Percentages

UK Secondary (7-11)

Percentages as Fractions

Lesson

Percentages and fractions are part of our every day lives, but did you know you can write percentages as 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?

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!

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!

You know that we can have improper fractions, which are fractions that are more than a whole, so does that mean we can have 'improper' percentages as well? Of course! Things such as $104%$104%, $560%$560% or even $1000000%$1000000% are all more than a whole, which is $100%$100%, and they all exist!

If we took one pizza as one whole, then here we have $275%$275%, as we have $2$2 whole pizzas plus $75%$75%of a third pizza.

But what does something like $104%$104% mean?

Well if we convert it into fractions we'll get $\frac{104}{100}=\frac{26}{25}$104100=2625

which is $1\frac{1}{25}$1125 as a mixed number, and we know this means $1$1 whole plus $\frac{1}{25}$125!

We can then just as easily compare percentages more than $100%$100% with fractions.

**Express **$5%$5% as a simplified fraction

**Think **about how to get rid of that decimal point in the without changing the answer

**Do: **

$\frac{5%}{100%}$5%100% | $=$= | $\frac{5}{100}$5100 | |

$=$= | $\frac{1}{20}$120 |

**Question: What is** $12\frac{4}{5}$1245 $%$% as a fraction?

**Think **about changing the mixed number into something simpler first

**Do**

$12\frac{4}{5}$1245 $%$% | $=$= | $\frac{64%}{5}$64%5 | turn to improper fraction |

$=$= | $\frac{64%}{5}\div100%$64%5÷100% | to change to fraction $[/]$[/] $100%$100% | |

$=$= | $\frac{64}{5}\div100$645÷100 | $%$%signs cancel out | |

$=$= | $\frac{64}{5}\times\frac{1}{100}$645×1100 | change to multiplication of reciprocal | |

$=$= | $\frac{64}{500}$64500 | evaluate | |

$=$= | $\frac{16}{25}$1625 | simplify |

Percentages to Fractions

Percentage → Fraction: divide by $100%$100%

Express $784%$784% as a fraction. Give your answer as a simplified mixed number.

Express $0.67%$0.67% as a fraction in simplest form.