Question

How do you convert $0.16(16{\text{ being repeated)}}$ to a fraction?

Answer 1

Hint: Here we can proceed by taking $0.16$ as any variable. As $x$ is recurring in two decimal places, we multiply it by ${10^2} = 100$ and then subtract the originally let equation, and this equation forms. In this way, we can solve this problem.

Complete step-by-step answer:
Here we are given the repeating number as $16$ in $0.16$ which means that it is a repeating decimal which is actually $0.1616161616......$ and we need to find the fraction whose result gives us such a recurring number in decimal.
So we can solve such problems by letting the given number be a variable.
So let $x = 0.16 - - - - - - (1)$
Now we need to notice the number of digits that are there that are repeating or recurring after the decimal and if they are equal to $n$ we need to multiply the equation (1) by ${10^n}$
So here we can see that in the given decimal we have two digits after the decimal that are repeating which is $16$ so we need to multiply the equation (1) with the value which is ${10^n}$ and here $n = 2$
So we will get the multiplication factor as ${10^2} = 100$ and we need to keep in mind that we need to keep the same repeating number after the decimal even after multiplication.
So multiplying equation (1) by $100$ we get:
$
  100x = 0.1616(100) \\
  100x = 16.16 - - - - - (2) \\
 $
Now subtracting equation (1) from (2) we will get our desired fraction that would give us the decimal
Which is $0.16(16{\text{ being repeated)}}$ upon solving.
$
  100x - x = 16.16 - 0.16 \\
\Rightarrow 99x = 16 \\
\Rightarrow x = \dfrac{{16}}{{99}} \\
 $
Hence we get $x = \dfrac{{16}}{{99}}$ as the fraction which on solving will give us the decimal answer $0.16(16{\text{ being repeated)}}$

Note: Here in these types of problems the student must keep in mind that we need to keep the same repeating number after the decimal as we have done here. We did not write that term after multiplication as $16.00$ but we wrote it as $16.16$ because we need to get an integer value after the subtraction.