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 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!
$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.
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.
Think: We 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: We will need to be careful with rounding. Consider 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 
Fraction → Percentage: multiply by $100%$100% then simplify
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?
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.
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 
Consider the following:
Express $75%$75% as a fraction in simplest form.
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:
Express $60%$60% as a decimal.
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!
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% .
Which point on the line is closest to $56%$56%?
$A$A
$C$C
$D$D
$B$B
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?
$10%\times454$10%×454
$20%\times454$20%×454
$10%\times400$10%×400
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.
To the nearest $1%$1%, what percentage of females are between $5$5 and $9$9 years of age?
$7%$7%
$2%$2%
$11%$11%
To the nearest $1%$1%, what percentage of males are between $30$30 and $34$34 years of age?
$7%$7%
$4%$4%
$2%$2%
The percentage of females between the ages of $20$20 and $29$29 is about:
$15%$15%
$7%$7%
$25%$25%
The percentage of males below $20$20 years of age is about:
$15%$15%
$10%$10%
$30%$30%
$50%$50%
Use ratio and rate reasoning to solve realworld 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.