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Question 16

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Number 16

The fractional form of the repeating decimal $3.15931593...$ is?

$\frac{1170}{37}$

$\frac{3510}{1111}$

$\frac{1177}{111}$

$\frac{1170}{999}$

$\frac{1177}{999}$

Explanation

Let $x = 3.15931593...$ .

Since there are $4$ repeating digits (which are $1593$ ), we multiply $x$ by $10000$ :

10000x = 31593.15931593...

Next, we subtract the original equation $x$ from this equation:

\begin{aligned} 10000x &= 31593.1593... \\ x &= \phantom{3159}3.1593... \\ \hline 9999x &= 31590 \end{aligned}

Thus, we obtain:

x = \frac{31590}{9999}

We simplify the fraction by dividing both the numerator and the denominator by $9$ :

x = \frac{31590 \div 9}{9999 \div 9} = \frac{3510}{1111}

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