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Specific Objectives

By the end of the topic the learner should be able to:

  1. Identify proper and improper fractions and mixed number.
  2. Convert mixed numbers to improper fractions and vice versa.
  3. Compare fractions;
  4. Perform the four basic operations on fractions.
  5. Carry out combined operations on fractions in the correct order.
  6. Apply the knowledge of fractions to real life situations.

Content

  1. Fractions
  2. Proper, improper fractions and mixed numbers.
  3. Conversion of improper fractions to mixed numbers and vice versa.
  4. Comparing fractions.
  5. Operations on fractions.
  6. Order of operations on fractions
  7. Word problems involving fractions in real life situations.

 

Introduction

A fraction is written in the form where a and b are numbers and b is not equal to 0.The upper number is called the numerator and the lower number is the denominator.

 

Proper fraction

In proper fraction the numerator is smaller than the denominator. E.g.

 

Improper fraction

The numerator is bigger than or equal to denominator. E.g.

 

Mixed fraction

ecolebooks.com

An improper fraction written as the sum of an integer and a proper fraction. For example

 

=

Changing a Mixed Number to an Improper Fraction

 

Mixed number – 4 Image From EcoleBooks.com(contains a whole number and a fraction)

Improper fraction – Image From EcoleBooks.com(numerator is larger than denominator)

 

Step 1 – Multiply the denominator and the whole number

Step 2 – Add this answer to the numerator; this becomes the new numerator

Step 3 – Carry the original denominator over

 

Example

3 Image From EcoleBooks.com = 3 × 8 + 1 = 25

 

=Image From EcoleBooks.com

 

Example

  1. Image From EcoleBooks.com = 4 × 9 + 4 = 40

     

    =Image From EcoleBooks.com

Changing an Improper Fraction to a Mixed Number

 

Step 1– Divide the numerator by the denominator

Step 2– The answer from step 1 becomes the whole number

Step 3– The remainder becomes the new numerator

Step 4– The original denominator carries over

Image From EcoleBooks.com

Image From EcoleBooks.com

Example

Image From EcoleBooks.com = 47 ÷ 5 or

5Image From EcoleBooks.com = 5 Image From EcoleBooks.com = 9 Image From EcoleBooks.com

2

Example

Image From EcoleBooks.comImage From EcoleBooks.com = 2 Image From EcoleBooks.com = 2 Image From EcoleBooks.com = 4 ½

Comparing Fractions

When comparing fractions, they are first converted into their equivalent forms using the same denominator.

Equivalent Fractions

To get the equivalent fractions, we multiply or divide the numerator and denominator of a given fraction by the same number. When the fraction has no factor in common other than 1, the fraction is said to be in its simplest form.

Example

Arrange the following fractions in ascending order (from the smallest to the biggest):

1/2 1 /4 5/6 2/3  

 

Step 1: Change all the fractions to the same denominator.

 

Step 2: In this case we will use 12 because 2, 4, 6, and3 all go into i.e. We get 12 by finding the L.C.M of the denominators. To get the equivalent fractions divide the denominator by the L.C.M and then multiply both the numerator and denominator by the answer,

For ½ we divide 12 2 = 6, then multiply both the numerator and denominator by 6 as shown below.

 

1 x 6  1x3  5 x2  2 x4

 2 x6  4 x3  6 x2  3 x4

 

Step3: The fractions will now be:  

6/12 3/12  10/12   8/12  

 

Step 4: Now put your fractions in order (smallest to biggest.)

 

3/12 6/12 8/12 10/12

 

Step 5: Change back, keeping them in order.  

 1/4 1/2 2/3 5/6

You can also use percentages to compare fractions as shown below.

Example

Arrange the following in descending order (from the biggest)

5/12 7 /3 11/5 9 /4

Solution

X 100 = 41.67%

X 100 = 233.3%

X 100 = 220%

X 100 = 225%

7/3, 9/4, 11/5, 5/12

 

Operation on Fractions

Addition and Subtraction

The numerators of fractions whose denominators are equal can be added or subtracted directly.

Example

2/7 + 3/7 = 5/7

6/8 – 5/8 = 1/8

When adding or subtracting numbers with different denominators like:

5/4 + 3 /6=?

2/5 – 2/7 =?

 

Step 1– Find a common denominator (a number that both denominators will go into or L.C.M)

Step 2– Divide the denominator of each fraction by the common denominator or L.C.M and then multiply the answers by the numerator of each fraction

Step 3– Add or subtract the numerators as indicated by the operation sign

Step 4 – Change the answer to lowest terms

Example

Image From EcoleBooks.com + Image From EcoleBooks.com = Common denominator is 8 because both 2 and 8 will go into 8

Image From EcoleBooks.com + Image From EcoleBooks.com=

Image From EcoleBooks.com Which simplifies to 1Image From EcoleBooks.com

 

Example  

4Image From EcoleBooks.comImage From EcoleBooks.com = Common denominator is 20 because both 4 and 5 will go into 20

 

 

 

4Image From EcoleBooks.com = 4Image From EcoleBooks.com

Image From EcoleBooks.com = Image From EcoleBooks.com

Image From EcoleBooks.com

  4Image From EcoleBooks.com

 

Or

4Image From EcoleBooks.comImage From EcoleBooks.com = =

Image From EcoleBooks.comMixed  numbers can be added or subtracted easily by first expressing them as improper fractions.

Examples

5

Solution

5 = 5 +

Example

Evaluate

Solution

 

Multiplying Simple Fractions

Step 1– Multiply the numerators

Step 2– Multiply the denominators

Step 3– Reduce the answer to lowest terms by dividing by common divisors

 

Example

Image From EcoleBooks.com × Image From EcoleBooks.com = Image From EcoleBooks.com which reduces to Image From EcoleBooks.com

 

Multiplying Mixed Numbers

 

Step 1– Convert the mixed numbers to improper fractions first

Step 2– Multiply the numerators

Step 3– Multiply the denominators

Step 4– Reduce the answer to lowest terms

 

Example

2Image From EcoleBooks.com × 1Image From EcoleBooks.com = Image From EcoleBooks.com × Image From EcoleBooks.com = Image From EcoleBooks.com

Which then reduces to 3Image From EcoleBooks.com

 

Note:

When opposing numerators and denominators are divisible by a common number, you may reduce the numerator and denominator before multiplying. In the above example, after converting the mixed numbers to improper fractions, you will see that the 3 in the numerator and the opposing 3 in the denominator could have been reduced by dividing both numbers by 3, resulting in the following reduced fraction:

 

Image From EcoleBooks.comImage From EcoleBooks.comImage From EcoleBooks.com Image From EcoleBooks.com × Image From EcoleBooks.com = Image From EcoleBooks.com = 3Image From EcoleBooks.com

 

 

Dividing Simple Fractions

Step 1– Change division sign to multiplication

Step 2– Change the fraction following the multiplication sign to its reciprocal (rotate the fraction around so the old denominator is the new numerator and the old numerator is the new denominator)

Step 3– Multiply the numerators

Step 4– Multiply the denominators

Step 5– simplify the answer to lowest terms

Example

Image From EcoleBooks.com ÷ Image From EcoleBooks.com = becomes Image From EcoleBooks.com × Image From EcoleBooks.com which when solved is Image From EcoleBooks.com

 

Dividing Mixed Numbers

Step 1 – Convert the mixed number or numbers to improper fraction.

Step 2 – Change the division sign to multiplication.

Step 3– Change the fraction following the multiplication sign to its reciprocal (flip the fraction around so the old denominator is the new numerator and the old numerator is the new denominator)

Step 4– Multiply the numerators.

Step 5– Multiply the denominators.

Step 6– Simplify the answer to lowest form.

 

Example

3Image From EcoleBooks.com ÷ 2Image From EcoleBooks.com = becomes Image From EcoleBooks.com ÷ Image From EcoleBooks.com becomes Image From EcoleBooks.com × Image From EcoleBooks.com =

 

Image From EcoleBooks.comImage From EcoleBooks.comWhich when solved is Image From EcoleBooks.com × Image From EcoleBooks.com = Image From EcoleBooks.com which simplifies to 1Image From EcoleBooks.com

Order of operations on Fractions

The same rules that apply on integers are the same for fractions

BODMAS

Example

15 (we start with of then division)

= 15

= 5

Example

=

Solution

1/3 – 1/4 = (we start with bracket)

(We then work out the outer bracket)

(We then work out the multiplication)

(Addition comes last here)

 

Example

Evaluate + ½

 

Solution

We first work out this first

 

 

 

Therefore + ½ = 25 + ½

= 25 ½

 

Note:

Operations on fractions are performed in the following order.

  • Perform the operation enclosed within the bracket first.
  • If (of) appears, perform that operation before any other.

Example

Evaluate: =

Solution

=

 

=

=

=

Example

Two pipes A and B can fill an empty tank in 3hrs and 5hrs respectively. Pipe C can empty the tank in 4hrs. If the three pipes A, B and C are opened at the same time find how long it will take for the tank to be full.

Solution

1/3 +1/5 -1/4 = 20+12-15

60

= 17/60

17/60=1hr

1= 1 x 60/17

60/17 = 3.5294118

= 3.529 hrs.

End of topic

 

Did you understand everything?

If not ask a teacher, friends or anybody and make sure you understand before going to sleep!

Past KCSE Questions on the topic

 

1.  Evaluate without using a calculator.

Image From EcoleBooks.com

2. A two digit number is such that the sum of the ones and the tens digit is ten. If the digits are reversed, the new number formed exceeds the original number by 54.

Find the number.

3.  Evaluate Image From EcoleBooks.com  

4.  Convert the recurring decimal lImage From EcoleBooks.cominto fraction

5.  Simplify Image From EcoleBooks.comwithout using tables or calculator  

6.  Evaluate without using tables or calculators

 Image From EcoleBooks.com

7.  Mr. Saidi keeps turkeys and chickens. The number of turkeys exceeds the number of chickens by 6. During an outbreak of a disease, ¼ of the chicken and 1/3 of the turkeys died. If he lost total of 30 birds, how many birds did he have altogether?

9.  Work out Image From EcoleBooks.com

10.   Evaluate  -4 of (-4 + -5 ¸15) + -3 – 4 ¸2)

Image From EcoleBooks.com  84 ¸ -7 + 3 – -5

11. Write the recurring decimal 0.Image From EcoleBooks.comCan as Fraction

12.  Evaluate Image From EcoleBooks.comwithout using a calculator.

 

13.  Without using tables or calculators evaluate.

 Image From EcoleBooks.com

14.  Without using tables or calculator, evaluate the following.

8 + (13) x 3 – (5)

1 + (6) ÷ 2 x 2

15.  Express Image From EcoleBooks.com as a single fraction  

 

Image From EcoleBooks.com16.  Simplify  ½ of 3½ + 1½ (2½ – 2/3)  

¾ of 2½ ¸ ½

 

17.  Evaluate:

Image From EcoleBooks.com 2/5¸ ½ of 4/9 – 11/10

1/81/6 of 3/8

18.  Without using a calculator or table, work out the following leaving the answer as a mixed number in its simplest form:-

Image From EcoleBooks.com  ¾ + 12/7 ÷ 3/7 of 21/3

(9/73/8) x 2/3

 

 

19.  Work out the following, giving the answer as a mixed number in its simplest form.  

Image From EcoleBooks.com  2/5¸ ½ of 4/9 – 1 1/10

 1/81/16 x 3/8

 

20.  Evaluate;  

Image From EcoleBooks.comImage From EcoleBooks.comImage From EcoleBooks.comImage From EcoleBooks.com

 

 

23.  Without using a calculator, evaluate:

Image From EcoleBooks.com   14/5 of 25/18¸12/3 x 24

  21/3 – ¼ of 12 ¸5/3  leaving the answer as a fraction in its simplest form

 

24.  There was a fund-raising in Matisse high school. One seventh of the money that was raised was used to construct a teacher’s house and two thirds of the remaining money was used to construct classrooms. If shs.300, 000 remained, how much money was raised  




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EcoleBooks | Mathematics Form 1-4 : CHAPTER SEVEN - FRACTIONS

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