Percentages

UK Secondary (7-11)

Fractions and Percentages I

Lesson

We've already learnt before how to convert fractions to percentages and back again, but now it's time to have a look at some more complicated percentages.

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 fraction, 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.

Sometimes we'll have a very confusing percentages with decimals and fractions already IN them, like $3.4%$3.4% or $40\frac{2}{7}$4027%.

So converting them into fractions It's important not to treat them any differently and still apply the technique of **dividing by 100%**.

$3.4%\div100%=\frac{3.4}{100}$3.4%÷100%=3.4100 which is the same as $\frac{34}{1000}$341000 (remember we can't have decimals in fractions!). We can then take out $2$2 from the numerator and denominator and simplify the fraction to get $\frac{17}{500}$17500.

It's a little trickier with mixed fraction percentages but let's see if we can convert them into fractions.

$40\frac{2}{7}$4027 $%$% divided by $100%$100% | $=$= | $40\frac{2}{7}\div100$4027÷100 |

$=$= | $\frac{282}{7}\div100$2827÷100 | |

$=$= | $\frac{\frac{282}{7}\times1}{100}$2827×1100 | |

$=$= | $\frac{282}{700}$282700 |

So now we can see that it's as easy as converting the mixed fraction into the improper fraction $\frac{282}{7}$2827 and multiplying the denominator by $100$100. But we have not simplified our answer! So $\frac{282}{700}$282700 can be rewritten as $\frac{141}{350}$141350 if we take $2$2 out from the top and the bottom.

Let's see what happens when we try to convert a fraction that's doesn't convert to a whole number when represented as percentage, for example $\frac{4}{7}$47. Of course let's first follow the usual steps to multiply it by $100%$100% to convert into a percentage. $\frac{4}{7}\times100%=\frac{400%}{7}$47×100%=400%7. Because this is a improper fraction percentage, it's hard to understand it when looking at it straight away, that's why it'll be easier to change it into a mixed fraction, which is $57\frac{1}{7}$5717%. Now we can look at it straight away and understand this is around $57%$57% but a tiny bit over.

Try and see if you can do the next one without looking at the worked example:

**Express **$11\frac{5}{9}$1159% as a fraction

**Think** of treating it as a normal conversion problem first, then try to deal with the mixed fraction part

**Do: **

$11\frac{5}{9}\div100$1159÷100 | $=$= | $\frac{104}{9}\div100$1049÷100 | convert to improper fraction |

$=$= | $\frac{104}{9}\times\frac{1}{100}$1049×1100 | dividing by a fraction is the same as multiplying by its negative reciprocal | |

$=$= | $\frac{104\times1}{9\times100}$104×19×100 | ||

$=$= | $\frac{104}{900}$104900 | simplifying by removing the common factor of 4 | |

$=$= | $\frac{26}{225}$26225 |

Express $49%$49% as a fraction over $100$100.

Express the fraction $\frac{12}{20}$1220 as a percentage.

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.