# How do you write the repeating decimal 0.175 where 175 is repeated as a fraction?

Oct 24, 2016

In mathematical exactitude, the value presentation is

${10}^{- 3} \frac{1}{1 - {10}^{- 3}} 175$

#### Explanation:

In mathematical exactitude, the value presentation is

${10}^{- 3} \frac{1}{1 - {10}^{- 3}} 175$

I have written this from the general formula

$V = v + {10}^{- m} / \left(1 - {10}^{- n}\right) P$ given in my answer to a similar

question, earlier. See

Here, V = value, v= value prefixing the repeating digits string, 10^(-m)

is the place value of the least significant digit (lsd ) in the

repeatend (reptend) in the infinitely-long string and n is the

number of digits in the reptend string P ( called Period )..

In this problem, v=0, m=3, n=3 and P=175.

Oct 24, 2016

$\frac{175}{999}$

#### Explanation:

We require to establish 2 equations with the same repeating fraction and subtract them to eliminate the repeating fraction.

Let $x = 0. \overline{175} \ldots \ldots \ldots . \left(A\right)$

The bar above the digits 175 denotes 175 is repeated.

To obtain the same repeating fraction after the decimal point, we require to multiply (A) by 1000.

$\Rightarrow 1000 x = 175. \overline{175} \ldots \ldots \ldots . \left(B\right)$

Now subtract (A) from (B)

$\Rightarrow 999 x = 175$

$\Rightarrow x = \frac{175}{999} \leftarrow \text{ in simplest form}$