Question

How do you convert $0.345$ ($345$ being repeated) to a fraction?

Answer 1

Hint:To solve this problem, we first assume $0.345$ ($345$ being repeated) as a variable $x$.Since $x$ is recurring in 3 decimal places, we multiply it by 1000. Finally, we divide both sides by 999 to get $x$ as a fraction.

Complete step by step answer:
To convert the given number to a fraction form, we will first assume the number $0.345$ ($345$ being repeated) as a variable $x$.
$x = 0.345$
It is given that $345$ being repeated which means three decimal places are repeating.Therefore, we multiply it by 1000.
$1000x = 345.345$
Now, we will subtract these two equations.
$1000x - x = 345.345 - 0.345 \\
\Rightarrow 999x = 345$
We will now divide both the sides by 999 to obtain the fraction form.
$ \Rightarrow x = \dfrac{{345}}{{999}}$
We can see that both numerator and denominator are divisible by 3.When we divide 345 by 3, we get 115.When we divide 999by 3, we get 333.
$ \therefore x = \dfrac{{115}}{{333}}$

Thus, the fraction form of $0.345$ ($345$ being repeated) is $\dfrac{{115}}{{333}}$.

Note:As we have seen in this problem, all recurring decimals can be represented as fractions. To find this fraction we need to generate two equations which have the same repeating part and subtracting one from the other to eliminate it. The general steps to convert the recurring decimals into the fraction form are:
Step 1: Let $x$ be recurring decimal in expanded form.
Step 2: Let the number of recurring digits be $n$.
Step 3: Multiply recurring decimal by ${10^n}$ .
Step 4: Subtract the first equation from the second equation to eliminate the recurring part.
Step 5: Solve for $x$, expressing your answer as a fraction in its simplest form.