Question

How do you find the repeating decimal $0.45$ with $45$ repeated as a fraction?

Answer 1

Hint : While converting decimal into fraction, make sure you evaluate all the decimal terms properly and then only multiply by $10$ to the numerator as well as denominator. While converting to percentage make sure you multiply by $100\%$ to the given term for the percentage. While reducing the terms, reduce them until they cannot be reduced any further.

Complete step-by-step answer :
We will start off by indicating a repeating decimal by placing a bar over the repeating pattern.
$0.\overline {45} = 0.454545...$
Now with that notation, we will what happens when we multiply $0.\overline {45}$ by $(100 - 1)$ .

$${(100 - 1)0.\overline {45} } = {100 \times {0.454545} - 1\times {0.454545} } \\\ {} = {45.\overline {45} - 0.\overline {45} } \\\ {} = {45} \;$$

So here, $100$ shifted our original decimal representation $2$ places to the left – the length of the repeating pattern.
Then subtracting the original cancelled out the repeating tail. Now further, we will divide both sides by $(100 - 1)$ and then simplify.
$= 0.\overline {45} \\\ = \dfrac{{45}}{{100 - 1}} \\\ = \dfrac{{45}}{{99}} \\\ = \dfrac{{5 \times 9}}{{11 \times 9}} \\\ = \dfrac{5}{{11}} \;$
So, the correct answer is “ $\dfrac{5}{{11}}$ ”.

Note : Percentage is always based off of $100\%$ , so decimals up to the hundredths place will always be integers, and the numbers after the hundredths place will be after the decimal. The tens place in the percent will always be in the tenths place in decimal form as well. To convert from decimals to percentage, you will multiply by $100$ , which gives you the percentage equivalent of the percent. While converting from decimal to fraction, we will multiply both the numerator and denominator by $10$ .