Express each repeating decimal as a fraction. a) 5.23 b) 1.37 c) 2.135

Question

Express each repeating decimal as a fraction.  a) 5.23  b) 1.37  c) 2.135

giskacu · Accepted Answer

Step 1
Definition used -
Converting repeating decimal to fraction:
First, we will equate the given decimal number to x and subtract the integer part from it, then multiply by 10, 100, or 1000 on both sides according to repeating digits. We solve for x and simplify fractions.
Step 2
Given:
a) 5.23
Let $x = 5.23$
$x - 5 = 0.232323$ .... (1)
$100 (x - 5) = 23.232323$ .... (2)
Subtracting equation (1) from the equation (2)
$99 (x - 5) = 23$
$x - 5 = \frac{23}{99}$
$x = 5 + \frac{23}{99}$
$= \frac{518}{99}$
Step 3
b) 1.37
Let $x = 1.37$
$x - 1.3 = 0.0777$ .......
$10 (x - 1.3) = 0.7777$ .......... (1)
$100 (x - 1.3) = 7.777$ .......... (2)
Subtracting equation (1) from equation (2)
$90 (x - 1.3) = 7$
$x - 1.3 = \frac{7}{90}$
$x = 1.3 + \frac{7}{90}$
$= \frac{124}{90}$
$= \frac{62}{45}$
Step 4
c) Let $x = 2.135$
$x - 2.1 = 0.03535$ ....
$10 (x - 2.1) = 0.3535$ ........ (1)
$1000 (x - 2.1) = 35.3535$ ...... (2)
Subtracting equation (1) from equation (2)
$900 (x - 2.1) = 35$
$x - 2.1 = \frac{35}{990}$
$x = 2.1 + \frac{35}{990}$
$= \frac{2114}{990}$
$= \frac{1057}{495}$

Express each repeating decimal as a fraction. a) 5.23 b) 1.37

Answered question

Answer & Explanation

New Questions in Pre-Algebra