Question

What is 1.3(3 repeated) as a fraction?

Answer 1

Hint: We must first check how many digits are being repeated in the given number, and if the number of such digits is n, we can multiply the number by ${10}^n$. By subtracting this new number with the original given number, we can get an equation free from repeating parts. Thus, we can solve further to convert it into fractions.

Complete step by step solution:
We know that a specific type of decimal numbers, in which the digits after the decimal point keep on repeating endlessly, are called non terminating, repeating decimals.
We can express the number 1.3 (3-repeated) as 1.333333… We represent these repeating decimals with a bar above the repeating number. So, 1.3 repeating can be expressed as $1.\bar{3}$.
We know that a division of 10 results in the shifting of the decimal point on the left side, and a multiplication by 10 results in the shifting of the decimal point on the right side.
So, let us assume $x=1.\bar{3}...\left(i\right)$
Let us multiply equation (i) by 10 on both sides. Thus, we get
$10x=10\times 1.\bar{3}$
By using the concept above, we get
$10x=13.\bar{3}...\left(ii\right)$
Now, subtracting (i) from (ii), we get
$10x-x=13.\bar{3}-1.\bar{3}$
Thus, on evaluating the above equation, we get
$9x=12$
We can now divide both sides of this equation by 9, to get
${\displaystyle \frac{9x}{9}}={\displaystyle \frac{12}{9}}$
So, now we have
$x={\displaystyle \frac{12}{9}}$
Both 12 and 9 have 3 as a common factor. So, by cancelling the factor from both 12 and 9, we get
$x={\displaystyle \frac{4}{3}}$
Hence, we can say that the non terminating repeating decimal $1.\bar{3}$ is equal to the fraction ${\displaystyle \frac{4}{3}}$.

Note: Here, in this question, we must note that in $1.\bar{3}$, only one digit 3 is repeating. If the number of repeating digits would have been n, then instead of multiplying by 10, we would have multiplied by ${10}^n$. We should know that repeating decimals are also called recurring decimals.