## DECIMALS AND PERCENTAGES

Are fractions of tenth, they are written using a point which is a result of division of a normal fraction

E.g. 0.34, 0.5, 0.333——–

In the fraction 0.2546 the place values are

Ones | 0 |

Tenth | 2 |

Hundredths | 5 |

Thousandths | 4 |

Ten Thousandths | 6 |

Decimals can be converted into fractions and vice versa

E.g. Change in to decimals

Solution:

= 0.75

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

E.g.

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 questions 4 to 12

1. ,

**Solution **

L.C.M = 3

3 = 2

3 = 1

2. ,

**Solution**

L.C.M = 63

63 = 7

63 = 9

3.

**Solution **

12 = 10

12 = 9

4. ,

**Solution **

20 = 16

20 = 15

5. ,

**Solution **

L.C.M of 20 and 4 = 80

80 = 60

80 = 140

6. ,

**Solution **

L. C. M of 4 and 4 = 4

4 = 1

4 = 3

7. ,

**Solution **

L. C. M of 5 and 6 = 30

30 = 12

30 = 5

8. ,

**Solution **

L. C. M of 9 and 6 = 18

18 = 16

18 = 15

9. Which numbers are denominators in each of the following fractions?

(a) 16 is the denominator.

(b) 93 is the denominator

(c) 3 5 is the denominator

10. Which numbers are numerators in each of the following fractions?

(a) Numerators is 3

(b) 3 Numerators is 4

(c) Numerators is 12

12. Which is greater

(a) or

**Solution **

Find the L.C.M of 5 and 4 = 20

20 = 12

20 = 15

(b) or

**Solution **

Find the L.C.M of 3 and 2 = 6

x 6 = 4

x 6 = 3

13. What is the condition for a fraction to be called improper?

The numerator is bigger than the denominator.

14. Change the following improper fractions into mixed numbers

(a) = 1

(b) = 4

(c) = 3

(d) = 1

16 15. Change the following mixed numbers into improper fractions

(a) 3

**Solution **

=

(b) 15

**Solution **

=

(c) 24

**Solution **

=

## 3.3 PERCENTAGES

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

Example: 12% means 12 =

70% = etc.

Examples: 1. convert the following percentage into fraction

(i) 65%

(ii)75%

(iii)12 %

Solution

(i) 65%

65 = =

=

(ii) 75%

75 = =

=

(iii) 12 %

12 = =

12 % =

2. Change

(i) 40% into decimal

(ii) 35% into fractions

(iii) 0.125 into percentage

**Solution
**(i) 40% = =

= 0.4

(ii) 35%

35 =

=

(iii) 0.125

**Solution**

0.125 = 100%

= 12.5%

**3. Change the recurring decimals into fractions**

(i)0

**Solution **

Let x = 0.……………………….. (i)

100x = 21.……………………. (ii)

Take away equation (i) from (ii)

100x = 21.

=

=

x =

(ii)

Solution

Let x = 0.9……………………….. (i)

10x = 9.…………………………. (ii)

100x = 93.3 ………………………(iii)

Take equation (ii) away from equation (iii)

100x = 93.3

=

=

x =

(iii)0.6

**Solution **

Let x = 0.6………………………….. (i)

1000x = 567.567 ……………………. (ii)

Take away equation (i) from (ii)

1000x = 567.567

X = 0.567

x =

(iv) 0.35

Solution

Let x = 0.35……………………….. (i)

10000x = 1352.1352 ……………………. (ii)

Take (ii) – (i)

10000x = 1352.1352

=

=

x =

(v) 0.1

Solution

Let x = 0.1………………………….. (i)

1000x = 219.219 ……………………. (ii)

Take away equation (i) from (ii)

1000x = 219.219

=

=

x =

(vi)0.8

Solution

Let x = 0.8………………………….. (i)

1000x = 186.186 ……………………. (ii)

Take away equation (i) from (ii)

1000x = 186.186

=

=

x =

(vii) 0.63

**Solution **

Let n = 0.63………………………….. (i)

10000n = 8634.8634 ……………………. (ii)

Take away equation (i) from (ii)

10000n = 8634.8634

=

=

n =

(viii) 0.7

Solution

Let x = 0.7……………………….. (i)

10x =0.7 …………………………. (ii)

1000x = 792. ………………………(iii)

Take away equation (ii) from equation (iii)

100x = 792.

1000x – 10x = 792. – 7.

990x = 785

x =

(ix) 0.4

**Solution **

Let y = 0.4………………………….. (i)

1000y = 645.4……………….. (ii)

Take away equation (i) from (ii)

1000y – y = 645-4

999y=645

=

y =

(x)0.

**Solution **

Let b = 0.………………………….. (i)

100b = 64. ……………………. (ii)

Take away equation (i) from (ii)

100b – b = 64.-0.

99b = 64

=

b =

(xi)0.2

**Solution **

Let m = 0.2………………………….. (i)

1000m = 627.2……………………. (ii)

Take away equation (i) from (ii)

1000m – m = 627.2– 0.2

999m = 627

=

m =

4.In question (i) to (v) change the fractions into decimals.

**Solution **

1 ÷ 3 =

= 0.33

ii.

**Solution **

5 ÷ 6 =

= 0.833

iii.

**Solution **

4 ÷ 11 =

= 0.3636

iv.

**Solution **

1 ÷ 9 =

= 0.111

v.

**Solution **

7 ÷ 13 =

= 0.538461

**Solution **

Let b = 0.2………………………….. (i)

1000b = 123.2……………….. (ii)

Take equation (i) away from equation (ii)

1000b – b = 123.2 – 0.2

999b = 123

–

b =

**Operations on Decimals
**

Operations with decimals are similar to operations with whole numbers:

**Addition**

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

**Multiplication**

**Note:**

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

**Division**

**Note:**

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

Example:- (i) Find (a) 68.32 ÷ 1.4

(b) 9.66 ÷ 0.23

**Solution**

(a) 68.32÷ 1.4 = 68.32 x 10 ÷1.4 x 10

682.2÷14

By long division

Therefore 68.32÷ 1.4 = 48.8

(b)

Therefore 9.66÷ 0.23 = 42

(c) 7.32 1.2 = 7.32 x 10 1.2 x 10

73.2

t

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 **

20,000

=9,600

20,000

– 9,600

11,600

∴The remained money was 11,600/=

**PERCENTAGES APPLIED TO REAL LIFE PROBLEMS**

The examples below show the wide range of application

Examples:-

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 is 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 the year 1976?