4.03 Percentage of quantities
One of the primary uses of percentages is to communicate parts of a whole. We see this in the percentages of ingredients in the food we buy and in statistics that we use to present information. Understanding what these percentages of quantities mean is important for interpreting this information.
Fractions as percentages
We know that another way to express parts of whole is through fractions. We will use this to help us find percentages of quantities.
Some fractions that we can easily convert to and from percentages are:
We can see from the table that $100%$100% is a whole, $50%$50% is a half and $25%$25% is a quarter. When applying this to percentages of quantities, we can use these equivalences to help us understand what we are looking for.
Worked example
example 1
What is $50%$50% of $30$30?
Think: We know that $50%$50% is equivalent to a half, so we can think of the problem as "what is half of $30$30?".
Do: After translating the question to be "what is half of $30$30?", we can find half of $30$30 by dividing it by $2$2, so the solution will be:
$\frac{30}{2}=15$302=15
Reflect: By connecting the percentage to a fraction, we interpreted the question to be about finding a fraction of the whole.
We can also double check our answer by confirming that $15$15 out of $30$30 is indeed $50%$50%.
Finding other percentages
A percentage might not be one of those in the table, but it might be a multiple.
Exploration
When doing research on the popularity of different public transports in Australia, Claris found a study that claims that $65%$65% of people drive to work. Out of the $60$60 people in her workplace, how many of them can Claris expect to be driving to work?
We know that Claris wants to find $65%$65% of $60$60.
While $65%$65% does not relate to any of our simple fractions, we can notice that $65%$65% is equal to thirteen lots of $5%$5%. If we can find $5%$5% in the same way as before, we can then just multiply the answer by $13$13.
We can find $5%$5% of $60$60 by dividing it by $20$20:
$5%$5% of $60$60 $=$= $\frac{60}{20}$6020 $=$= $3$3
Then we multiply this answer by $13$13:
$65%$65% of $60$60 $=$= $13\times3$13×3 $=$= $39$39
So Claris can expect $39$39 of the people in her workplace to be driving to work.
If we can write the percentage we want to find as some multiple of a smaller percentage, we first find that smaller percentage. Then we multiply the result by the number of smaller quantities that fit into the original percentage.
Worked examples
Example 2
What is $70%$70% of $120$120?
Think: We want to break up $70%$70% into smaller percentages that are easier to find.
Do: Notice that $70%$70% is equal to seven lots of $10%$10%. Since $10%$10% of $120$120 is equal to a tenth of $120$120, we calculate:
$10%$10% of $120$120 $=$= $12$12
We then multiply our result by $7$7 to find $70%$70%:
$70%$70% of $120$120 $=$= $7\times12$7×12 $=$= $84$84.
So $70%$70% of $120$120 is $84$84.
Example 3
What is $43%$43% of $300$300?
Think: We want to break up $43%$43% into smaller percentages that are easier to find.
Do: Notice that $43%$43% is equal to forty three lots of $1%$1%. Since $1%$1% of $300$300 is equal to a hundredth of $300$300, we calculate:
$1%$1% of $300$300 $=$= $3$3
Then we multiply to find the answer:
$43%$43% of $300$300 $=$= $43\times3$43×3 $=$= $129$129.
So $43%$43% of $300$300 is $129$129.
Reflect: We found a more difficult percentage of a quantity by writing it as the multiple of an 'easier to find' percentage. After finding the easier percentage, we multiplied it by the required amount to get the more difficult percentage.
Practice question
Question 1
Suppose we want to find $28%$28% of a quantity.
Which of the following is the same as $28%$28%?
seven groups of $100%$100%
Atwenty eight groups of $100%$100%
Bseven groups of $4%$4%
Ceight groups of $4%$4%
DWhat is $4%$4% of $175$175?
Hence or otherwise, find $28%$28% of $175$175.
Finding percentages directly
In the case where we can't break up a percentage into smaller, easier to find pieces, we can always calculate the percentage directly. We can do this by converting our percentage into a fraction or decimal and applying that directly to the quantity.
For example, we can write $48%$48% of $60$60 as $\frac{48}{100}\times60$48100×60 or $0.48\times60$0.48×60.
But why does this work?
So far we have been dividing the whole to find a smaller part of it and showing that this corresponds to the percentage we were looking for, like dividing by $2$2 to find $50%$50% of a quantity. However, another way to divide by $2$2 is to multiply by $\frac{1}{2}$12 or $0.5$0.5. Notice that both $\frac{1}{2}$12 and $0.5$0.5 are equivalent to $50%$50%.
We can find percentages of quantities directly by multiplying the quantity by either the fraction or decimal equivalent to the percentage.
Worked examples
Example 4
Find $23%$23% of $40$40.
Think: We can solve this problem directly by converting $23%$23% into a decimal and multiplying the quantity of $40$40 by it.
Do: We can write $23%$23% as the decimal $0.23$0.23 which we can then multiply by $40$40:
$0.23\times40=9.2$0.23×40=9.2
So $9.2$9.2 is equal to $23%$23% of $40$40.
Example 5
Find $24%$24% of $45$45.
Think: We can solve this problem directly by converting $24%$24% into a fraction and multiplying by $45$45.
Do: We can write $24%$24% as the fraction $\frac{24}{100}$24100 which we multiply by $45$45:
| $\frac{24}{100}\times45$24100×45 | $=$= | $\frac{6}{25}\times45$625×45 |
Removing common factor of $4$4 |
| $=$= | $\frac{6}{5}\times9$65×9 |
Removing common factor of $5$5 |
|
| $=$= | $\frac{54}{5}$545 |
Multiplying |
|
| $=$= | $10\frac{4}{5}$1045 |
Writing as mixed fraction |
Reflect: In both cases, when solving the problem directly we converted the problem into a single multiplication that we evaluated to find the solution.
Notice that the solutions to the two problems above are not integers. This is because there was no application of our method that would allow us to work with integers.
The main advantage of finding a percentage of a quantity directly is that there is only one calculation to do.
Practice question
Question 2
What is $23%$23% of $16$16?
Write your answer as a decimal.
Finding percentages greater than 100%
So far, the percentages that we have found have always been below $100%$100%. As a result, the solution has always been smaller than the original quantity. This is because we have been taking a fraction of the whole.
However, if the percentage we are looking for is greater than $100%$100% then we are taking more than one whole. In other words, we should end up with more than our original quantity.
We can apply some of our calculation methods to check if this is true.
Worked example
Example 6
Find $120%$120% of $70$70.
Think: We can break up $120%$120% into twelve lots of $10%$10%, so we can use our previous method to find the solution.
Do: Since $10%$10% of $70$70 is equal to $7$7 and $120%$120% is equal to twelve lots of$10%$10%, we can calculate $120%$120% of $70$70 to be:
$12\times7=84$12×7=84
So $84$84 is equal to $120%$120% of $70$70.
Reflect: Regardless of whether the percentage we are looking for is above or below $100%$100%, we apply our methods in the same way. As expected, since we were taking a percentage greater than $100%$100%, our solution of $84$84 was greater than our original quantity of $70$70.
It is also worth noticing that both $120%$120% of $70$70 and $70%$70% of $120$120 were equal to $84$84.
When finding a percentage of a quantity, swapping the values of the percentage and the quantity does not affect the final solution.
Practice question
Question 3
What is $350%$350% of $16$16?