Question

Express the following in p\q form.

1. 0.3187(178 is recurring).

2. 32.1235( 35 is reccuring).

3. 0.407 (7 is recurring).

Answer 1

1.

let x=0.3178(178 bar)

x=0.3178178....----------(1)

Multiply 1000 on both sides as its periodicity is 3

1000*x=1000*0.3178178..

1000x=317.8178...------------(2)

(2)-(1)

1000x - x=317.8178...- 0.3178...

999x = 317.5

x = 317.5/999

x = 3175/9990

2.

Let x = 32.12353535...(1)

Periodicity = 2

Multiply both sides by 100

=> 100x = 3212.353535......(2)

Now subtract (1) from (2) equation

=> (100x = 3212.353535...) - ( x = 32.12353535...)

=> 99x = 3018

=> x = 3018/99

=> x = 1006/33


3.

To find out what is 0.407… as a fraction

x = 0.40777...

10x = 4.0777...

10x - x = (4.0777...) - (0.40777...)

9x = 3.67

x = 3.67/9

x = 367/900