How to Add Fractions

Posted on September 14, 2013 by Civil Service Reviewer

Fractions whose denominators are the same are called similar fractions. Fractions that
are not similar are called dissimilar fractions. Hence, the fractions , , and are similar
fractions, while the fractions and are dissimilar fractions. In this post, we are going to
learn how to add fractions.
How to Add Similar Fractions
Adding similar fractions is very easy. In adding similar fractions, you just add the
numerator and copy the denominator. Here are a few examples.
Example 1

Example 2

Example 3

In most cases, improper fractions or fractions whose denominator is less than its
numerator such as the third example is converted to mixed form. The mixed form of
is . We will discuss how to make such conversion in the near future.
How to Add Dissimilar Fractions
Addition of dissimilar fractions is a bit more complicated than adding similar fractions.
In adding dissimilar fractions, you must determine the least common multiple (LCM) of
their denominator which is known as the least common denominator. Next, you have to
convert all the addends to equivalent fractions whose denominator is the LCM. Having
the same denominator means that the fractions are already similar. Here are a few
examples.
Example 1

Solution
a. Get the least common multiple (LCM) of 2 and 3.

Multiples of 2: 2, 4, 6, 8, 10, 12
Multiples of 3: 3, 6, 9, 12, 15
LCM of 2 and 3 is 6.
b. Convert the fractions into fractions whose denominator is the LCM which is 6.
First Addend:
.
So, the equivalent of is .

Second Addend:

So, the equivalent fraction of is .

c. Add the equivalent fractions
.

So,

Example 2

Solution
a. Get the LCM of 3 and 5.
Multiples of 3: 3, 6, 9, 12, 15, 18
Multiples of 5: 5, 10, 15, 20
Therefore, the LCM of 3 and 5 is 15.
b. Convert the given fractions into equivalent fractions whose denominator is 15.
First Addend:

So, the equivalent fraction of is .

Second Addend:

So, the equivalent fraction of is .

c. Add the equivalent fractions

So,
Example 3

Solution
a. Get the LCM of 3, 6 and 8.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
Multiples of 6: 6, 12, 18, 24, 30
Multiples of 8: 8, 16, 24, 32, 40
LCM of 3, 6 and 8 is 24.
b. Convert the given fractions into equivalent fractions whose denominator is 24.
First Addend:
.
Therefore, the equivalent fraction of is

Second Addend:

Therefore, the equivalent fraction of

is

Third Addend:

Therefore, the equivalent fraction of is .

c. Add the equivalent fractions

In the next post, we will have more examples and exercises regarding addition of similar
and dissimilar fractions. I will also give you some tips in getting the least common
multiple of two or more numbers without listing.

How to Get the Least Common Multiple of Numbers

Posted on September 10, 2013 by Civil Service Reviewer

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In mathematics, a multiple is a product of any number and an integer. The numbers 16,
-48 and 72 are multiples of 8 because 8 x 2 = 16, 8 x -3 = -48 and 8 x 9 = 72. Similarly,
the first five positive multiples of 7 are the following:
7, 14, 21, 28, 35.
In this post, we will particularly talk about positive integers and positive multiples. This
is in preparation for the discussions on addition and subtraction of fractions.
We can always find a common multiple given two or more numbers. For example, if we
list all the positive multiples of 2 and 3, we have
2, 4, 6, 8, 10, 12, 14, 16, 18, 20

and
3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
As we can see, in the list, 6, 12 and 18 are common multiples of 2 and 3. If we continue
further, there are still other multiples, and in fact, we will never run out of multiples.
Can you predict the next five multiples of 2 and 3 without listing?
The most important among the multiples is the least common multiple. The least
common multiple is the smallest among all the multiples. Clearly, the least common
multiple of 2 and 3 is 6. Here are some examples.
Example 1: Find the least common multiple of 3 and 5
Multiples of 3: 3, 6, 9. 12, 15, 18
Multiples of 5: 5, 10, 15, 20, 25,30
As we can see, 15 appeared as the first common multiple, so 15 is the least common
multiple of 3 and 5.
Example 2: Find the least common multiple of 3, 4, and 6.
In this example, we find the least multiple that are common to the three numbers.
Multiples of 3: 3, 6, 9, 12, 15
Multiples of 4: 4, 8, 12, 16, 20
Multiples of 6: 6, 12, 18, 24, 30
So, the least common multiple of 3, 4, and 6 is 12.
Example 3: Find the least common multiple of 3, 8 and 12.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
Multiples of 4: 4, 8, 12, 16, 20, 24,
Mulitples of 12: 12, 24, 36, 48, 60
So, the least common multiple of 3, 4 and 6 is 24.
In the next part of this series, we will discuss about How to Add Fractions.

Introduction to the Concept of Percentage

Posted on February 25, 2014 by Civil Service Reviewer

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Now that we have already studiedfractions and decimals, we discuss percentage. You
are likely to be aware that the concept of percentage is very useful in daily life. We
always go to stores where there are discounts and we do not want loans with high
interest. These calculations involve the concept of percentage.
What is percentage really?
Percentage is a number ratio expressed as a fraction of 100. When we say 10 percent,
what we really mean is 10 out of 100, or in fraction notation 10/100. Therefore, when we
see that a shirt is sold for a 50 percent discount, we actually say 50 out of 100 or 50/100.
Notice that 50/100 when reduced to lowest terms is 1/2 which means that we only have
to pay half of the price of the shirt. As we all know, we use the symbol % to denote
percent.
Converting Percent to Fractions for Faster Calculations
Numbers in their percent form can be converted to fractions for quicker calculations.
For example, when we say that a Php2400.00 wristwatch has a 25% discount, we can
easily calculate by converting 25% to fraction. The equivalent of 25% discount is 1/4 in
fraction, so, we deduct 1/4 of 2400 (which is equal to Php600) from Php2400. This
means that we can buy the watch for only Php 1800.00
Percents, fractions, and decimals can be converted to one another, to whichever
representation is more convenient for calculations. In examinations such as the Civil
Service Exam, in most cases, fraction is the easiest to use but the problem is conversion
also takes time. Therefore, it is also good to familiarize yourself with the conversion of
the most commonly used fractions in problems such as shown in the table below. You
can memorize them if you want, but the conversion method is fairly easy that you can do
them mentally.

In the next few post, will discuss how to convert fractions, percents, and decimals to one
another. Then, we will also discuss common percentage problems like discounts and

interests. These types of problems usually appear in the Math Word Problem Solving
section of the Civil Service Examination.

How to Convert Percent to Fraction

Posted on March 5, 2014 by Civil Service Reviewer

In Civil Service Examinations, as well as other examinations in basic mathematics,

knowing how to convert percent,fractions, and decimals to each other is very
advantageous especially if you can do it mentally. Let us try with the following example.
A P640 shirt is marked 25% discount. How much will you have to pay for it?
It seems that you need a pencil for this problem, but you can actually do it in your head.
Read it to believe it.

The equivalent of 25% in fraction is 1/4, therefore, you have to take away the fourth of
the price. Now, 1/4 of 640 seems difficult but what if we try to split it to 600 + 40? Now,
1/4 of 600 is 150, which means that from the 600, you have 450 left. Now, 1/4 of 40
is 10, which means that you have 30 left. So, 450 + 30 is 480 and that is the discounted
price of the t-shirt.
Now, with a little bit of practice, you would be able to do this on your own and you wont
have to use a pen to perform calculations for problems such as this.
How to Convert Percent to Fraction

There is one important concept to remember when converting percent to fraction. That
is, when you say percent, it means per hundred. The word cent comes from the Latin
wordcentum which means hundred. In effect, when you say, 60%, it means 60 per
hundred, 0.4% means 0.4 per hundred, 125% means 125 per hundred. When you
say x per hundred, you can also represent it by the fraction x/100. This means that the
percentages above can be represented as

respectively. Now, all we have left to do is to convert these fractions to lowest terms.
Example 1:
Recall that to convert a fraction to lowest terms, we find the greatest common
factor (GCF) of its numerator and denominator and then divide them both by the GCF.
The GCF of 60 and 100 is 20, so

Therefore, the equivalent of 60% in fraction is .

Example 2:
In this example, we have a decimal point at the numerator and a whole number at the
denominator. We have to get rid of the decimal point. To do this, we can multiply both
the numerator and the denominator by 10 (since 0.4 x 10 = 4). Therefore, we have
.
Now, the greatest common factor of 4 and 1000 is 4, so we divide both the numerator
and the denominator by 4. The final result is .
Therefore, the equivalent fraction of 0.4% is
Example 3:

The greatest common factor of 125 and 100 is 25, so we divide both the numerator and
the denominator by 25. In doing this, we get .
Therefore, the equivalent fraction of 125% is
Summary
There are three steps to remember in converting percent to fractions.
1.

Make a fraction from the given percent with the given as numerator and 100 as
denominator.

2.

Eliminate the decimal points (if there are any) by multiplying the numerator and
denominator by the same number which is a power of 10 (10, 100, 1000 and so on).

3.

Reduce the resulting fraction to lowest terms.

Thats it. You can now convert any given percent to fraction.

In the previous post, we have learnedhow to convert percent to fraction. In these series
of posts, we learn the opposite: how to convert fraction to percent. I am going to teach
you three methods, the last one would be used if you forgot the other two methods, or if
the first two methods would not work. Please be reminded though to understand the
concept (please do not just memorize).
The first method can be used for fractions whose denominators can be easily related to
100 by multiplication or division. Recall that from Converting Percent to
Fraction, I have mentioned that when we saypercent it means per hundred. In effect,
n% can be represented by n/100. Therefore, if you have a fraction and you can turn it
into n/100 (by multiplication/division), then you have turned it into percent.
Example 1: What is the equivalent of 1/5 in percent?

How do we relate the denominator 5 to 100? By multiplying it by 20. Therefore, we also

multiply its numerator by 20:

Now, since we have 100 as denominator, the answer in percent is therefore the
numerator. Therefore, the equivalent of 1/5 in percent is 20%.
Example 2: What is 3/25 in percent?
Again, how do you related 25 to 100? By multiplying it by 4. Therefore,

Therefore, the equivalent of 3/25 in percent is 12%.

Example 3: What is 23/200 in percent?
In this example, we can relate 200 to 100 by dividing it by 2. So, we also divide the
numerator by 2. That is

Therefore, the answer is 11.5%

There are two important things to remember in using the method above.
(1) in changing the form the fractions to n/100, the only operations that you can use are
multiplication and division and
(2) whatever you do to the numerator, you also do to the denominator.
Note that multiplying the denominator (or dividing it) by the same number does not
change its value, it only change its representation (fraction, percent or decimal).
Why It Works
When you are relating a fraction a/b to n/100, you are actually using ratio and
proprotion. For example, in the first example, you are actually solving the equation

.
The equation will result to
which is equal to 20. Now, this is just the same as
multiplying both the numerator and the denominator by 20.
Note that the method of relating to 100 by multiplication or division can only work
easily for denominators that divides 100 or can be divided by 100. Other fractions (try
1/7), you have to use ratio and proportion and manual division.

In the Part 1, we have learned how to convert fraction to percent by relating the denominator to
100 by multiplication or division. In this post, we do its algebraic version. This method is a
generalized method to the previous post especially for numbers that do not divide 100 or cannot
be divided by 100 easily. However, to see the relationship between the two methods, let us do
the first example in Part 1 of this series.

Example 1: What is the equivalent of 1/5 in percent.

Recall that in Part 1, we multiplied both the numerator and the denominator by 20, to
make the denominator 100. That is,

Now, notice how it is related to the new method. In this method, we related 1/5 to
n/100. That is, what is the value of in

.
To simplify the equation, we multiply both sides of the equation by 100, and we get

Simplifying and switching the position of the expressions, we get the

that
.

. This means

Of course, Part 1 seems to be easier, but the good thing about putting it into equation is
that it applies to all fractions. For instance, it is quite hard to convert 7/12 using the
method in part 1.
Example 2: What is the equivalent of
We set up the equation with

in percent?

on the left.

To eliminate the fraction, multiply both sides by denominator. This results to

or about 58.33%.
The curly equal sign means approximately equal to since 3 is a non-terminating decimal.
Now, try to examine the expression

because this is where they derived the rule. Recall the rule in converting fraction to
percent: Divide the fraction and then multiply the result to 100. That is exactly
it.
So, when you have the fraction,
just divide it manually, and then multiply the result
to 100. That is,
.
Do not forget though that the divisor during division is the denominator (5 in 2/5). as
shown below.

Thats it. I think we dont have to have the third part, since we already derived the rule
here.

A Tutorial on Solving Equations Part 1

Posted on March 8, 2014 by Civil Service Reviewer

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Solving equations is one of the most fundamental concepts that you should learn to
be able to solve a lot of mathematical problems such as those in the Civil Service
Examinations. For example, for you to be able to solve aword problem, you need to
translate words into expressions, set up the equation, and solve it. Therefore, you should
learn this post and its continuation by heart.
In this series of posts, we are going to learn how to solve equations and then learn how
to solve different types of problems (number, age, coin, Geometry, motion, etc). These
types of problems usually appear in the Civil Service Examinations.
So, what is an equation really?
An equation are two expressions (sometimes more) with the equal sign in between. The
equation

means that the algebraic expression on the left hand side which is
has the same
value as the numerical expression on the right hand side which is . Now, you can think
of the equal sign as a balance. If you put two different objects and they balance, it
means if you take away half of the object on the left, you also have to take half of the
object on the left. Or, if you double the amount (or weight) of the object on the left, you
also double whats on the right to keep the balance.

The fancy name of this principle in mathematics is Properties of Equality. It

basically means that whatever you do on the left hand side, you also do on the right
hand side of the equation. Here are a few examples to illustrate the idea.
How to Solve Equations
Example 1:
There is really nothing to solve in this example. What will you add with to get . Of
course . However, we use the Properties of Equality future reference. The idea is to
isolate on one side and all the other numbers on the other side. Since, is on the left
hand side, we want to get rid of . So, since was added to , we have to subtract from
both sides to get rid of it. So,
.
This gives us
Example 2:

This example can be again solved mentally. What will you multiply with to get , of
course, its . But, solving it as above, to get rid of 3 in , since it is multiplication, we
divide it by .
.
Of course, if you divide the left hand side by , you also divide the right hand side of the
equation by .
This gives us

Example 3:
In this example, is a fraction which mean that we have to get rid of 5. To do this, we
multiply both sides by . That is,

Therefore,

Like Examples 1 and 2, this can be solved mentally.

Example 4:
In this example, we have 2 times and then added to . Well, intuitively, we can
eliminate first by subtracting it from both sides. That is

which results to
.
Now, its multiplication, so we eliminate
by . That is

by on the left hand side by dividing both sides

This results to
Example 5:
We first need to eliminate 3 from the left hand side. Since it is subtraction, to eliminate
it, we have to perform addition (because
) on both sides of the equation.
Doing this, we have

Now, we solve for

by dividing both sides by . That is

.

That is,

or

in mixed fraction or

in decimals.

In the next part of this series, we are going to learn how to solve more complicated
equations.

A Tutorial on Solving Equations Part 2

Posted on March 9, 2014 by Civil Service Reviewer

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This is a continuation of Solving Equations Part 1. As I have mentioned in that post,

being able to solve equations is very important since it is used for solving more
complicated problems (e.g. word problems).
In this post, we are going to solve a slightly more complicated equations. We already
discussed 5 examples in the first post, so we start with our sixth example.
Example 6:

As I have mentioned in the previous examples, we need to isolate on one side of the
equation and all the numbers on the other side. Here, we decide to put all s on the left
hand side, so we remove
on the right hand side. To do this, we subtract
from both
sides of the equation.

Of course,

, so, simplifying, we have

Then, we want to eliminate on the left hand side. Since it is multiplication, we

therefore divide both sides of the equation by .

Therefore,

Example 7:
In this example, we want to avoid a negative , so it is better to put all s on the right
hand side of the equation. This means that we have to eliminate from the left hand
side. So, we subtract from the left hand side, and of course, the right hand side as well.

Next, since we want to eliminate all the numbers on the right, the easiest to eliminate
first is
. To do this, we just add on both sides of the equation.

.
Next, we only have one number on the right hand side which is . To eliminate it, we
divide by . Of course, we also need to divide the other side by .

Therefore, the answer is

Notice also that we can add and subtract immediately resulting

to
making the process faster. You will be able to
discover such strategy on your own if you solve more equations.
Example 8:
In this example, we have the form
in the left hand side of the equation. To
simplify this, we simply distribute the multiplication of over
. That is
.
This is called the distributive property of multiplication over addition.
So, solving the problem above, we have

Adding

to both sides of the equation, we hhave

Dividing both sides of the equation by

we have
.

Example 9:
In equations with fractions, the basic strategy is to eliminate the denominator. In this
example, the denominator is . Since means
divided by , we cancel out by
multiplying the equation by 5. Notice how 5 is distributed over the left hand side.

which is the same as

.
Simplifying, we have
Subtracting

from both sides, we have

Dividing both sides by 3, we have

Example 10:

We eliminate fraction by multiplying both sides of the equation by 2. That is

In the left hand side, cancels out , so only

distributive property.

Subtracting

Subtracting

is left. On the right hand side, we use

from both sides, we have

from both sides, we have

Thats it. In the next post, we solve more equations, particularly those that involve
fractions.

A Tutorial on Solving Equations Part 3

Posted on March 13, 2014 by Civil Service Reviewer

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This is the third part of the series of tutorials on solving equations. In this part, we will
solve more complicated equations especially those that contain fractions. The first
part and the second part of this series discuss 10 sample equations. We start with the
11th example.
Example 11: -5x 3 = -4x + 12
This example deals with the question of what if is negative? Let us solve the equation.
We want on the left and all the numbers on the right. So, we add 4x to both sides.
-5x 3 + 4x = -4x + 4x + 12
-x - 3 = 12
Next, we add 3 to both sides to eliminate -3 from the left hand side of the equation.
-x - 3 + 3 = 12 + 3
-x = 15
You cannot have a final equation like this where there is a negative sign on x. To
eliminate the negative sign on x, multiply both sides by -1. That is
(-1)(-x) = (-1)(15)
So, x = -15 is the final answer.
Example 12:
This example highlights the distributive property. Notice that distributive property is
also needed on equations with fractions. The idea is that if you have an expression that
looks like
; that is, a multiplied by the quantity
, you must distribute
over them. That is,
and

Solving the equation above, we have

Notice on the right hand side that is not distributed to the second
second is outside the parenthesis. We now simplify.

because the

.
Next, we simplify the expression

on the right hand side.

Now, we want to put on the left and all the numbers on the right. We do this
simultaneously. We subtract
from the right hand side and add 6 on the left hand side,
so we add
to both sides of the equation. You can do this separately if you are
confused.

On the
side,

left

hand
and

This gives us
have

side:

and

. Multiplying both sides by

On

the

right

hand

, as we have done in Example 11, we

as the final answer.

Example 13:

This type of equation usually appears in work and motion problems which we will
discuss later. Just like in solving fractions, all you have to do is get the least common
denominator. Now, the least common denominator of 2, 3, and 4 is 12. So, all we have to
do is to multiply everything with 12. That is

Dividing both sides by

, we have

.
Example 14:
This is almost the same the above example. We get the least common denominator of
and which is equal to . Then, we multiply everything with . That is

Now, on the left hand side,

and on the right hand side

. This gives us

.
Simplifying the left hand side, we have

Now,
gives us
gives us the final answer

. Multiplying both sides by

to make

positive

.
Example 15:
This example discusses the question what if is in the denominator? If is just in the
denominator just like this example, the solution is quite similar to Example 13.
However, if is both found in the numerator and denominator, this will result to a

quadratic equation (something with ). This seldom comes out, and we will discuss this
separately. For now, let us solve this example.
The strategy here is to get the least common denominator of the numbers and then
include during the multiplication. In this example, we want to get the least common
denominator of and which is . Now, we include and the least common
denominator of the equation above is
. Now, we multiply everything with
. That
is,

. Therefore, the answer is

This ends the third part of this series, in the next part of this series (I am not sure if I
will discuss this soon), we will discuss about dealing equations with radicals (square
root and cube root).