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2.09 Working with percentages


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?


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 learned, $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!


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 repeating 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.


From a fraction to a mixed number percentage

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 number, which is $57\frac{1}{7}$5717%. Now we can look at it right away and understand this is around $57%$57% but a tiny bit over.


Worked examples

question 1
Convert $\frac{16}{3}$163 into a percentage.


Think: We can have percentages more than $100$100


$\frac{16}{3}\times100%$163×100% $=$= $\frac{1600%}{3}$1600%3
  $=$= $533\frac{1}{3}$53313 $%$%


question 2

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

Think: We will need to be careful with rounding. Consider whether you need to round up or round down.


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

Multiply numerators

  $=$= $30.7692$30.7692 ... $%$%


  $=$= $30.77%$30.77%

Round to $2$2 decimal places


Fractions to percentages

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


Practice questions

question 3

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

question 4

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.

  1. What percentage of the article have they checked altogether?

  2. What percentage still needs to be checked?


Applications of percentages

Percentages are used for a variety of things, usually when we want to describe how much of something there is. For example, perhaps you only want $50%$50% of the juice in you cup or when the car dashboard says that the fuel tank is only $20%$20% full. However, $50%$50% of the water in a $100$100L swimming pool is obviously very different to $50%$50% of the $2$2L milk in your fridge. Let's take a look at how we can figure out how much there actually is when we hear about percentages.


Finding percentages of a quantity

We already know how to find a fraction of a quantity through multiplication. For example, we know to find $\frac{2}{3}$23 of $60$60 all we do is multiply the two numbers together, so $\frac{2}{3}\times60=40$23×60=40 is our answer. We can do the same with percentages as we know how to turn them into fractions with $100$100 as the denominator.

For example, we want to find what $25%$25% of $84$84 is, so let's multiply them together. $25%\times84$25%×84 can be rewritten as $\frac{25}{100}\times84$25100×84, and we can simplify the fraction and get $\frac{1}{4}\times84=\frac{84}{4}$14×84=844 = $21$21.

$25%\times84$25%×84 $=$= $\frac{25}{100}\times84$25100×84
  $=$= $\frac{1}{4}\times84$14×84
  $=$= $\frac{84}{4}$844
  $=$= $21$21


Practice questions


Consider the following:

  1. Express $75%$75% as a fraction in simplest form.

  2. Beth was given $20$20 minutes in which to solve a Rubik's Cube. She only needed $75%$75% of the time to finish it. How many minutes did she take?


Consider the following:

  1. Express $60%$60% as a decimal.

  2. Hence find $60%$60% of $90$90 kilograms.


A lot of the time it's hard for us to accurately calculate percentages of amounts in real life, so we'll have to estimate! Because percentages are expressed as something out of a hundred, we can also express them in diagrams of $5,10,100$5,10,100 things or more! 



Worked example

Question 7

Someone has been eating the brand new $10\times10$10×10 square block of chocolate! Can you figure out how much of the original chocolate block is left in percentages?

Think about the chocolate block as a fraction first

Do: We can see that there used to $10\times10=100$10×10=100 blocks of chocolate here, and now there are $67$67 blocks. So the fraction that represents how much is left of the original is $\frac{67}{100}$67100 . This is easily translated into a percentage as the denominator is already $100$100 , so the answer is $67%$67% .


Practice questions


Which point on the line is closest to $56%$56%?

A horizontal number line ranging from 0% on the far left to 100% on the far right. This number line is divided into equal segments, each representing a 10% interval, but only the 0% and 100% markers are labeled. Four arrows, labeled A, B, C, and D, are positioned above the line, each pointing downwards to specific intervals on the scale. (The arrow labeled A points between 20% and 30%, B's arrow points between 30% and 40%, C's arrow is between 50% and 60%, and D's arrow is between 80% and 90%. The location of the arrows is not explicitly given, so just use them as hints for the students. Remember that the students are asked which of the letters is closest to the given number.)
  1. $A$A









Ellie bought a $454$454 mL drink that claimed to be orange juice. In the ingredients list it said that orange juice made up $17%$17% of the drink. To estimate the amount of orange juice in the drink, which of the following would give the closest answer?

  1. $10%\times454$10%×454







In a census, people are asked their gender and age. The graph shows the results: the percentage of females and males in each age group.

  1. To the nearest $1%$1%, what percentage of females are between $5$5 and $9$9 years of age?






  2. To the nearest $1%$1%, what percentage of males are between $30$30 and $34$34 years of age?






  3. The percentage of females between the ages of $20$20 and $29$29 is about:






  4. The percentage of males below $20$20 years of age is about:











Use ratio and rate reasoning to solve real-world and mathematical problems, e.g. By reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.


Find a percent of a quantity as a rate per 100 (e.g. 30% Of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent.

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