Percentages

Lesson

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

$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.

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.

**Convert **$\frac{16}{3}$163 into a percentage

**Remember **that you can have percentages more than $100$100

**Do:**

$\frac{16}{3}\times100%$163×100% | $=$= | $\frac{1600%}{3}$1600%3 |

$=$= | $533\frac{1}{3}$53313 $%$% |

**Express **$\frac{4}{13}$413 as a percentage, rounded to $2$2 decimal places

**Think **about whether you need to round up or round down

**Do**

$\frac{4}{13}\times100%$413×100% | $=$= | $\frac{400%}{13}$400%13 | multiply numerators |

$=$= | $30.7692$30.7692 ... $%$% | evaluate | |

$=$= | $30.77%$30.77% | round to $2$2 decimal places |

Fractions to Percentages

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

Convert $\frac{3}{4}$34 into a percentage.

Xanthe and Jimmy are spellchecking an article before it is printed. Xanthe checks $\frac{3}{5}$35 of the article and Jimmy checks $34%$34% of the article.

What percentage of the article have they checked altogether?

What percentage still needs to be checked?