Question

What is 0.4 repeating as a fraction?

Answer 1

Hint: We need to convert \[0.44.....\] (4 repeating) as a fraction, we have a particular method to convert a repeating decimal into a fraction. First, we will consider this decimal as \[x\] and this will serve as an equation. Then we will multiply it with 10 to obtain another equation. We will subtract these two equations, by solving these equations we get the required result.

Complete step by step answer:
Now let us follow the step-by-step method.
Let us assume that
\[x=0.44....\]
Now we will multiply both sides with 10 and get
\[\begin{align}
  & 10x=10\times 0.44.... \\
 & 10x=4.44.....\to \to \left( i \right) \\
\end{align}\]
So, we have our assumption and equation (i) at hand.
Now, we will subtract them to get rid of the repeating part.
\[ 10x-x=4.44… - 0.44…\]
\[\Rightarrow 9x=4\]
Taking \[9\] to the right hand side, we get value of \[x\] as
\[\Rightarrow x=\dfrac{4}{9}\]
Since we had assumed \[x=0.44...\] therefore, we can say that it is also the same as \[x=\dfrac{4}{9}\].

Note: Let us see what is repeating decimal, these are the decimals in which any particular number of digits keeps on repeating for infinite times, there is another term also for repeating numbers that is recurring numbers, note that repeating numbers are rational numbers only because they can be converted into fractional form. We have a single-digit repeated that is 4, so we can also multiply by 10 and then 100. So, we will get another set of equations which can then be subtracted to get the value of x. So, even if we compute using this way, we will get the same value of x.