# Benchmark Percentages & Fractions

Lesson

Percentages and fractions are part of our everyday lives, but did you know you can write percentages and fractions, and fractions as percentages? For example, you probably know that $50%$50% is the same as a half, or $\frac{1}{2}$12, but WHY?

## From Percentages to Fractions

Every percentage can be thought of as a fraction with a denominator of $100$100. In fact, that's what the percent sign means! Doesn't it look like a strange mixed up little $100$100, or even a fraction with a $0$0 on top and and a $0$0 on bottom? Even cooler is the fact that the word percent actually comes from per centum, which is Latin for per one hundred! For example, $3%$3% would mean $3$3 per $100$100, which is a fancy way of saying $3$3 out of $100$100. This is why we can write it as the fraction $\frac{3}{100}$3100, which is ALSO like saying $3$3 out of $100$100

So to convert any percentage to a fraction all you have to do is to take the number in front of the percent sign and put it as the numerator of a fraction with a denominator of $100$100, or in other words, divide by $100$100.

But how did we go from $50%$50% to $\frac{1}{2}$12? Well, using what we just learnt, $50%=\frac{50}{100}$50%=50100. Can you see that we can simplify this fraction by dividing top and bottom by $50$50? $50\div50=1$50÷​50=1, and $100\div50=2$100÷​50=2, so $\frac{50}{100}=\frac{1}{2}$50100=12, voila!

## From Fractions to percentages

Getting back is a little more difficult but we just have to remember that all percentages are fractions with $100$100 as the denominator. For example, to find $\frac{4}{5}$45 as a percentage, we're really just finding $\frac{4}{5}$45ths of $100%$100% - which is one whole - and we know to find a certain fraction of another we just multiply them together. $\frac{4}{5}\times100%=\frac{4\times100%}{5}$45×100%=4×100%5 = $\frac{400%}{5}=80%$400%5=80%

## Special cases

$33\frac{1}{3}$3313% and $66\frac{2}{3}$6623% are special percentages, can you guess what they'll be as fractions? Try and put $\frac{1}{3}$13 and $\frac{2}{3}$23 into your calculator and seeing what decimal it becomes! Now try putting those percentages in! That's right, all four values turn into one of two recurring decimals $0.3333$0.3333... and $0.6666$0.6666... So it's important to remember that $33\frac{1}{3}$3313% = $\frac{1}{3}$13 and $66\frac{2}{3}$6623% = $\frac{2}{3}$23, and later you'll learn why that's so when you encounter these strange decimals.

On sale now!

You might have seen percentages in a lot of shops and markets when there're special sales and deals. Have a look at the picture below and try converting them into fractions!

#### Worked Examples

##### QUESTION 1

Convert $90%$90% into a fraction. Give the fraction in simplest form.

##### QUESTION 2

Convert $\frac{5}{10}$510 into a percentage.

##### QUESTION 3

A student survey found that $\frac{1}{5}$15 of students play Saturday sport, and that $\frac{3}{5}$35 of students play Sunday sport.

1. What percentage of students play a weekend sport?

2. What percentage of students do not play weekend sport?