## Decimals

LESSON OBJECTIVES

#### By the end of this lesson you should be able to:

convert recurring decimals into fractions accurately.

LESSON OBJECTIVES

#### By the end of this lesson you should be able to:

convert recurring decimals into fractions accurately.

convert a fractions into a decimals.

__
Exercise2
__

Express the following as fractions in their simplest forms.

1. a) 3.3636''''''

b) 0.0909''''''

c) 2.7272''''''

d) 0.46846'''''..

e) 1.313313'''''.

f) 0.522522'''''..

CASE II

Convert 0.45.. into a fraction in its simplest form.

Let r stand for the recurring decimal given i.e.

r=0.45..

Let the above equation be equation I

r=0.45.. I

Multiply both sides of equation by 100

100r=45.4545..

Let this be equation II

100r=45.4545;. II

Now subtract equation I from equation II

110r=45.4545.. II

-r= 0.4545.. I

99r=45

99r=45

99 99

r=5/11

but r=0.4545.

Therefore 0.4545. =5/11

NB: In both case I and case II the decimal point is

just before the recurring number or numbers.

In the case I only one number is recurring hence

we multiply by 10.

In case II two numbers re recurring and hence

& we multiply by 100.

If three numbers just after the decimal pointare

recurring we need to multiply by 1000.

In general if n numbers just after the decimal point a

re recurring we multiply by 10n.

Example: convert 0.123123. to a fraction in its

simplest form.

Let r stand for the recurring decimal given that

r=0.123123.

Let the above equation be equation I

r=0.123123. I

Multiply both sides of equation I by 1000 since three numbers

are recurring after the decimal point.

1000r=123.123123.

Let this equation be equation II.

1000r=123.123123. II

Now subtract equation I from equation II

1000r=123.123123..

- r= 0.123123..

999r=123

999r/999=123/999

r=123/999

But r=0.1231223.

Therefore 0.123123. = 123/999

__
CASE III__

Convert 2.8333''''. to a fraction in the simplest form.

When there are non recurring digit(s) after the decimal point,

the decimal point is moved between the last non recurring

digit and the first recurring digit(s).

Let r stand for the recurring decimal

r=2.833333333'''' I

Let this equation be equation I

Multiply by 10 to move the decimal point just before 3

10r=28.333333''''..

Let this be equation II

10r=28.333333''''' II

Multiply equation II by ten since one digit is recurring.

100r=283.333''''.

Let this be equation III

100r=283.333''''. III

Subtract equation II from III

100r=283.333''''.

10r=28.333333'''''

90r=255.0000''''''.

90r=255

r=255/90

r=2 75/90

r=2 5/6

But r=2.83

Therefore 2.83=2 5/6

b) Convert t3.45676767''''' to a fraction in it simplest form.

Let r stand for 3.4567'''''.

r=3.456767'''''

let this be equation I

r=3.456767'''''I

Multiply equation I by 100 to move the decimal point to be

between the last non recurring digit and the first recurring digit.

100r=345.676767''''''

Let this be equation II

100r=345.676767'''''' II

Multiply equation II by 100 since digits are recurring.

10000r=34567.676767''''''

Let this be equation III

10000r=34567.676767'''''' III

Subtract equation II from equation III

10000r=34567.676767'''''' III

-100r=345.676767''''''' II

9900r=34222

r=34222/9900

r=3 4522/9900

r=3 2261/4950

But r=3.45676767'''''

Therefore 3.45676767''..=3 2261/4950

CASE III

Convert 2.8333.. to a fraction in the simplest form.

When there are non recurring digit(s) after the decimal point,

the decimal point is moved between the last non recurring

digit and the first recurring digit(s).

Let r stand for the recurring decimal

r=2.833333333...... I

Let this equation be equation I

Multiply by 10 to move the decimal point just before 3

10r=28.333333...

Let this be equation II

10r=28.333333....... II

Multiply equation II by ten since one digit is recurring.

100r=283.333.......

Let this be equation III

100r=283.333......... III

Subtract equation II from III

100r=283.333........

10r=28.333333..........

90r=255.0000...........

90r=255

r=255/90

r=2 75/90

r=2 5/6

But r=2.83

Therefore 2.83=2 5/6

b) Convert t3.45676767....... to a fraction in it simplest form.

Let r stand for 3.4567.......

r=3.456767.......

let this be equation I

r=3.456767'''''I

Multiply equation I by 100 to move the decimal point to be

between the last non recurring digit and the first recurring digit.

100r=345.676767.......

Let this be equation II

100r=345.676767......... II

Multiply equation II by 100 since digits are recurring.

10000r=34567.676767.......

Let this be equation III

10000r=34567.676767......... III

Subtract equation II from equation III

10000r=34567.676767...... III

-100r=345.676767........... II

9900r=34222

r=34222/9900

r=3 4522/9900

r=3 2261/4950

But r=3.45676767...........

Therefore 3.45676767.........=3 2261/4950

__
Exercise
3__

Express the following s fractions in their simplest form

1. a) 0.222222222''''.

b) 2.8888888888'''...

c) 0.2727272727''''.

d) 2.4545454545''''.

e) 0.315315315'''''

f) 2.815815815'''''

g) 5.26666666'''''.

h) 1.633333333'''''

i) 2.163636363'''''.

j) 2.134343434'''''.

EXERCISE 1

Using the method given in the above examples

, convert the following into fractions to their simplest form.

(1) 0.8

(2) 3.4

(3) 0.6

DECIMALS

There many different types of decimals in mathematics.

Terminating, Non Terminating and Recurring decimals

We shall deal with reccuring decimals in this lesson.

.

Decimals

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