Quantitative Reasoning-1

Department Of Mathematics

LCWU
Contents:

• Real Number System

• Set
• Matrices
• Algebra
• Sequences
• Introduction to Factorial, Permutation and Combination
• Trigonometry
• Basic Geometry
• Ratio and Proportion
• Percentage
• Practical life scenarios involving parts and whole.
• Money management
• Average
Real Number System:

Sets:

Human beings share a desire to organize and classify. Ancient astronomers classified stars into groups
called constellations. Modern astronomers continue to classify stars by such characteristics as color,
mass, size, temperature, and distance from Earth. In mathematics it is useful to place numbers with
similar characteristics into sets. The following sets of numbers are used extensively in the study of
algebra.

Natural numbers {1 , 2 ,3 ,4 ,5 , 6 , 7 ,………}

Integers {………, -5 ,-4 ,-3 ,-2 ,-1 ,0 ,1 ,2 ,3 ,4 ,5 , ……….}

Rational Numbers {all terminating and repeating decimals }

Irrational Numbers {all non-terminating and non-repeating decimals}

Real Numbers {all rational and irrational numbers}

Integers:

In primary school, we have learnt about whole numbers such as 0,1,2,3, ….. , etc. The numbers 1,2,3,…
are called positive integers while examples of negative integers is ……,-3,-2,-1,0,1,2,3,…… . 0 is neither a
positive nor a negative number.

Number Line:

Representing Numbers that we have learnt on a number line. The markings are equally spaced, and
arrow indicate the positive direction.
Also from figure,

Exercise 1:

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:
Addition and Subtraction involving Negative Numbers:

Exercise 2:

Question 1:

Question 2:

Question 3:

Question 4:
Question 5:

Multiplication and Division involving negative number:

Multiplication involving negative number:

We will now show how to carry out multiplication involving negative numbers.

Division involving negative numbers:

Square roots and cube roots:

To find the square root of a number, we need to find a number which when multiplied twice by itself
gives the original number. Similarly, to find to cube root of a number we need to find a number which
when multiplied three times by itself give the original number.

Combined operations on numbers:

Exercise 3:

1- 3-

2-

6- Evaluate each of the following:

3 3 3 3
(a) √27 (b) − √64 (c) √8 (d) √216
8.

Rational Numbers and Real Numbers:

Fractions and Mixed Numbers:

In a primary school, we have learnt about fractions (e.g., ¾); improper fraction (e.g. 5/3 and 2/2) and
1
mixed numbers (e.g., 5 4 ). These numbers are positive, but they can be extended to include negative
fractions and negative mixed numbers.

The rules for performing the four basic operations on negative fractions and negative mixed numbers
are the same as those for positive fractions and positive mixed numbers.

To add or subtract fractions with different denominators, i.e., unlike fractions, we must first express the
fractions in the same denominators, i.e., like fractions, using the idea of equivalent fractions. We may
use of the lowest common multiple (LCM) of the denominators.
Rational Numbers:

Figure below illustrates the relationships between the different types of numbers.
Real Numbers:
3
We have learnt about 𝜋 in primary school and about square roots and cube roots, such as √7 and √5 in
3
previous section. The number 𝜋 , √7 and √5 are called irrational numbers because they cannot be
𝑎
expressed in the form 𝑏 , where a and b are integers and 𝑏 ≠ 0.

Real numbers are made up of rational numbers and irrational numbers. Figure illustrates the
relationships among these three types of numbers.

Exercise 4:

Question 1:

Question 2:

Question 3:
Question 4:

Question 5:

Summary:
Prime Numbers and Composite Numbers:

Prime Factorization:

Note:

Index Notation:
Exercise 5:

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:
Question 6:

Highest Common Factor and Lowest Multiple Factor:

Highest Common Factor (HCF):

Worked Example:
Lowest Common Multiple (LCM):
Finding LCM using Prime Factorization:

Exercise 6:

Question 1:
Question 2:

Question 3:

Question 4:

Question 5:

Question 6:
Set

Introduction:
Exercise 1:
Operation on sets:
Exercise 2:

Properties:
Exercise 3:
Matrices
Introduction:
Addition of Matrices:

Transpose of a matrix:

Scalar Multiplication:

Subtraction of Matrices:

Multiplication of two Matrices:

Note:

Determinant of a 2 x 2 matrix:
Singular and non-singular Matrices:

Adjoint of a 2x2 matrices:

Inverse of a 2x2 Matrix:

Exercise 1:
Properties of Matrix Addition, Scalar Multiplication and Matrix Multiplication:

If A, B and C are n x n matrices and c and d are scalars , the following properties are true:
Exercise 2:
Homogeneous and Non-homogeneous Linear Equation in x and y:

Cramer’s Rule:
Exercise 3:
Algebra:

Introduction:

Note:

Algebraic Expressions:

Table shows some examples of linear and non-linear expressions:

Addition and Subtraction of Linear Expressions:

Note that like terms can add and subtract only in to like terms.

Exercise 1:

Question 1:

Question2:
Question 3:

Question 4:

Expansion and simplification of Linear Expressions:

Worked example:
Exercise 2:

Question 1:

Question 2:

Question 3:

Factorization:
Exercise 3:

Question 1:

Question 2:

Linear Equations:
Below table shows some examples of linear and non-linear expressions.

Exercise 4:

Question 1:

Question 2:
Question 3:

Question 4:

Question 5:

Quadratic Equation:
Solution of Quadratic Equation:

Note:

The Quadratic Formula is given by with respect to General Quadratic Equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 , is

Exercise 5:
Exercise 6:
Sequences:

Introduction:
Exercise 1:

Arithmetic Progression (A.P.):

Exercise 2:
Geometric Progression (G.P.):

Exercise 3:
Arithmetic Mean:

Exercise 4:
Introduction to Factorial, Permutation and Combination:

Factorial:

Exercise 1:

Permutation:
Exercise 2:

Combination:
Complementary Combination:

The complementary Combination is given by,

Exercise 3:
Trigonometry:

Introduction:

Units Of Measurements of Angles:

Concept Of an Angle:

Sexagesimal System: (degree, minute and second).

Conversion from 𝑫° , 𝑴′ , 𝑺′′ to a decimal form and vice versa:

Circular System (Radian):

Relation between the length of an arc of a circle and the circular measure of its central angle:

The relation is given by,

Conversion of radian into degree and vice versa:

Exercise 1:
Trigonometric Ratios:

These Trigonometric ratios with angle 𝜃 are defined as follows,

We observe useful relation between these trigonometric ratios as follows,

Fundamental Identities:

For any real number 𝜃 , we have following three fundamental identities:

Exercise2:

Prove the following Identities:

Basic Geometry:

Introduction:

In geometry, we study shapes, relative positions of figures and properties of space. We will be taking
close look at geometry in this section.

Points:

The most basic geometric figure is a point. All other geometric figures are made up of a collection of
points. A point has a position but it has neither size nor shape. We use dot or cross to make a point. We
normally use capital letters to name points, for example, point A and point B, as shown in figure.

Lines:

If we extend the line segment AB in above figure in both directions indefinitely, we get a line.

Sometimes, an arrowhead is drawn at each end of the line in above figure to indicate that the line
continues indefinitely.

Intersecting Lines:

Two lines say AB and PQ that have a common point X and both lines intersect at X is called the point of
intersection.
Planes:

Angles:

Types of angles:
Perpendicular Lines:

We say two lines AB and PQ are perpendicular to each other if they intersecting at right angles. F is
known as a foot of the perpendicular from P to AB.

Complementary Angles:

Supplementary Angles:

Angles on a straight line:

Example:

Note:

Angle at a point:

Note:

Vertically Opposite Angles:

Two straight lines AB and CD intersect at point O. 𝐴𝑂̂𝐶 and 𝐵𝑂̂𝐷 are called vertically opposite angles.
𝐵𝑂̂𝐶 and 𝐴𝑂̂𝐷 are also called vertically opposite angles.

From Figure,
Note:

Exercise 1:

Question 1:

Question 2:

Question 3:

Question 4:
Question5:

Question 6:

Question 7:
Parallel Lines:

When two lines lying on the same plane do not intersect, are called parallel lines. Parallel lines are
represented either single or double arrowheads pointing in the same direction.

Corresponding Angles, Alternate Angles and Interior Angles:

Below Figure shows a line PQ which cuts two parallel lines AB and CD. PQ is known as a transversal.

Exercise 2:

Question no 1:
Question no 2:

Question no 3:

Question no 4:
Question no 5:

Triangles:

Below figure shows the triangle ABC that has three sides AB, BC and AC. The points A, B and C are called
the vertices of the triangle. 𝐵𝐴̂𝐶 , 𝐴𝐵̂𝐶 𝑎𝑛𝑑 𝐴𝐶̂ 𝐵 are known as the interior angle of the triangle ABC.

Classification of triangles:

Triangles can be classified according to the number of equal sides they have,
Angle Sum of the triangle:

Exterior angle of the triangle:

Exercise 3:
Quadrilaterals:

Below figure shows the quadrilateral ABCD. A quadrilateral is a closed plane figure that has four sides,
four vertices and four interier angles. The line segment BD that joins the vertices , B and D, is a diagonal
of the quadrilateral ABCD.

Properties of Special Quadrilaterals:

Exercise 4:

Question no 1:
Question no 2: Question no 3:

Question no 4: Question no 5:

Question no 6:

Polygons:

A polygon is a closed plane figure with three and more straight line segments as its sides.
The first two polygons in the figure are called simple polygons because their boundaries do not cross
themselves, that is the line segment do not intersect one another, unlike the third polygon.

The first polygon in above figure is called a convex polygon because all its interior angles are less than
180 degree.

A simple polygon in which one of its interior angle is more than 180 degree is called a concave polygon
as given in second polygon of the above figure.

The third polygon is neither convex nor concave because it is not a simple polygon.

In this section we will study only convex polygons.

Regular polygons:

Sum of Interior angles of the polygons:

Moreover,

Exercise 5:
Perimeter and area of the basic plane figures:

Some Important formulas are provided in the table below.

Exercise 6:

Question no 1:

Question no 2:

Question no 3:

Consider the diagram ∆ 𝐽𝐿𝐾 𝐽𝐿 = 39 𝑚 , 𝐿𝐾 = 25𝑚 , 𝐽𝐾 = 40 𝑚 , 𝐾𝑀 = 24𝑚

Question no 4:

Find the perimeter and the area of a square, if we know the length of its diagonal d = 4.2
m.
Question no 5:

A right-angled isosceles triangle has the area of 32 cm 2. What is its perimeter?

Question no 6:

A parallelogram ABCD has the area of 40 cm 2, |AB| = 8.5 cm and |BC| = 5.65 cm. Find
the length of its diagonals.
Question no 7:

A square on the picture has 8 cm long side. Find the area of the colored part of a circle.

Question no 8:
Question no 9:

Question no 10:

The poster has a border whose width is 2 cm on each side. If the printed
material has the length of 6 cm and width of 8 cm, then find the area of the
border.
Ratio and Proportion:
Ratio:

Equivalent Ratios:

In General,

Ratio Involving three Quantities:

Exercise 1:

Question 1:

Question 2:

Question 3:

Question4:
Question 5:

Question 6:

Question 7:

Proportion:

Direct Proportion:

In a Singapore, if we borrow books from a public library and are late in returning the books, we will be
fined 15 cents per day for each overdue of book. Table shows the fine for over due of the book.

No of 1 2 3 4 5 6 7 8 9
days (x)
Fine (y 15 30 45 60 75 90 105 120 135
cents)
 1- If number of days a book is overdue increases, will the fine increase or decrease? (See table).
2- If the number of days the book is overdue is doubled, how will the fine change? (See table)
3- If the number of days book overdue is tripled, what will happen to the fine? (See table)
4- If the number of days of book is overdue is halved how will the fine change? (See table)
5- If the number of days a book is overdue is reduced is 1/3 of the original number what will
happen to the fine? (See table)

From the investigation, we notice that as the number of days, x, a book is overdue increases the fine, y
cents, increases proportionally, i.e., if x is doubled, then y will be doubled, if x is tripled then y will be
tripled.

Similarly, as the number of days, x, a book is overdue decreases, the fine y cents, decreases
proportionally .i.e., if x is halved ,then y will be halved, if x is reduced to 1/3 of its original value. This
relationship is known as direct proportion. We say that the fine y cents, is directly proportional to the
number of days, x , a book is overdue.

The relationship is known as direct proportion. We say that the fine, y cents is directly proportional to
the number of days, x , book is overdue.

Note:
𝑦 𝑦2 𝑥2 𝑦2
If y is directly proportional to x , then 𝑥1 = 𝑥2
𝑜𝑟 𝑥1
= 𝑦1
1

where 𝑥1 , 𝑥2 represent quantities with respect to 𝑥. 𝑦1 , 𝑦2 represent quantities with respect to y.

Algebraic Representation of Direct Proportion:

𝑦
If y is directly proportional to x , then 𝑥 = 𝑘 𝑜𝑟 𝑦 = 𝑘𝑥 , where k is a constant and 𝑘 ≠ 0.

Exercise 2:

Question no 1:

108 identical books have a mass of 30 kg. Find

i)- the mass of 150 such books.

ii)- the number of such books that has a mass of 20 kg.

Question no 2:

In a bookstore, 60 identical books occupy a length of 1.5 m on a shelf. Find

i)- the length occupied by 50 such books on a shelf.

ii)- the number of such books needed to completely occupy a shelf that is 80 cm long.

Question no 3:

If x is directly proportional to y and x = 4.5 when y = 3, Find

i)- equation connecting x and y.

ii)- the value of x when y = 6.

iii)- the value of y when x =12

Question no 4:

If Q is directly proportional to P and Q = 28 when P = 4

i)- Express Q in terms of P.

ii)- Find the value of Q when P = 5.

iii)- Calculate the value of P when Q=42

Question no 5:

Find the cost of

i)- 10 kg of tea leaves when 3 kg of tea leaves cost 18 Rs.

ii)- ‘a’ kg of sugar when ‘b’ kg of sugar cost S Rs.

Question no 6:

5/9 of a piece of metal has a mass of 7 kg. What is the cost of 2/7 of the piece of metal….

Question no 7:

If z is directly proportional to x and z=12 when x=3. Find the value of x when z = 18.

Question no 8:

If B is directly proportional to A and B=3 , when A = 18. Find the value of B when A = 24.

Question no 9:

If y is directly proportional to x and y = 20, when x =5

i)- Find an equation connecting x and y.

Inverse Proportion:

The table below shows the time taken for a car to travel a distance of 120 km at different speeds.

Speed (x 10 20 30 40 60 120
km/h)
Time Taken 12 6 4 3 2 1
(y hours)

1- The speed of the car increases will the time taken increases or decrease.
2- If the speed of the car is doubled, how will time have taken change?
3- If the speed of the car is tripled, what will happen to time taken?
4- If the speed of the car is halved, how will time taken change?
5- If the speed of the car is reduced to 1/3 to its original speed , what will happened to time taken?
 From the investigation, we notice that as the speed of the car x km/h, increases, the time taken
y hours, decreases proportionaly.i.e., if x is doubled y will be halved.
If ‘x’ is tripled, then y will be reduced to 1/3 of its original value.
Similarly, as the speed of the cars x km /h, decreases the time taken y hours, increases
proportionally i.e., x is halved, then y will be doubled, if x is reduced to 1/3 of its original value,
then y will be tripled.
The relationship is known as inverse proportion, we say that the speed of the car, ‘x’ km/h is
inversely proportional to the time taken.

Note:
𝑦2 𝑥1
If y is inversely proportional to x , then 𝑦1
= 𝑥2
𝑜𝑟 𝑥1 𝑦1 = 𝑥2 𝑦2

Algebraic Representation of In Direct Proportion:

If y is inversely proportional to x , then xy=k or y = k/x , where k is constant and k is not equal to zero.

Exercise 3:

Question no 1:

If x is inversely proportional to y and x = 40 when y =5 . Find

i)- the value of x when y =25

ii)- an equation connecting x and y.

iii)- the value of y when x =40.

Question no 2:

If Q is inversely proportional to P and Q = 0.25, when P = 2

i)- Express Q in terms of P.

ii)- Find the value of Q when P = 5.

iii)- Calculate the value of P when Q = 0.2

Question no 3:

A consignment of fodder can feed 1260 cattle for 50 days. Given that all the cattle consume for fodder
at the same rate, find.

i)- the number of cattle an equal consignment of fodder can feed for 75 days.

ii)- the number of days an equal consignment of fodder can last it is used to feed 1575 cattle.

Question no 4:

At a sport camp, there is sufficient food for 72 athletes to last 6 days. If x athletes are absent from the
camp how many more days can the food last for the other athletes. State the assumption made.
Question no 5:

If z is inversely proportional to x and z = 5, when x = 7. Find the value of x when z =70.

Question no 6:

If B is inversely proportional to A and B = 3.5 when A = 2. Find the value of B when A = 1.4
Percentage:

Introduction:

Percentages, Fractions and Decimals:

Figure shows the grid divided in to 100 equal parts. 75 parts are shaded.

Expressing one Quantity as a percentage of the other:

Exercise 1:
9-

10-

11-

12-
Percentage Change:
Exercise 2:

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:
Question 6:

Question 7:
Parts and whole:
Exercise 1:

1- A garden has an area of 2/5 hectare. The owner buys an extra 1/3 hectare of land to increase
the size of the garden. What is the new size of the garden.
2- A large company make a profit of ¾ million dollar in one year and 2/3 million dollar the next
year. Find the total profit of the two-year period.
3-

4-
5-

6-

7-

8-

9-

10-

11-

12-

13-
Money Management.
Profit, Loss, and Management:

Profit is the difference amount when a person sells a product at a higher rate than cost price and loss is
the difference amount when a person sells a product at a lower rate than cost price. Every commodity,
product or item has a cost price and selling price and depending on the values of these prices, we
compute the profit gained or the loss incurred for an individual product.

Profit: When a person sells a product at a higher rate than cost price, then the difference between
both amounts.

Profit Formula = Selling Price – Cost Price

Loss: When a person sells a product at a lower rate than the cost price, then the difference between the
two amounts

Loss = Cost Price – Selling Price

Terms used in Profit and Loss:

Cost Price: Cost price is a price at which a person purchases a product. For example, if Amna purchased
a book for 250 rupees, this is the cost price for that particular book. Cost price is abbreviated as C.P.

Selling Price: Selling price is a price at which a person sells a product. For example, if Ali sold a book for
350 rupees, then this is thought to be the selling price of the book. The selling price is abbreviated as
S.P.

Market Price: It is the price that is marked on an article and commodity. It is also known as a list price or
tag price. If there is no discount on the market price, then selling price is equal to the market price.

Markup: It is the amount by which cost price is increased to reach the market price.

Mark up = Market price – cost price.

Discount: The reduction offered by a merchant on a marked price is called discount.

Profit and Loss Formulas:

Profit and loss formula is employed in math’s to determine the price of an entity in the market and
comprehend how advantageous a business is.

If the selling price > cost price, then difference between S.P. and C.P. is called the profit.

Similarly, if the selling price < cost price, then the difference between C.P. and S.P. is called the loss.

Profit and Loss Terms Meaning Formulas

Profit or Gain The selling price of an object > Profit = Selling Price (S.P) – Cost
than its cost price Price (C.P)
 Loss The cost price of the object > Loss = Cost Price (C.P) – Selling
than its selling price Price (S.P)
Selling Price The price for which a 100+𝑃𝑟𝑜𝑓𝑖𝑡%
S.P = ( 100
) × 𝐶𝑃
commodity is sold is said to be
the selling price of the
Or
particular item denoted as S.P.
100−𝐿𝑜𝑠𝑠%
S.P = ( 100
) × 𝐶𝑃
Cost Price The expense at which the object 100
CP = (100+𝑝𝑟𝑜𝑓𝑖𝑡%) × 𝑆𝑃
is bought is termed as a cost
price for the object and is
Or
denoted by C.P.
100
C.P = (100−𝐿𝑜𝑠𝑠% ) × 𝑆𝑃

Discount To manage the competitors in

the industry and promote the Discount = MP – SP (Marked
sale of goods, vendors offered Price – Selling Price)
discounts to consumers

Profit Percentage and Loss Percentage:

The profit (%) as well as a loss (%) is obtained with the help of the below mentioned formulas. Along
with the profit percentage (%) and loss (%) other percentage – relate formulas are also discussed below.

Profit and loss terms Formulas in Percentage

Profit percentage (%) Profit = (SP) – (CP)

𝑝𝑟𝑜𝑓𝑖𝑡
Profit Percentage % = ( ) × 100
𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒

Loss percentage (%) Loss = (CP) – (SP)

𝐿𝑜𝑠𝑠
Loss Percentage % = (𝐶𝑜𝑠𝑡 𝑃𝑟𝑖𝑐𝑒 ) × 100

Discount (%) 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡

( ) × 100
𝑀𝑎𝑟𝑘𝑒𝑡 𝑃𝑟𝑖𝑐𝑒

𝑚𝑎𝑟𝑘𝑢𝑝
( ) × 100
Mark Up (%) 𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒

Note:
For false weight, the profit percentage can be determined by the formula:
𝐸𝑟𝑟𝑜𝑟
Gain% = 𝑇𝑟𝑢𝑒 𝑉𝑎𝑙𝑢𝑒−𝐸𝑟𝑟𝑜𝑟 × 100%

Exercise 1:

1- Marked price of a cricket bat is Rs 1000 and its sold price is Rs 800. Find the discount
percentage.
2- Marked price of a product is Rs 240 and 25% discount is provided on it. Find the selling price.
3- A T-shirt sold after providing two successive discount of 20%. If the marked price of the T- shirt
is Rs 200 then find the selling price.
4- A man gain 30 % by selling an article for a certain price. What will be the profit percentage.
5- If A bought an article od Rs 200 and sold it to B at 20% profit. Again, B sold the article at 10%
profit to C. Find the amount paid by C
6- A man sold two bicycles at the same selling price. One at 20% loss and other at 20% profit. Find
the overall profit and loss percentage.
7- If the cost price of 5 oranges is equal to sale price of 4 oranges, then find the profit percentage.
8- 10 pens cost Rs 100 each. If half of the pens are sold at 10% loss find at what price remaining
each pen should be sold for making no profit no loss.
9- Goods are purchased for Rs 1500. If one fifth of the goods are sold at a profit of 5% and the
remaining four-fifth of the goods at a profit of 10%, find the net percentage.
10- A trader brought a product for Rs 2000. If he marks his good for 20% above the cost price and
gives the discount of 10% for cash, find the profit percentage.
11- A person want to get 20% profit after selling his object at 20% discount. Find the required
percentage increase in marked price.

Zakat:
There are five pillars of Islam. Zakat is one of the pillars of Islam. It imposes on those Muslims who have
certain amount of wealth the whole year. The purpose of zakat is to help the poor and needy among the
Muslims to create welfare Muslim state. The Muslim pay zakat if their annual savings reaches a certain
level.

Nisab of Zakat:

Nisab is a minimum amount of annual savings on which zakat has to be paid. Nisab for zakat is 7.5 tola
(87.48 grams) gold and 52.5 tola (612.36 grams) silver or equivalent amount.

Rate of Zakat:

The rate of zakat is 2.5% of the total wealth.

The amount of zakat is calculated by the following formula:

Amount of Zakat = Rate of Zakat x Total amount

Exercise 2:
1- Aslam saved rupees 2000000 for one year in his account. Calculate the amount of zakat Aslam
has to pay.
2- Sadia paid zakat of Rs. 1500 on gold. Find the total price of gold.
3- Find the amount of zakat on 15 tola gold if value of 1 tola gold is Rs 12000.
4- Find the amount of zakat on 80 tola silver if value of 1 tola silver is Rs. 1500.
5- Reheel paid zakat worth Rs 47000, on gold and his savings. Find the price of gold if his savings is
Rs. 1000000.
6- Zahid paid zakat of Rs 23500. Find his savings.
Averages:
In Mathematics an average of a list of data is the expression of the central value of a
set of data. Mathematically, it is defined as the ratio of summation of all the data to the
number of units present in the list. In terms of statistics, the average of a given set of
numerical data is also called mean. For example, the average of 2, 3 and 4 is (2+3+4)/3
= 9/3 =3. So here 3 is the central value of 2,3 and 4. Thus, the meaning of average is to
find the mean value of a group of numbers.

So, Average = sum of values/ number of values.

Exercise:

Question 1:

The average age of 24 boys and the teacher is 15 years. When the teacher’s age is

excluded the average decreases by 1. What is the age of the teacher?

Question 2:

The average age of seven boys sitting in a row facing North is 26 years. If the average

age of the first three boys is 19 years and the average age of the last three boys is 32

years, what is the age of the boy who is sitting in the middle of the row?
Question 3:

The average age of the mother and her six children is 12 years which is reduced by 5

years if the age of the mother is excluded. How old is the mother?
Question 4:

A batch of 60 students made an average score of 50 marks and another batch of 40

students made it only 45. What is the overall average score?

Question 5:

The average of eight numbers is 14. The average of six of these numbers is 16. The

average of the remaining two numbers is

Question 6:
The average of 50 numbers is 38. If two numbers 45 and 55 are discarded, the average of

the remaining set of numbers is ?

Question 7:

In a cricket team of 11 boys, one player weighing 42 Kg, is in jured and replaced by

another player. If the average weight of the team is increased by 100 gm as a result of

this, then what is the weight of the new player?

Question 8:

The average of six numbers is 20. If one number is removed, the average becomes 15.

What is the number removed?

Question 9:

Ten years ago the average age of P and Q was 20 years. Average age of P, Q and R is 30

years now. After 10 years, the age of R will be ?

Question 10:

The average of 17 numbers is 10.9. If the average of first nine numbers is 10.5 and that of

the last 9 numbers is 11.4, the middle number is?