Benchmark Percentages & Fractions

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}$45​ths 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

QUESTION 2

QUESTION 3