MATHEMATICS O LEVEL(FORM ONE) NOTES – FRACTIONS

FRACTIONS

A fraction is a number that can be written in the form of where a and b are integers and b 0.

The number written on top of the fraction is called the Numerator and the bottom is called the Denominator, e.g.

Types of Fractions

• Proper fractions: A proper fraction is one in which the numerator is less than the denominator, e.g. etc.
• Improper fractions: These are fractions where the numerator is greater than the denominator, e.g. etc.
• Mixed fractions or mixed numbers: These are formed after improper fractions are divided completely, e.g. = 2 , 6 , 9 .
• Equivalent fractions: These are two or more fractions that can be simplified to the same lowest fraction.

Example: Change the following into mixed numbers

1. = 7
2. = 3
3. = 6

Fractions can be represented on number lines, e.g. represent on a number line.

Exercise 1

1. (i) Which of the following are:

• (a) Proper fractions
• (b) Mixed fractions
• (c) Improper fractions

(ii) List four equivalent fractions:

1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 3

Solution

(a) Proper fractions:

, , , , , , , , , , .

(b) Improper fractions:

, , , , .

(c) Mixed fractions:

1 , 3 .

Four equivalent fractions are:

2. Write the following fractions in words

• (a) three quarters
• (b) A half
• (c) A third
• (d) Five over six
• (e) Nine over Ten
• (f) A quarter

2. Write the name of the fraction of the shaded part in figures ABCD and EFGH

One over three (A third) which is a proper fraction.

E H

F G

= one over four (A quarter) which is a proper fraction.

Comparison of fractions

Fractions can be compared by using two methods:

1. Number line
2. L.C.M of the denominators

(I) Number line

Example 1: Show and on a number line and find which is greater.

(II) L.C.M of the denominators

Determine which fraction is greater between and .

Solution:

1st find the L.C.M of 5 and 7:

L.C.M of 5 and 7 = 35

2nd multiply each fraction by that L.C.M:

35 = 14

35 = 20

Conclusion: The fraction with the bigger number after multiplication with L.C.M is greater.

Operations on Fractions

Addition and Subtraction of Fractions

NOTE:

1. Add the numerators together if each fraction has the same denominator.
2. If the fractions have different denominators, find the smallest number that each denominator divides into exactly (LCM).
3. When adding fractions, do not add the denominators.

Example: Evaluate the following fractions

Multiplication

NOTE:

1. Before multiplying, convert mixed numbers into improper fractions.
2. Multiply the numerators and multiply the denominators.

Examples:

Dividing Fractions

1. When dividing fractions, invert the second fraction then multiply the first fraction by the inverted fraction.
2. Before dividing, convert mixed numbers into improper fractions.

DECIMALS AND PERCENTAGES

Decimals are fractions of tenths, written using a decimal point which results from division of a normal fraction.

Examples: 0.34, 0.5, 0.333…

In the decimal 0.2546 the place values are:

Decimals can be converted into fractions and vice versa.

Example: Change into decimals.

Solution:

= 0.75

This fraction which ends after dividing is called a terminating fraction. Other fractions do not end; these are called recurring or repeating decimals.

Example:

Conversion of Repeating Decimal into Fractions

Solution:

0.3 = 0.333…

Subtract (i) from (ii):

9t = 3.0

t =

Exercise 1

Insert or between each pair of fractions in questions 4 to 12.

1. ,

Solution:

L.C.M = 3

3 = 2

3 = 1

3. PERCENTAGES

Percentages are fractions expressed out of 100. That is, they are fractions whose denominator is one hundred, denoted by (%) called percent.

Example: 12% means 12 100.

70% = etc.

Examples:

1. Convert the following percentages into fractions:

1. 65%
2. 75%
3. 12%

Solution:

(i) 65%

65 100 =

(ii) 75%

75 100 =

(iii) 12%

12 100 = 12.5%

2. Change

1. 40% into decimal
2. 35% into fractions
3. 0.125 into percentage

Solution:

(i) 40% = 0.4

(ii) 35% =

(iii) 0.125 = 12.5%

3. Change the recurring decimals into fractions

(i) 0.

Solution:

Let x = 0.

100x = 21.

Take away equation (i) from (ii):

99x = 20

x =

Operations on Decimals

Operations with decimals are similar to operations with whole numbers:

Addition

Note: The decimal points must be aligned. Put zeros at the end to give the same number of decimal places in each number.

Multiplication

Note:

1. When multiplying decimals, the answer must have the same number of decimal places as the total number of decimal places in the numbers being multiplied.
2. First carry out the multiplication in the usual way, without any decimal points, then place the decimal point to the total decimal places.

Division

Note:

It is not easy to divide by a decimal, so multiply each number by a power of 10 so that you are dividing by a whole number.

Example:

(a) 68.32 ÷ 1.4

Solution:

68.32 ÷ 1.4 = 68.32 × 10 ÷ 1.4 × 10 = 683.2 ÷ 14

By long division:

Therefore 68.32 ÷ 1.4 = 48.8

(b) 9.66 ÷ 0.23

Solution:

9.66 × 100 ÷ 0.23 × 100 = 966 ÷ 23

Therefore 9.66 ÷ 0.23 = 42

(c) 7.32 ÷ 1.2

Solution:

7.32 × 10 ÷ 1.2 × 10 = 73.2 ÷ 12

Therefore 7.32 ÷ 1.2 = 6.1

Mariam was given 20,000 shillings by her father. She spent 48% of it to buy shoes. How much money remained?

Solution:

48% of 20,000 = 9,600

20,000 – 9,600 = 11,600

∴ The remaining money was 11,600/=

PERCENTAGES APPLIED TO REAL LIFE PROBLEMS

The examples below show the wide range of applications.

Example 1: In one week, Flora earned 48,000/=. She spent 4,000/= on travel to and from work. What percentage of her money was left?

Solution:

Percentage of a quantity

When finding a percentage of a quantity, it often helps to change the percentage to a decimal and multiply it by the quantity.

Example: Find (a) 20% of 840,000

Percentage increase and decrease

There are two steps to calculate percentage increase (or decrease).

Example: In 1975 the population of a village was 90. It increased by 30% the following year. What was the population in 1976?