CONVERTING REPEATING DECIMALS TO FRACTIONS

Convert the following recurring decimals to fractions :

Problem 1 :

0.333….

Solution :

Let x = 0.333… ---> (1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 0.333….

10x = 3.333… --- > (2)

From (2) - (1)

10x - x = 3.333… - 0.333…

9x = 3

`x = 3/9

x = 1/3

So, 0.333… = 1/3

Problem 2 :

0.444….

Solution :

Let x = 0.444… ---> (1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 0.444….

10x = 4.444… ---> (2)

From (2) - (1)

10x - x = 4.444… - 0.444…

9x = 4

`x = 4/9

So, 0.444… = 4/9

Convert the following recurring decimals to fractions:

Problem 3 :

1.0909….

Solution :

Let x = 1.0909… ---> (1)

Since two digits are repeating, we will multiply by 100.

100x = 100 × 1.0909….

100x = 109.09… ---> (2)

From (2) - (1)

100x - x = 109.09… - 1.0909…

99x = 108

`x = 108/99

So, 1.0909… = 108/99

Problem 4 :

2.0909….

Solution :

Let x = 2.0909… --->(1)

Since two digits are repeating, we will multiply by 100.

100x = 100 × 2.0909….

100x = 209.09… ---> (2)

From (2) - (1)

100x - x = 209.09… - 2.0909…

99x = 207

`x = 207/99

So, 2.0909… = 207/99

Problem 5 :

0.5333….

Solution :

Let x = 0.5333… ---> (1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 0.5333….

10x = 5.333… ---> (2)

From (2) - (1)

10x - x = 5.333… - 0.5333…

9x = 4.8

`x = 4.8/9

x = 4.8 × 10/9 × 10

x = 48/90

So, 0.5333… = 48/90

Problem 6 :

1.7333….

Solution :

Let x = 1.7333… ---> (1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 1.7333….

10x = 17.333… ---> (2)

From (2) - (1)

10x - x = 17.333… - 1.7333…

9x = 15.6

`x = 15.6/9

x = 15.6 × 10/9 × 10

x = 156/90

So, 1.7333… = 156/90

Problem 7 :

0.9444….

Solution :

Let x = 0.9444… ---> (1)

10x = 10 × 0.9444….

10x = 9.444… --- > (2)

From (2) - (1)

10x - x = 9.444… - 0.9444…

9x = 8.5

`x = 8.5/9

x = 8.5 × 10/9 × 10

x = 85/90

So, 0.9444… = 85/90

Problem 8 :

2.0555…

Solution :

Let x = 2.0555… --- > (1)

10x = 10 × 2.0555….

10x = 20.555… --- > (2)

From (2) - (1)

10x - x = 20.555… - 2.0555…

9x = 18.5

`x = 18.5/9

x = 18.5 × 10/9 × 10

x = 185/90

So, 2.0555… = 185/90

Problem 9 :

a) Write 1/7 as a repeating decimal. How many digits repeat? 

b) Write the fractions

2/7, 3/7, 4/7, 5/7 and 6/7

in decimal form. What patterns do you see? Explain how the circle of digits can help you write these fractions as decimals.

Solution :

a) 1/7 = 0.142857142857 ................

The repeating digits are 142857. So six digits are repeating.

b) Other fractions and patterns 

• 2/7: 0.285714 (starts at '2' in the cycle).
• 3/7: 0.428571 (starts at '4' in the cycle).
• 4/7: 0.571428 (starts at '5' in the cycle).
• 5/7: 0.714285 (starts at '7' in the cycle).
• 6/7: 0.857142 (starts at '8' in the cycle). 

Patterns and the "Circle of Digits" 

• Pattern: All fractions (1/7 to 6/7) use the exact same six digits (1, 4, 2, 8, 5, 7) in their repeating cycle, just starting at a different point.

Problem 10 :

0.5454........ x 0.555...........

Solution :

0.5454........ x 0.555...........

Let x = 0.5454........ -----(1)

Since two digits are repeating, we have to multiply both sides by 100.

100x = 54.54.......... -----(2)

(2) - (1)

99x = 54.5454.......... - 0.5454............

99x = 54

x = 54/99

x = 18/33

x = 6/11

Let x = 0.555555........ -----(1)

Since two digits are repeating, we have to multiply both sides by 10.

10x = 5.555555......... -----(2)

(2) - (1)

9x = 5.55555.... - 0.55555......

9x = 5

x = 5/9

Multiplying the fraction form of each repeating decimals, we get

= (6/11)(5/9)

= 10/33

So, the answer is 10/33.

x = 18/33

x = 6/11