Question

What is 1.6 repeating as a fraction?

Answer 1

Hint: To find the fraction of repeating decimal 1.6, we have to equate the repeating decimal to x. We have to consider only a few positions of the decimals. This will be equation (i). We have to place this repeating digit to the left of the decimal point. This is done by multiplying both sides by 10 to the required power. This will be equation (ii). Then we have to subtract equation (i) from (ii) and solve for x.

Complete step by step solution:
We have to find the fraction of repeating decimal 1.6. Repeating decimals are the one, which has a set of terms in decimal to be repeated uniformly. We can write repeating 1.6 as 1.66666…, that is, the number after the decimal repeats. This can also be written as $1.\bar{6}$ .
To convert a repeating decimal into fraction, we have to equate the repeating decimal to x. Let us consider only a few positions of the decimals. Hence, we can write
$x=1.666666...\left(i\right)$
We know that the repeating digit is 6. We have to place this repeating digit to the left of the decimal point. For this, we have to multiply 1.666666 with 10. When we multiply a number on one side, we have to multiply it on the other side also.
$\begin{array}{rl}& \Rightarrow 10x=1.666666\times 10\\ & \Rightarrow 10x=16.66666...\left(ii\right)\end{array}$
Let us subtract (i) from (ii).
$\begin{array}{rl}& \Rightarrow 10x-x=16.66666-1.666666\\ & \Rightarrow 9x=15\end{array}$
Let us take 9 from LHS to RHS to find the value of x.
$\Rightarrow x={\displaystyle \frac{15}{9}}={\displaystyle \frac{5}{3}}$

Hence, the fraction of repeating 1.6 is ${\displaystyle \frac{5}{3}}$.

Note: Students must be aware that they must subtract equation (i) from (ii) not (ii) from (i). We can also do this problem in an alternate method.
The fraction of repeating decimal of the form $0.\overline{abcd}$ is given by ${\displaystyle \frac{\text{Repeated term}}{\text{Number of 9 }{}^{\prime }\text{ s for the repeated term}}}+\text{Number before decimal}$
We have 1.6. The repeating term is only 6. Therefore, the number of 9’s for the repeated term will be only one 9 since only one repeating term is there.
Therefore, fraction of repeated 1.6 is ${\displaystyle \frac{6}{9}}+1={\displaystyle \frac{2}{3}}+1={\displaystyle \frac{5}{3}}$