fixed income securities in the
Philippines?
A. Government National Mortgage Association (GNMA)

B. Philippine Dealing and Exchange Corporation (PDEx)
C. PSE Composite Index (PSE)
D. Local Government Units (LGUs)
5. What is the main platform for stocks securities in the Philippines?
A. Government National Mortgage Association (GNMA)

B. Philippine Dealing and Exchange Corporation (PDEx)
C. PSE Composite Index (PSE)
D. Local Government Units (LGUs).
POSTTEST
Performance Task: (To be submitted on or before January TBA, 2020)
Make your own infographic about how the stock market works or bonds
work. Include the following information:
a) Define stock or bond
b) How it works (show the flow of trading)
c) How traders/investors make money
d) How traders/investors lose money
Infographic Component of Project
Purpose: To verbally and visually represent the data to help others to clearly
understand the information you would like to present. Infographics aim to
allow the reader to draw conclusions. Audience: Your audience is broader
and more general than your research paper. Target SENIOR HIGH
SCHOOL students in general.
Process:
A. Explore online to see the various infographics. Become familiar with

what infographics are and what they look like. a. A good basic overview
of the history and use of information graphics is available on Wikipedia:
https://en.wikipedia.org/wiki/Infographic
B. Know your subject. Become a Content Expert on your topic. Do

thorough research on your topic.
C. Plan the “Story” your Infographic will tell. Draw a rough sketch of the
infographic. The infographic must have a beginning, middle, and end.
Consider developing a concept map, flow diagram, or wireframe
(shown below) to depict your infographic plan. Check out the good
infographic examples: https://www.designyourway.net/blog/graphic-
design/infographic-examples
D. Think Visual:
a. Identify ways to convert text to images. Try to convert as
much of your data and text into visual imagery by using charts,
graphs, diagrams, maps, flowcharts, and other elements.
b. Determine the desired look you are trying to achieve for your
infographic. The visual approach you want helps you determine
the color scheme, font types, and structure. Keep things simple
with only 2-3 fonts, sizes, and colors.
E. Create your infographic using the website:

https://create.piktochart.com/infographic
BASIC CONCEPTS OF BUSINESS LOANS AND

CONSUMER LOANS
OVERVIEW
help you understand business and consumer loans. The scope of this
module permits it to be used in many different learning situations. The
lessons are arranged to follow the standard sequence of the course. But
the order in which you read them can be changed to correspond with the
textbook you are now using. The module focuses on achieving this
learning competency:
Illustrates and distinguish between business and consumer loans.

• identify business or consumer loan, and
• differentiate business loan from consumer loan or vice versa.
PRETEST
1. This refers to pay the entire loan on time

A. Documentation
B. Collateral
C. Follow-up
D. Term
2. These documents are needed for consumer loans EXCEPT:
A. Bank statements
B. Employee pay slips
C. Certificate of employment
D. Company financial statements
3. Mrs. Gozon owns a restaurant business. She wants an expansion

on her place to accommodate more customers. She decided to
have a loan to expand her business place. This is an example of:
A. Consumer Loans
B. Business Loans
C. Bank Loans
D. Microloans
LOOKING BACK
Let us recall the following terms:
A loan is a financial obligation of paying someone a certain amount for the

use of his or her money. Loan proceeds can be used for a variety of
purposes. It can be used to finance a business, buy a new car, expansion of
business, for house improvement.
Amortization is a gradual disappearance of debt, principal and interest by

series of equal periodic payments or installment payments due at the end of
equal intervals of time.
There are different types of loans. If we want to purchase goods and

merchandise or we want to open a small business, or we want to expand
your business this involves a big sum of money. Either you will borrow money
from the bank in which a collateral is needed. It may be a land title, real estate
or other investments that you have. Or you want to buy a brand-new car thru
bank financing or in-house financing. Let us familiarize with the following
terms to understand more about which type of loan are best to purchase a
new car, expand your business or open a new business, or for mortgages.
Definition of Terms:
• Business Loan – money lent specifically for a business purpose. It

may be used to start a business or to have a business expansion.
• Consumer Loan – money lent to an individual for personal or family
purpose
• Collateral – assets to secure the loan. It may be real estate or other
investments.
• Guarantor – is a person who guarantees to pay for someone else’s
financial obligation if the borrower fails to do so,
• Term of the Loan – time to pay the entire loan.
Difference between business loan and consumer loan
Business Loans Consumer Loans
Terms
The terms for business loan are The term for consumer loan is longer
generally shorter with an inclusion of with lower interest rate.
greater interest rate.
Collateral
For business loans collateral may In consumer loans may include also
include real estate or investment. In real estate or investments depends
addition, they can use equipment, on the loans applied.
fixtures or furniture as collateral. To
secure the assets of the business, a
business loan may also require that
the business owners make personal
assets available as well.
Guarantor
A business owner of the business A consumer loan typically does not
required to sign themselves as require a guarantor to sign the loan.
guarantor of the loan.
Documentation
For business loans, the lendee For consumer loans, the bank or the
must submit a credit report for lending company or institution may
evaluation, income tax returns and require a credit report, bank
company financial statements to be statements, and an income tax
compiled and documented by a return, and if the lendee is employed
certified accountant. a certificate of employment and
employee pay slips.
Follow up
With business loans, annual However, with a consumer loan,
reviews of the relationship are often once the money has been
conducted by the bank. Many banks distributed, there is generally no
will require businesses to submit further follow-up needed by the
annual financial reports for review. bank as long as payments are
This can alert the bank to any made as agreed.
impending issues with the business
that could threaten repayment of
the loan.
Loans
Business loans on the other hand, is Consumer loan is when an
a loan particularly meant for individual borrows money for
business reasons or purposes. personal or family purpose to
Business loans includes microloans, secured or unsecured funds or
invoice financing, bank loans, asset- money from a lender. Consumer
based financing, cash flow loans as loan includes credit cards,
well as cash advances. mortgages, home equity lines of
credit, refinances, auto loans,
student loans as well as personal
loans.
ACTIVITY
A. Fill in the blanks.

1. ___________ is an asset to secure the loan.
2. A kind of loan that the term of the loan usually shorter and include
a higher interest rate is__________.
3. A person who guarantees to pay for someone else’s financial
obligation if the borrower fails to pay. _____________
4. Once the money has been distributed, there is no further follow-up
needed by the bank if payments are agreed upon. This refers to
____________.
5. Money lent to an individual for personal or family purpose is called
___________.
B. Identify whether the following is a consumer or business loan.
1. Due to pandemic, Mrs. Reyes needs to follow the protocol imposed by

IATF in order to operate her salon. She needs to renovate her Nail Art
and Foot SPA Salon and buy other equipment needed. In order to
pursue her plan, she wants to borrow money from the bank.
2. Darren owns a milk tea store at their subdivision. He wants to put
another branch in a new mall near in their subdivision. He decided to
have a loan to set up the new business as soon as possible.
3. Ivan plans to buy a motorbike. He got a loan of P80,000.
4. Andrew decided to expand his Korean Food Grocery. He decided to
borrow money from the bank to pursue the expansion.
5. Mr. Nestor Dela Cruz decided to change his laptop and buy other
gadgets which he will need for his online class. He uses his credit card
to purchase these items payable for six months.
REMEMBER
To distinguish Business loans from Consumer loans:
Business Loan – is usually made to fund a company’s operating

expenses and other financial needs such as microloans, invoice
financing, bank loans, asset-based financing, cash flow loans as well as
cash advances
Consumer loans – are used for personal, family or household purposes
such as credit cards, mortgages, home equity lines of credit, refinances,
auto loans, student loans as well as personal loans.

paper.
1. The following are example of business loans EXCEPT

A. Invoice Financing
B. Credit Card
C. Bank loan
D. Microloans
2. Mr. and Mrs. Cabuslay want to borrow money from the bank to finance
the college education of her daughter. This is an example of which type
of loan?
A. Business Loan
B. Consumer Loan
C. Bank loan
D. Microloans
3. Which of this is not true for a consumer loan?

A. Does not require a guarantor
B. Include real estate or investments depends on the loans applied.
C. It is particularly meant for asset-based financing.
D. If loan has been distributed, follow-up is not needed.
4. Ana wants to have some improvements on their house. She needs

to elevate her house because they are always prone in floods and
she wants to build a new room for her son and daughter. She will
borrow money from the bank to finance her plan. This is an example
of which type of loan?
A. Business Loan
B. Consumer Loan
C. Bank loan
D. Microloans
5. Which of these is NOT true for business loans?

A. Does not require a guarantor
B. Include real estate or investments depends on the loans
applied.
C. It is particularly meant for asset-based financing.
D. If loan has been distributed, follow-up is not needed.
POSTTEST
Choose the letter of the best answer. Write the chosen letter on
a separate sheet of paper.
1. Which of the following is an example of consumer loans?

A. Invoice Financing C. Bank loan
B. Credit Card D. Microloans
2. Which of the following uses refers to business loans?

A. Purchasing new or used equipment
B. Opening a new location
C. Repairs or renovations
D. All of the above
SOLVING PROBLEMS ON AMORTIZATION
OVERVIEW
help you understand solving problems on business and consumer loans.
Solve problems on business and consumer loans (amortization and

mortgage)

• solve problems involving business or consumer loans; and
• differentiate amortization from mortgage or vice versa.
PRETEST
1. A debt of P45,000 with an interest rate of 8% compounded

quarterly will be amortized for 1 year and 6 months. How much
is the quarterly amortization?
A. P5,128.60 C. P6,088.62
B. P7,046.64 D. P8,033.66
2. Mr. Marcelo is required to pay 5 annual installments of P25,000

each for a loan at 10% compounded annually. How much is the
loan?
A. P90,596.38 B. P92,600.47
B. P93,432.59 D.P94,769.68
3. A man borrowed P60,000 to be amortized by equal payments at

the end of six months for 3 years at 10% interest compounded
semi-annually. How much is the semi-annual amortization?
A. P14,523.22 B. P13,672.20
B. P12,713.10 D. P11,821.05
LOOKING BACK
Let us recall the following formulas:
1. Simple interest 𝑰𝒔 = 𝑷𝒓𝒕 is usually used for short borrowing of

money.
2. Compound interest 𝑭 = 𝑷(𝟏 + 𝒓)𝒏 is usually used by banks in
calculating interest for long term investments and loans such as
savings account and time deposit.
𝟏−(𝟏+𝒋)−𝒏
3. Ordinary annuity 𝑷 = 𝑹 𝒋
is usually used for a series of equal
and regular payments in which each payment is made at the end of
the payment period.
The trend now in buying goods and merchandise, availment of services,

and the like are made easy and convenient through credit cards. One
important approach in annuities is the repayment of debts by periodic or
installment payments.
When a debt is amortized by equal payments at equal intervals the debt

becomes the present value of an annuity.
Amortization Method - A method of paying a loan (principal and interest)

on installment basis, usually of equal amounts at regular interval. This can
only be used for intangible assets such as licenses, trademarks, patents
and loans.
Examples: Solve problems involving amortization.
1. Mr. Martinez bought a LED TV set for P60, 000. He made a 15%
down payment and paid P3,150 a month for 1½ years. Find the total
amount (cost) and the total charges (interest, service/finance
charges and other fees) Mr. Martinez paid for the TV set.
Given:
Price of TV set = P60,000
Down payment = 15% of price of TV set
Monthly payment = P3,150
Solutions:
Down payment = 15% x Price of TV set = 15% x 60,000 = P9,000
Total amount of Monthly Payment = Monthly payment x No. of years
= 3,150 x 1½ years
3
= 3,150 x ( x 12)
2
= 3,150 x 18
Total Amount of Monthly Payment = P56,700
Total Cost of TV set = Down payment + Total Amount of Monthly
Payment
= 9,000 + 56,700
= P65,700
Total Charges = Total Cost of TV set – Price of TV set
= 65,700 – 60,000
Total Charges = P5,700
2. A loan of P100,000 is to be amortized by equal payments at the end

of each quarter for 18 months. If interest is 10.5% compounded
quarterly, find the periodic payment and construct an amortization
schedule.
Given:
P = P100,000 i= 10.5% m = 12 t = 18 mos.
Solution:
j = 10.5% ÷ 4 n = 18 ÷ 12 = 1.5
= 2.625% = 1.5 x 4
j = 0.02625 n=6
1−(1+𝑗)−𝑛
Let us use the ordinary annuity formula 𝑃 = 𝑅 and
𝑗
substitute all the known value.
1 − (1 + 𝑗)−𝑛
𝑃=𝑅
𝑗
1 − (1 + 0.02625)−6
100,000 = 𝑅
0.02625
1 − (1.02625)−6
100,000 = 𝑅
0.02625
1 − 0.8560142
100,000 = 𝑅
0.02625
0.14398578
100,000 = 𝑅
0.2625
100,000 = 𝑅(5.48517266)
100,000 5.48517266
=𝑅
5.48517266 5.48517266
100,000
𝑅=
5.48517266
𝑹 = 𝑷𝟏𝟖, 𝟐𝟑𝟎. 𝟗𝟕
To compute the following

1. Interest paid = Unpaid Balance x i = 100,000 x 0.02625 = P2,625
2. Principal Repaid = Periodic Payment – Interest paid
= 18,230.97 – 2,625 = P15,605.97
Payment No. = Previous Unpaid Balance – Principal Repaid
Payment No. 2 = 100,000 – 15,605.97 = P84,394.03
Amortization Schedule:
Periodic 10.5% Interest Principal Unpaid
Payment Payment at the Paid at the Repaid at the Balance at the
Period end of 3 end of 3 end of 3 end of 3
months months months months
0 - - - P100,000
1 18,230.97 2,625 15,605.97 84,394.03
2 18,230.97 2,215.34 16,015.63 68,378.40
3 18,230.97 1,794.93 16,436.04 51,942.37
4 18,230.97 1,363.49 16,867.48 35,074.88
5 18,230.97 920.72 17,310.25 17,764.63
6 18,230.97 466.32 17,764.65 0.00
TOTAL P109,385.82 P9,385.80 P100,000.02
Note: You can make the Amortization Schedule using excel.
ACTIVITY
Solve the following problems. Round amounts to the nearest

centavo.
1. Sandra is required to pay 24 equal installments of P2,300 payable at

the end of each month for a loan at 7.5% compunded quarterly. Find
the amount of his loan.
2. Mrs. Cabuslay obtained a housing loan of P1,800,000, which she
would like to pay on equal installment at the end of each month for 15
years. She pay the 10% downpayment of the amount of the loan. Find
his periodic installment if she is required to pay 7.5% interest
compounded monthly. Construct an amortization schedule for his first
10 periodic payments.
REMEMBER
To solve problems involving amortization:

1. Read and analyze the problem.
2. Determine the given. Use an appropriate formula.
3. Substitute the known values in the formula. Solve.
To construct an amortization schedule:
1. Construct a table for payment number, unpaid balance, interest paid,
periodic payment and principal prepaid.
2. Know the monthly/periodic payment of the loan.
3. Starting in the first month, write the amount of the loan in unpaid
balance column.
4. Multiply the unpaid balance by the interest rate of the loan to get the
interest paid.
5. Get the difference of the periodic payment and the Interest paid to
get the principal prepaid
6. Subtract the principal prepaid from Previous unpaid balance to get
the 2nd month unpaid balance and so on.
7. By the end of the set loan term, the principal amount should be zero.
(Note: a discrepancy of .01 -.05 centavo will not affect the loan
payment)
I. Choose the letter of the correct answer. Write it on a separate sheet

of paper.
1. Darren obtain a loan of P50,000 from PAGIBIG Fund to be

amortized by equal payment at the end of each 6 months for 3 years
at 8% interest compounded semi-annually. Find the periodic
payment.
A. P9,538.10 B. P8,676.15 C. P7,728.20 D. P6,936.25
2. Mr. Salvador is required to pay 12 quarterly installments of P12,500

each for a loan of 8% compounded quarterly. How much is his loan?
A. P213,219.98 C. P132,191.77
B. P175,204.65 D. P121,178.54
For questions #3 – 5, read the passage below:

Mr. Cruz bought a car for P900,000. After deducting the down
payment, her total loan amount is P825,000. She amortized the
loan by paying P18,500 monthly for 5 years with 5% interest
compounded monthly.
3. How much is the amount of the car after 5 years?
A. P1,008,500 C. P1,058,792
B. P1,028,654 D. P1,048,827
4. How much is the total interest?
A. P108,500 B. P128,654 C. P158,792 D. P148,827
5. How much was the down payment made by Mr. Cruz?
A. P55,000 B. 65,000 C. P75,000 D. P85,000

II. Read and analyze the given problem. Write your answer on a
Suppose your father applied a housing loan of P750,000 in

PAGIBIG Fund, which he would like to pay on equal installment at
the end of each month for 15 years. Find his periodic payment
installment if he is required to pay 6.75% interest compounded
monthly. Construct an amortization schedule for his first 18 periodic
payments.
POSTTEST
Choose the letter of the best answer. Write the chosen letter on
a separate sheet of paper.
1. It is a debt investment in which an investor loans money to an

entity which borrows the funds for a defined period of time at a
variable of fixed interest rate.
A. Amortization C. Mortgage
B. Bond D. Stock
2. A loan of 6 semi-annual payments of P4,500 are to be made to

pay for a loan at 9% compounded semi-annually. Find the value
of the loan.
A. P23,210.44 C. P25,390.32
B. P24,300.50 D. P26,480.10
3. Mr. Roel made a loan of P40,000 is to be amortized by equal

payments at the end of each quarter for 18 months. If interest is
10% compounded quarterly, find the periodic payment.
A. P7,262 C. P8,375
B. P9,463 D.P10,587
SOLVING PROBLEMS ON MORTGAGES
OVERVIEW
help you understand solving problems on mortgages. The scope of this
module permits it to be used in many different learning situations. The
lessons are arranged to follow the standard sequence of the course. But the
order in which you read them can be changed to correspond with the
textbook you are now using. The module focuses on achieving this learning
competency:
Solve problems on business and consumer loans (amortization and

mortgage)

• solve problems involving mortgages.
PRETEST
1. If a condominium unit is purchased for P6,200,000 and the bank

requires 30% down payment, how much is the mortgaged amount?
A. P2,430,400 C. 4,340,000
B. P3,440,000 D. 5,440,300
2. In relation to QUESTION 1, if the mortgaged amount will be paid for

10 years at 12% compounded quarterly, how much is the quarterly
payment?
A. P197,858,64 C. P177,658.84
B. P187,758.74 D. P166,558.94
3. In relation to QUESTION 2, how much is the total amount of the

loan at the end of 10 years?
A. P7,914,345.60 C. P7,106,353.60
B. P7,510,349.60 D. P6,662,357.60
LOOKING BACK
In our previous lesson we learned on how to construct an amortization
1−(1+𝑗)−𝑛 𝑃
schedule using the Ordinary annuity 𝑃 = 𝑅 or 𝑅 = 1−(1+𝑗)−𝑛
to solve
𝑗
𝑗
for the periodic payment. We will also use this formula to solve for
mortgages.
Let us familiarize with the following terms.
Mortgage - A loan, secured by a collateral, that the borrower is obliged to

pay at specified terms.
Example of this is purchasing a house and lot or a real estate property
There are different types of mortgages

1. Fixed-rate mortgage – the rate of interest is the same throughout the
term of mortgage. This is the most common type of mortgage.
2. Adjustable-rate mortgage – the interest rate goes with the situation
of the economy and the lending rates of the prime banks.
3. Graduated-payment mortgage – the buyer pays a smaller payment
at the beginning, then makes bigger payments towards the end of the
payment period.
Chattle Mortgage - is a mortgage on a movable property.

Collateral – assts used to secure the loan. It may be real-estate or other
investments.
Mortgagor – is the borrower in a mortgage.
Mortgagee – is the lender in a mortgage.
A mortgage is a business loan or a consumer loan that is secured with a

collateral. For example, if a house and lot is purchased or a real estate
property, the purchased property will be used as a mortgaged property or
collateral. A down payment is usually required before the mortgage is
granted to the buyer. However, some government lending institutions, such
as Pag-ibig Fund, GSIS and some private offices do not require down
payments for their members/employees.
Let us solve problems involving mortgages for further understanding.
Examples: Solve problems involving mortgage.

1. Mr. Alfonso borrowed P1,500,000 to buy a house and lot. Find the total
amount of interest he will pay if his monthly payment for a 25-year
mortgage is P8,400.
Given:
P = P1,500,000 Monthly Payment = P8,400
Mortgage Year = 25 Find: Total interest.
Solution:
Total amount to be paid = Monthly mortgage x No. months per year x years
= 8,400 x 12 months x 25 years
Total Amount to be paid = P2,520,000
Total Interest Paid = Amount to be paid – Amount borrowed
= 2,520,000 – 1,500,000
Total Interest Paid = P1,020,000
The above problem is example of Chattel Mortgage.
2. Mr. Castro bought a car that cost P1,318,000. The downpayment is

15% of the cost. After paying the downpayment, the remaining amount
is payable for 5 years with 5% interest compounded monthly. How
much is the monthly payment? How much is the amount of car after 5
years? How much is the total interest?
Given:
P = P1,318,000 Down payment = 15% of present value
n = 5 years i = 5% m = 12 R=? F=? j=?
Solution:
Compute the Down payment = 15% of Present Value
= 15% x 1,318,000
Down payment = P197,700
Present Value = Principal Amount – Downpayment
= 1,318,000 – 197,000
Present Value = P1,121,000
n = mt = (12)(5) = 60 j = i÷m = 5% ÷ 12 = 0.0041667
𝑷
Solve the monthly payment using 𝑹 = 𝟏−(𝟏+𝒋)−𝒏
[ ]
𝒋
1,121,000
𝑅=
1 − (1 + 0. 0041667)−60
[ ]
0.0041667
1,121,000
𝑅=
1 − (1. 0041667)−60
[ ]
0.0041667
1,121,000
𝑅=
1 − 0.77920384
[ 0.0041667 ]
1,121,000
𝑅=
0.22079616
[ 0.0041667 ]
1,121,000
𝑅=
52.9906545
𝑹 = 𝑷𝟐𝟏, 𝟏𝟓𝟒. 𝟔𝟕
Hence, the monthly payment for 5 years is P21,154.67
Total Monthly Payment at the end of 5 years = R x n

= 21,154.67 x 60
Total Monthly Payment at the end of 5 years= P1,269,280.20
F = Total Monthly Payment at the end of 5 years + Downpayment

F = 1,269,280.20 + 197,000
Total Amount of car at the end of 5 years = P1,466,280.20
Total interest Paid = Amount of car at the end of 5 years – Principal Amount
= 1,466,280.20 – 1,318,000
Total Interest Paid = P148,280.20

ACTIVITY
A. Fill in the blanks.

1. A__________ is a mortgage with stable interest rate through out of
the term.
2. A __________ are secured loans attached to a personal movable
property.
3. A __________ is an entity that lends money to a borrower for the
purpose of purchasing real estate.
4. A __________ is a loan, secured by a collateral that the borrower is
obliged to pay at specified terms.
5. In a mortgage transaction, the borrower is known as the _______.
B. Solve the following problems. Write your solution in separate sheet of

paper
1. How much is the mortgaged amount of a house and lot that has a
cash value of P750,000 if the bank offers a minimum amount of 20%
down payment?
2. Supposed your sister obtained a condominium unit worth
P2,600,000 and the bank requires a 35% down payment, how much
is the mortgaged amount.
3. Mr. Morales borrowed an amount of P120,000 is to be amortized by
paying a monthly in 1.5 years. If money is worth 10% compounded
monthly, how much is the monthly installment?
REMEMBER
Mortgages, more than any other loans, come a lot of variables starting with
what must be paid and when. The mortgagor should work with a
mortgagee expert to get the best deal on what investment the
mortgagor wants in life.
To solve problems involving mortgages:

1. Read and analyze the problem.
2. Determine the given.
3. Use an appropriate formula such as Ordinary Annuity formula.
4. Substitute the known values in the formula.
5. Solve.

paper.
1. Supposed that a house and lot is purchased for P2,600,000 and the
bank requires 25% down payment. How much is the loan mortgaged
amount?
A. P2,000,000 C. P1,900,000
B. P1,950,000 D. P1,850,000
2. Dela Cruz family obtained a P1,000,000 mortgage. If the monthly

payment is P38,000 for four years, how much is the total interest paid?
A. P934,000 C. P714,000
B. P824,000 D. P604,000
3. How much is the monthly payment of Mr. and Mrs. Dela Vega on a
P1,800,000 mortgage at 9% interest for 20 years?
A. P16,195.07 B. P17,290.10 C. P18,395.13 D. P19,400.15
4. Mr. Ignacio obtained a 15-year mortgage for P5,400,000. If his monthly

payment is P42,500, how much is the total amount paid?
A. P6,750,000 C. P8,550,000
B. P7,650,000 D. P9,450,000
5. In problem No. 4, how much is the total interest paid?
A. P1,350,000 C. P3,150,000
B. P 2,250,000 D. P4,040,000
POSTTEST
1. A man obtained a mortgage of P680,000. If his monthly payment is

P4,795, find the total amount of interest he paid for a 20-year period.
A. P1,150,800 C. P695,800
B. P 925,800 D. P470,800
2. In problem no. 1, how much is the total monthly payment?
A. P1,150,800 C. P695,800
B. P 925,800 D. P470,800
3. Mr. Co acquired a mortgage for P980,000 at 12% compounded

monthly for 10 years. Find Mr. Co’s monthly payment.
A. P15,070.20 C. P13,050.10
B. P14,060,15 D. P12,040.05
ILLUSTRATE, SYMBOLIZE, AND DISTINGUISH BETWEEN

SIMPLE AND COMPOUND PROPOSITIONS
OVERVIEW
help you understand how to illustrate and distinguish simple and compound
logic. The scope of this module permits it to be used in many different
level of students. The lessons are arranged to follow the standard sequence
of the course. But the order in which you read them can be changed to
Illustrates and symbolizes propositions and distinguishes between

simple and compound propositions.

• identify propositions;
• Represent proposition using symbol; and
• distinguish between simple and compound propositions.
PRETEST
1. Which of the following is a proposition?
A. Hey!
B. Clean up your room.
C. Do you want to go to the movies?
D. The Earth is farther from the sun than Venus.
2. The following are propositions EXCEPT:
A. All cows are brown.

B. There is life in Mars.
x
C. ≤ 4x is a rational expression.
x+2
D. Multiply 7 from 9.
3. The following are compound proposition EXCEPT:
A. I am reading a book, or I am writing a novel.

B. Juan is good in English and Mathematics
C. If 3x + 5 = 11, then x = 2
D. The product of 3x + 2 = 6
LOOKING BACK
As we all know communication is an important activity of humanity. And
correct reasoning plays a vital role not just in our decision making but in the
information of our beliefs and opinions as well. Logical reasoning or correct
inference is the study of the principles and methods used to distinguish valid
arguments from those that are not valid.
What is logic?
Logic has been studied since the classical Greek period (600-300BC). The
Greeks, most notably Thales, were the first to formally analyze the reasoning
process. Aristotle (384-322BC), the “father of logic”, and many other
Greeks searched for universal truths that were irrefutable. A second great
period for logic came with the use of symbols to simplify complicated logical
arguments. Gottfried Leibniz (1646-1716) began this work at age 14 but
failed to provide a workable foundation for symbolic logic. George Boole
(1815-1864) is considered the “father of symbolic logic”. He developed
logic as an abstract mathematical system consisting of defined terms
(propositions), operations (conjunction, disjunction, and negation), and rules
for using the operations. Boole’s basic idea was that if simple propositions
could be represented by precise symbols, the relation between the
propositions could be read as precisely as an algebraic equation. Boole
developed an “algebra of logic” in which certain types of reasoning were
reduced to manipulations of symbols. LOGIC is the systematic study of
valid rules of inference such as the relations that lead to the acceptance of
one proposition (the conclusion), in other word logic is the analysis and
appraisal of arguments.
Logic also applied in computer science such as in the design of computing
machines, the specification of systems, artificial intelligence, computer
programing and program languages.
In logic we study sentences and relationships between certain kind of

sentences. We restrict our study of logic to declarative sentences that are
unambiguous statements or propositions.
PROPOSITIONS is a sentence that could be either true (T) or false (F) but
NOT both. Specifically, it is a declarative sentence, and no other forms of
sentences. It declares, informs, or states something and is always ends with
a simple period. It is derived from a statement that contains a subject and
predicate. In symbolic logic we use variables or letters such as p, q, r, s, and
t to symbolize propositions.
Examples:
1. Open your book to page 100. - - - - - - - - - -an imperative
2. Did you enjoy the show? - - - - - - - - - - - - - an interrogative
3. What an exciting game! - - - - - - - - - - - - - -an exclamatory
4. The sun rises in the East - - - - - - - - - - - - -true, declarative
5. Four plus three is ten - - - - - - - - - - - - - - - -false, declarative
6. That was a good movie - - - - - - - - - - - - - - an ambiguous
7. Manila is the capital of the Philippines. - - - true, declarative
8. The square root of two is less than two - - - true, declarative
9. Negative 100 is an integer - - - - - - - - - - - -true, declarative
√2
10. 𝑓(𝑥) = is rational function - - - - - - - - - false, declarative
𝑥+1
We can use letters or variables to represent propositions. Thus, sentences

4, 5, 7 to 10 can be represented as p, q, r, s, t and u respectively. In symbolic
logic as follows:
4. p: The sun rises in the East.

5. q: Four plus three is ten.
7. r: Manila is the capital of the Philippines
8. s: The square root of two is less than two
9. t: Negative 100 is an integer
√2
10. u: 𝑓(𝑥) = is rational function
𝑥+1
Let us now compare sentences 1, 2, 3 and 6 with sentences 4, 5, 7 to 10.

Sentences 1, 2, 3, and 6 are not declarative sentences. Hence sentences 1,
2, 3 and 6 are not propositions.
Sentences 4, 5, 7 to 10 are example of simple proposition. A proposition is

simple if it cannot be broken down any further into other component
propositions.
Let us determine the truth value for 4, 5, and 7 to 10 propositions.
p: The sun rises in the East. (True)

q: Four plus three is ten. (False)
r: Manila is the capital of the Philippines. (True)

s: The square root of two is less than two (True)
t: Negative 100 is an integer (True)
𝑥 2 + 𝑥√2 −3𝑥−5
u: 𝑓(𝑥) = is rational function (False, since the
𝑥+1
numerator is not polynomial even there are many terms)
As discussed, a proposition is a declarative sentence that could be classified

as either true (T) or false (F) and can be simple or compound proposition.
A compound proposition is the result of combining simple propositions or

producing a new proposition from existing propositions. We combine
propositions by using logical connectors or operators Some logical
connectors involving propositions p and/or q may be expressed as follows:
not p, p and q, p or q, if p then q.
Examples
pI: I am reading a book, or I am writing a novel.
p2: It is not the case that √2 is a rational number.
P3: Either logic is fun and interesting, or boring.
P4: If you study hard, then you will get good grades.
We can determine the simple propositions that make the propositions p1, p2,
p3, and p4
Propositions Simple components
p1 r: I am reading a book.
f: I am writing a novel.
p2 i: √2 is a rational number
p3 b: Logic is fun.
g: Logic is interesting.
a: Logic is boring.
p4 h: You study hard.
d: You get good grades.
Then, the compound propositions may be expressed as:

pI: r or f
p2: not i
p3: b and g, or a
p4: If h then d
ACTIVITY
A. Determine whether each of the following statements is a

proposition or not. If a proposition give its truth value.
1. Students are excited to learn logic.
2. An angle that measure exactly 900 is a right angle.
3. President Ramos was the first President of the Philippines.
4. Noynoy Aquino was a great President.
5. In 2023, February will have 29 days,

6. How did you get the correct answer?
7. Ferdinand Magellan did not arrive the Philippines in 1521.
8. If an integer is odd, then its square is also odd.
9. 2 is even and prime.
10. You can have ice cream or cake for snack.
11. If x = 2, then x + 2 = 2x
12. Is 3 a square of some number?
13. If x is an even integer, then x + 1 is an odd number.
14. X is greater than 7
15. Are you okay?
B. From its proposition in A, identify whether it is simple or

compound propositions. If it is a compound, identify its primitive
component.
REMEMBER
Proposition is a sentence that could be either true (T) or false (F) but NOT
both. Specifically, it is a declarative sentence, and no other forms of
sentences. It declares, informs, or states something and is always ends
with a simple period. It is derived from a statement that contains a subject
and predicate. In symbolic logic we use variables or letters such as p, q, r,
s, and t to symbolize propositions.
A proposition is simple if it cannot be broken down any further into other
component propositions.
A compound proposition is the result of combining simple propositions
or producing a new proposition from existing propositions. We combine
not p,
p and q,
p or q,
if p then q.

paper.
A. Bicol is an island in the Philippines.

B. What is the range of the function?
C. That was a hard examination!
D. Smile.
2. The truth value in QUESTION 1 is ….
A. True C. Both true and false

B. False D. Neither true nor false
3. Which of the following is NOT a proposition?
A. Twice a number equals the sum of two and a number.

B. The constant π is an irrational number.
C. Find a real number whose absolute value is 5.
D. Four is a natural number.
4. Which of the following is a compound proposition?
A. 3.75 is an integer.
B. Smile at your classmates.
C. Life is too short so enjoy it.
D. The ground is not wet.
5. The truth value for this statement “x + 2 = 2x when x = −2” is…
A. True
B. False
C. Both true and false
D. Neither true nor false
POSTTEST
1. It is a declarative sentence that is either true or false.
A. Conditional C. Inverse
B. Proposition D. Logic
2. The following are proposition EXCEPT:
A. Gen. Math is easy. C. It is Gen. Math.

B. Gen. Math is difficult. D. Study Gen. Math
A. Mabuhay! C. A priest is a male.

B. How are you? D. Smile
PERFORMS THE DIFFERENT TYPES OF OPERATIONS ON

PROPOSITIONS
OVERVIEW
help you to perform the different types of operations on propositions. The
scope of this module permits it to be used in many different learning
students. The lessons are arranged to follow the standard sequence of the
course. But the order in which you read them can be changed to correspond
with the textbook you are now using. The module focuses on achieving this
learning competency:
Performs the different types of operations on propositions.

• perform the operations on propositions, and
• represent proposition using symbol.
PRETEST
1. Given: “If I have a Lhasa Apso, then I have a dog.” What is the
biconditional of the following conditional statement?
A. If I do not have a Lhasa Apso, then I do not have a dog.

B. If I do not have a dog, then I do not have a Lhasa Apso.
C. I have a dog if and only if I have a Lhasa Apso.
D. I have a Lhasa Apso if and only if I have a dog.
2. p ↔ q means what?
A. Conditional C. Conjunction
B. Bi-conditional D. Disjunction
3. What is the conclusion of the following statement: The angles are

supplementary, if they add up to 180°?”
A. if they add up to 180° C. they are supplementary

B. they add up to 180° D. there is no conclusion
LOOKING BACK
A compound proposition is the result of combining simple propositions or
producing a new proposition from existing propositions. We combine
not p, p and q, p or q, if p then q.
Logical connectors involving propositions 𝑝 and/or 𝑞 are expressed as

follows:
The simple logical operator is the negation operator denoted by

¬𝒑 𝒐𝒓 ~𝒑 (read as not p).
Examples: What is the negation of the following statement?

1. p: √2 is a rational number. ------- ~𝒑: √2 is not a rational number.
or ~𝒑: √2 is an irrational number.
2. q: 6 is not an odd number. ------- ~𝒒: 6 is an odd number.
~𝒒: 6 is an even number.
The conjunction operator is the binary operator which when applied to two
propositions p and q denoted by 𝒑 ∧ 𝒒 (read as p and q). The conjunction
p∧q of p and q is the proposition that is true when both p and q are true and
false otherwise.
The disjunction operator is the binary operator which, when applied to two
propositions p and q, yields the proposition “p or q”, denoted 𝒑 ∨ 𝒒. The
disjunction p ∨ q of p and q is the proposition that is true when either p is
true, q is true, or both are true, and is false otherwise. Thus, the “or” intended
here is the inclusive or. In fact, the symbol ∨ is the abbreviation of the Latin
word vel for the inclusive “or”.
Implication/Conditional operator is the proposition 𝒑 → 𝒒 that is often read

“if p then q. 𝒑 → 𝒒 is also read as “p implies q”. p is called the hypothesis
and q is called the conclusion.
The biconditional of propositions p and q is denoted by p ↔ q: (p if and only
if q). The proposition may also be written as “p iff q”. The propositions p and
q are the components of the biconditional.
Some Special Negation:

The De Morgan’s Laws
(𝑝 ∧ 𝑞 ) = ~𝑝 ∨ ~𝑞
~(𝑝 ∨ 𝑞) = ~𝑝 ∧ ~𝑞
Examples:
Perform the following operations. Write the answer in statement form.
Let p and q be the following propositions:

𝑝: Andrew is good in English.
𝑞: Andrew is good in Mathematics.
1. ~𝑝
2. 𝑝 ˅ 𝑞
3. 𝑞 ∧ 𝑝
4. ~𝑝 ˄ 𝑞
5. ~(~𝑝 → 𝑞)
6. 𝑝 ↔ 𝑞
7. ~(𝑝 ∧ 𝑞)
8. ~(𝑝 ∨ 𝑞)
Solution:
1. Andrew is not good in English.
2. Andrew is good in English or Andrew is good in Mathematics.
3. Andrew is good in Mathematics and Andrew is good in English.
4. Andrew is not good in English and Andrew is good in Mathematics.
5. If is not the case that Andrew is not good in English, then Andrew is
good in Mathematics.
(The presence of double negation leads to a similar statement which
is “If Andrew is good in English, then he is good in Mathematics.”
6. Andrew is good in Mathematics, if and only if, he is good in English.
7. Andrew is not good in English or, he is not good in Mathematics.
8. Andrew is not good in English and he is not good in Mathematics.
ACTIVITY
A. Express the following propositions in symbols, where p, q, and r are

defined as follows:
p: The sun is shining.
q: It is raining.
r: The ground is wet.
1. If it is raining, then the sun is not shining.
2. It is raining and the ground is wet.
3. The ground is wet if and only if it is raining and the sun is shining.
4. The sun is shining, or it is raining.
5. The ground is not wet.
B. Translate the following symbolic form in statement where:

p: President Duterte is a good President.
q: Government officials are corrupt.
r: People are happy.
1. 𝑝 → ~𝑞
2. 𝑝 ˅ ~𝑞 → 𝑟
3. 𝑝 → ~𝑞
4. ~𝑝 ∧ 𝑞 → ~𝑟
5. 𝑝 ↔ ~𝑞 ∧ 𝑟
REMEMBER
Logical Connectors
Logical Connectives Symbolic Counterpart
or Operators
Negation not p ¬𝑝 𝑜𝑟 ~𝑝
Conjunction p and q 𝑝∧𝑞
Disjunction p or q 𝑝 ∨𝑞
Implication/Conditional if p then q 𝑝→𝑞
Biconditional P if and only if 𝑝↔𝑞

paper.
1. Consider the proposition, “If the sum of two angles is 90°, then they
are complementary.” Which of these is the hypothesis?
A. If
B. Then
C. They are complementary
D. The sum of two angles is 90°
For number 2 – 3: Write the proposition in symbol with the first proposition
as p and the second proposition as q.
2. The sum of the interior angles of a triangle is 180° and the polygon is
a triangle. In symbol:
A. p∧q C. p→q
B. p ∨q D. p↔q
3. If the sum of the interior angles of a triangle is not 180°, then the
polygon is not triangle. In symbol:
A. p∧q C. p→q
B. p ∨q D. p↔q
4. Negate: Darren is good in volleyball.
A. Darren is good.
B. Darren is lousy in volleyball.
C. Darren is not good in volleyball.
D. Volleyball is good for Darren.
5. What is the conclusion of the statement, “If the polygon is a triangle

then, the sum of the interior angles of a triangle is 180°”?
A. If
B. Then
C. The triangle is a polygon.
D. The sum of the interior angle of a triangle is 180°.
Suppose we look at the Instagram world of four girls: Janella, Julia, Kathryn
and Liza. We summarize their Instagram dynamics-who follows who-in a
table such as the following.
Janella Julia Kathryn Liza

Janella
Julia
Kathryn
Liza
A check in a cell of table means that the girl named at the beginning of the
row follows on Instagram the girl at the head of the column.
Instruction: Make the following propositions true by checking the

appropriate cell.
Liza follows Kathryn but does not follow Janella.

Either Julia follows Kathryn, or Julia follows Liza.
While Janella follows everyone that Julia follows, Janella does not
follow Liza.
Kathryn follows everyone who follows her.
Nobody follows herself.
There are several ways that this can be done.

POSTTEST
1. In propositional logic, this implies that p is the hypothesis and q is the

conclusion.
A. Conditional C. Conjunction
B. Proposition D. Disjunction
2. If 𝑝 → 𝑞, then which of the following is logically equivalent to it?
A. 𝒒→𝒑 C. ∼𝒒→∼𝒑
B. ∼𝒑→∼𝒒 D. ∼𝒑→𝒒
3. Aside from 𝑝 ↔ 𝑞, which of the following illustrates a biconditional?
A. (𝒑 → 𝒒) ∧ (~𝒑 → ~𝒒) C. (𝒑 → 𝒒) ∨ (~𝒑 → ~𝒒)

B. (𝒑 → 𝒒) ∧ (𝒒 → 𝒑) D. (𝒑 → 𝒒) ∨ (𝒒 → 𝒑)
CONSTRUCTING TRUTH TABLES
OVERVIEW
help you to determine the truth values of propositions and illustrates the
different forms of conditional propositions. The scope of this module
permits it to be used in many different learning situations. The language
used recognizes the diverse vocabulary level of students. The lessons are
arranged to follow the standard sequence of the course. But the order in
which you read them can be changed to correspond with the textbook you
are now using. The module focuses on achieving this learning
competency:
Determines the truth values of propositions and illustrates the

different forms of conditional propositions.

• construct a truth value;
• determine the truth values of propositions; and
• illustrates the different forms of conditional propositions.
PRETEST
1. When can a biconditional statement be true?
A. When the inverse and the converse are both true

B. When the converse is true.
C. When the original statement (conditional statement) and the
contrapositive are both true
D. When the original statement (conditional statement) and the
converse are both true.
2. When taking the inverse, we _____ the hypothesis and

conclusion.
A. negate C. switch and negate

B. switch D. negate and switch
3. What is the contrapositive of 𝑝 → 𝑞?
A. q →p C. ~p → ~q
B. ~q → ~p D. p → ~q
LOOKING BACK
In the previous lesson we learned in the preceding discussion that a
proposition is a statement that is either true or false. We also identified the
statements as compound proposition and defined the corresponding
logical operators below:
Logical Connectives Logical Connectors Logical Operators

Negation not p ¬𝑝 𝑜𝑟 ~𝑝
Conjunction p and q 𝑝∧𝑞
Disjunction p or q 𝑝 ∨𝑞
Implication/Conditional if p then q 𝑝→𝑞
Biconditional P if and only if 𝑝↔𝑞
where p and q are propositions.
Let say that p and q are two simple propositions, and that the truth values
of a simple proposition which is either true or false are T and F (or 1 and
0), respectively. Using a truth table, the truth value of p and q are shown
below:
The 2-column truth table
p Negation ~p p ~p
tells us that if p is true, ~𝑝
T F T F is not true, or false. If p is
F T F T false, ~𝑝 is not false, or true.
The truth table for p ∧ q (conjunction of p and q) is shown below:

p q 𝒑 ∧ 𝒒 The truth table shown on the side tells us that if
T T T there are two simple propositions, p and q, there
T F F four possible arrangements or assignments of
F T F their truth values. The 3rd column means that the
F F F conjunction 𝒑 ∧ 𝒒 is true in only one instance, and
that is when p and q are both true.
The truth table for 𝒑 ∨ 𝒒 (Disjunction of p or q) as shown below:

p q 𝒑 ∨ 𝒒 In disjunction, there are also four arrangements
T T T for their truth values. Observe that 𝒑 ∨ 𝒒 is false
T F T only if both p and q are false. This means that the
F T T disjunction is true if at least one of the simple
F F F propositions is true.
The truth table for 𝒑 → 𝒒 (conditional of p implies q) is shown below:

p q 𝒑→𝒒 The 3rd column for this table shows that the
T T T conditional 𝒑 → 𝒒 is false only if p is true and q is
T F F false. Otherwise, such conditional statement is
F T T true.
F F T
The truth table for 𝒑 ↔ 𝒒 (biconditional of p iff q) is shown below

p q 𝒑↔𝒒 The table shows that the biconditional statement
T T T 𝒑 ↔ 𝒒 is true if both p and q are true, and that it is
T F F false, if both p and q are false.
F T F
F F T
Order of Operations are essential in constructing a truth table:

- Grouping: (), [], {}
- ~
- ∧, left to right
- ∨, left to right
- →
- ↔
Examples:
1. Determine the truth value of each of the following if p is true, q is

false and r is false. (Follow the order of operations.)
a. ~𝑝 ∨ 𝑞 = F ∨ F = F (False)
b. (𝑝 ∨ 𝑞 ) ∧ (~𝑞 →∼ 𝑟) = (T ∨ F) ∧ (T → T) = T ∧ T = T (True)
c. ~(p ∧ q) ↔ ( ~p ∨ r) = (~p ∨ ~q) ↔ (~p ∨ r)
= (F ∨ T) ↔ (F ∨ F)
=T↔F
~(p ∧ q) ↔ (~p ∨ r) = F (False)
2. Construct the truth table for (𝒑 ∨∼ 𝒒) → (∼ 𝒑 ∧ 𝒒)
p q ∼𝒑 ∼𝒒 𝒑 ∨∼ 𝒒 ∼𝒑 ∧𝒒 (𝒑 ∨∼ 𝒒) → (∼ 𝒑 ∧ 𝒒)
T T F F T F F
T F F T T F F
F T T F F T T
F F T T T F F
Forms of Conditional Propositions

• Conditional proposition if p then q in symbol (𝒑 → 𝒒) takes
several forms. These forms, known as inverse, converse, and
contrapositive can be expressed as follows:
Let p and q are propositions. Given the implication 𝒑 → 𝒒
Inverse of 𝒑 → 𝒒 ∼ 𝒑 →∼ 𝒒
Converse of 𝒑 → 𝒒 𝒒→𝒑
Contrapositive 𝒑 → 𝒒 ∼ 𝒒 →∼ 𝒑
Examples: Say p and q are propositions. Given the implication 𝒑 →

𝒒
1. Give the inverse, converse and contrapositive of each of the

following:
a. Conditional 𝒑 → 𝒒
If this book is interesting, then I am staying at home.
Answer:
b. Inverse ∼ 𝒑 →∼ 𝒒:
If this book is not interesting, then I am not staying at home.
c. Converse 𝒒 → 𝒑:
If I am staying at home, then this book is interesting.
d. Contrapositive∼ 𝒒 →∼ 𝒑:
If I am not staying at home, then this book is not interesting.
2. Construct the truth table for the above example.
p q ∼𝒑 ∼𝒒 ∼ 𝒑 →∼ 𝒒 𝒒→𝒑 ∼ 𝒒 →∼ 𝒑
T T F F T T T
T F F T T T F
F T T F F F T
F F T T T T T
ACTIVITY
A. Express the following propositions in symbols, where p, q and r are
defined as follows:
p: The fish is cooked.

q: Dinner is ready.
r: I am hungry.
1. The dinner is ready, if and only if I am hungry.

2. The fish is cooked or if dinner is ready, then I am hungry.
3. It may or may not the case that I am hungry.
4. If the fish is not cooked, then the dinner is not ready, and I am
hungry.
5. I am not hungry, or dinner is ready if and only if the fish is cooked.
B. Express the following proposition in words, using the given

proposition in Activity A.
1. 𝑝 ∧ (~𝑝)
2. 𝑝 ∧ (~𝑞)
3. ~(𝑞 ∧ 𝑟)
4. ((~𝑝) ˅ 𝑞 ) ∧ 𝑟
5. ~𝑝 ∧ 𝑞 → ~𝑟
6. ~𝑝 ↔ (~𝑞) ∧ 𝑟
7. ∼ 𝑝 → ~𝑞
8. (𝑟 → ~𝑞) ˅ 𝑝
9. ∼ 𝑞 → (∼ 𝑝 ∧ 𝑟)
10. ∼ (𝑞 → 𝑟)
REMEMBER
Rules of Logic
• Negation – change the truth value.
• Conjunction – True only if both are true.
• Disjunction – False only if both are false or true if at least one is
true.
• Conditional – False only when p is true and q is false.
• Bi-conditional – True if the truth values are the same.
Summary of truth tables of logical connectives.
𝑝 𝑞 ∼𝑝 ~𝑞 𝑝 ∧ 𝑞 𝑝 ∨ 𝑞 𝑝 → 𝑞 𝑝 ↔ 𝑞 𝑞 → 𝑝 ∼ 𝑝 →∼ 𝑞 ∼ 𝑞 →∼ 𝑝
T T F F T T T T T T T
T F F T F T F F T T F
F T T F F T T F F F T
F F T T F F T T T T T

of paper.
Consider the truth table below to answer items numbers 1 – 4.
p q 𝒑 ∧𝒒 𝒑 ∨𝒒 ∼ (𝑝 ∧ 𝑞) (𝒑 ∧ 𝒒) ∨ ∼ (𝒑 ∨ 𝒒)
T T
T F
F T
F F
1. If the logical values of the third column is arranged in order, what

is the correct order?
A. 𝑻𝑭𝑻𝑭 C. 𝑻𝑭𝑭𝑭
B. 𝑭𝑻𝑭𝑻 D. 𝑻𝑻𝑻𝑭
2. If the logical values of the fourth column is arranged in order, what

A. 𝑻𝑭𝑻𝑭 C. 𝑻𝑭𝑭𝑭
B. 𝑭𝑻𝑭𝑻 D. TTTT
3. If the logical values of the fifth column is arranged in order, what is

the correct order?
A. 𝑭𝑭𝑻𝑻 C. 𝑻𝑭𝑭𝑭
B. 𝑭𝑭𝑭𝑻 D. 𝑻𝑻𝑻𝑭
4. If the logical values of the sixth column is arranged in order, what

A. 𝑻𝑭𝑻𝑭 C. 𝑻𝑭𝑭𝑻
B. 𝑭𝑻𝑭𝑻 D. 𝑭𝑻𝑻𝑭
5. Aside from 𝑝 ↔ 𝑞, which of the following illustrates a biconditional?
A. (𝒑 → 𝒒) ∧ (~𝒑 → ~𝒒) C. (𝒑 → 𝒒) ∨ (~𝒑 → ~𝒒)

B. (𝒑 → 𝒒) ∧ (𝒒 → 𝒑) D. (𝒑 → 𝒒) ∨ (𝒒 → 𝒑)
POSTTEST
Multiple Choice. Choose the letter of the best answer. Write the
chosen letter on a separate sheet of paper.
1. In propositional logic, this implies that p is the hypothesis and q is

the A truth table involving 𝑛 propositions has how many rows?
A. 𝟐𝒏 B. 𝒏𝟐 C. 𝟐𝒏 D. 𝒏𝒏
2. If 𝑝 → 𝑞, then which of the following is logically equivalent to it?
A. 𝒒→𝒑 C. ∼𝒒→∼𝒑
B. ∼𝒑→∼𝒒 D. ∼𝒑→q
3. What does ∼p→∼q mean?
A. Conditional Statement C. Inverse Statement

B. Converse Statement D. Contrapositive Statement

2ND Quarter Modules Gen Math

Uploaded by

Copyright:

Available Formats

2ND Quarter Modules Gen Math

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

2ND Quarter Modules Gen Math

Uploaded by

Copyright:

Available Formats

SDO MALABON CITY 11

G11 SLEM # 1 – WEEK 1 – 2nd QUARTER

ILLUSTRATING SIMPLE AND COMPOUND INTEREST

Illustrate simple and compound interest

After going through this module, you are expected to:

1. What do you call to a person or institution who owes the money

2. What is the amount of time in years the money is borrowed or

3. What do we call the amount of money or goods which are usually

A. Compound Interest C. Interest

This is a function in the form of 𝑓 (𝑥) = 𝑚𝑥 + 𝑏, where 𝑚 is the slope

➢ Exponential Function – is a one-to-one function whose exponent

This function is in the form of 𝑓 (𝑥) = 𝑎 𝑥 , where 𝑎 > 0, 𝑎 ≠ 1 and

BRIEF INTRODUCTION OF THE LESSON

❖ Repayment date or maturity date – date on which the money

Illustration of Simple and Compound Interest

Investment 1 Bank A: Simple interest, with annual rate 𝒓.

Principal Simple Interest (𝑰𝒔 ) Amount after t years

Investment 1 Bank B: Compound interest, with annual rate 𝒓.

Comparing the two investments:

Simple Interest (in pesos): 5,900 – 5,000 = P900

It can be seen from the illustration above that as investment period

Use an interest table (like what was presented above) to illustrate

1. Aling Nette borrowed P108,500 at an annual interest rate of

2. Thess deposited P30,000 in a bank that pays 0.6% interest

CHECKING YOUR UNDERSTANDING

Choose the letter of the correct answer. Write it on a separate sheet

1. If Juana invested money in a bank and after a period, she

A. Principal C. Future Value

2. Due to COVID-19 pandemic, Mark affected his financial status,

3. After 5 working days, Mark received his emergency loan

A. Loan Date C. Term

4. Mark computed the amount of his loan to be paid in 3 years at

A. Principal C. Future Value

5. Which of the following is not mentioned in the given situation

A. Loan Data C. Term

1. Which of the following are used by banks in calculating interest

A. Compound Interest C. Interest

2. Which of the following are NOT considered exponential?

A. Bumbay System C. Cooperative Loan

3. What should be added to the principal in order to get the future

A. Maturity Value C. Present Value

G11 SLEM # 2 – WEEK 1 – 2nd QUARTER

COMPUTE SIMPLE INTEREST

Compute interest, maturity value, future value, and present value in

After going through this module, you are expected to:

1. What should be added to the principal in order to get the future

A. Maturity Value C. Downpayment

2. How much should be paid if P12 000.00 is borrowed from a bank

3. Which of the following is the correct simple interest yield of P

Examples: Change the following percent in decimal.

BRIEF INTRODUCTION OF THE LESSON

Simple interest is usually used for short borrowing. An annual interest is

➢ Principal which is the amount invested or borrowed.

The formula for annual simple interest, 𝐼𝑠 , is given by:

where: 𝐼𝑠 is the simple interest

The maturity value or the future value, 𝐹, is given by:

where: 𝐹 is the maturity or future value

If 𝐼𝑠 = 𝑃𝑟𝑡 will be substituted in the previous formula, 𝐹 = 𝑃 + 𝐼𝑠 , another