Percentages are useful measures because they tell us how much of the whole we currently have. If we have $50%$50% then we have half the amount. If we have $100%$100%, what we have is equal to the amount. And if we have $200%$200% then we have double that amount.
But hold on... How can we have double the whole?
How much is 'the whole'?
When looking at percentages, the 'whole' is our reference number that our percentage amount refers to.
For example: $14$14 is equal to half of $28$28, so $14$14 is $50%$50% of $28$28.
But also: $14$14 is equal to double of $7$7, so $14$14 is $200%$200% of $7$7.
In both cases our amount is $14$14. The numbers that change are the percentage value and our reference number- the whole.
Practice question
Question 1
$20%$20% of a quantity is the same as:
one twentieth of that quantity
Aone fifth of that quantity
Bone tenth of that quantity
Cone quarter of that quantity
D
Using fractions to find the whole
Let's consider the case where we don't know the whole, or reference number:
$14$14 is $50%$50% of some number. What is this number?
To work out the whole, we can use the percentage value to figure out how much of the whole we currently have. Since $50%$50% is equal to $\frac{1}{2}$12 when written as a fraction, we know that $14$14 is half of our missing number. If we let $x$x represent our missing number, we can express this information as the equation:
$14=x\times\frac{1}{2}$14=x×12
We can solve for $x$x by multiplying both sides of the equation by $2$2, and find that $x=28$x=28, so we know the 'whole' we are looking for is $28$28.
We can apply this same technique to any amount and percentage, as shown below:
Worked Example
$39$39 is $30%$30% of some number. What is this number?
Think: This tells us that some number, let's call it $x$x, multiplied by the fraction equivalent to $30%$30% will be equal to $39$39.
Do: We can convert $30%$30% into the fraction $\frac{3}{10}$310 and express our information as the equation:
$39=x\times\frac{3}{10}$39=x×310
We can solve for $x$x by multiplying both sides of the equation by $\frac{10}{3}$103 and then simplify the result to find $x=130$x=130.
This means the number we are looking for is $130$130.
Practice question
Question 2
$20%$20% of a number is equal to $7$7. What is the number?
The fraction method is quick and easy for percentages that convert into easy-to-use fractions. But what about percentages that don't give us nice fractions?
The unitary method
Let's look at the worked example from before again:
$39$39 is $30%$30% of some number. What is this number?
Another approach to solving this would be to find $1%$1% of the number first. From there we can find the whole, or indeed any other percentage amount.
We know that $30%$30% of the number is $39$39, so if we divide $39$39 by $30$30, we now know $1%$1% of the number. Multiplying this amount by $100$100 will then tell us $100%$100% or the whole of the amount.
$x=\frac{39}{30}\times100$x=3930×100$=$=$1.3\times100$1.3×100$=$=$130$130
This is known as the unitary method.
We can find the whole by:
- Dividing the given amount by the percentage number. This finds $1%$1% of the whole.
- Multiplying the result by $100$100. This will give us $100%$100% of the whole which is the number we are looking for.
Since the order of operations allows us to perform either multiplication and division in any order, we can do whichever operation is easier first.
Worked Example
$33.9$33.9 is equal to $113%$113% of some number. What is this number?
Think: Since we cannot convert $113%$113% into a nice fraction, we can use the unitary method.
Do: Following the two steps of the unitary method, we get:
| $33.9\div113=0.3$33.9÷113=0.3 | Divide the amount by the percentage number to find $1%$1% | ||
| $0.3\times100=30$0.3×100=30 | Multiply the result by 100 to find the whole |
As such, we find that the number we are looking for is $30$30.
The unitary method is particularly useful as it can be used to find the whole using any amount and percentage.
Practice question
Question 3
$9%$9% of a number is $72$72.
Fill in the blanks:
Because $9%$9% of the number is $\editable{}$, we know that $1%$1% of the number is $\editable{}$.
What is the number?