2.02 Quantities as percentages
Percentages are useful because $1%$1% always represents $1$1 part out of $100$100. This helps us compare different quantities out of different wholes because, after we convert them to percentages, we can easily compare them.
Finding parts of a whole
Before we convert quantities into percentages, we first need to know what two quantities we are comparing. A good place to start is with fractions.
Exploration
Alice and Martin both take part in an archery competition.
Alice takes $25$25 shots and hits the target $18$18 times.
Martin takes $10$10 shots and hits the target $7$7 times.
Alice claims victory - she hit the target more times. But Martin says he was more accurate because he missed fewer shots. To settle their argument, they decide to compare their accuracies using percentages.
Since Alice hit the target $18$18 times out of $25$25, her accuracy can be expressed as the fraction $\frac{18}{25}$1825.
Similarly, Martin's accuracy can be expressed as the fraction $\frac{7}{10}$710.
We get these fractions by comparing the total number of shots they took, written as the denominator, to the number of shots they hit, written as the numerator.
By converting these fractions into percentages we find that Alice's accuracy as a percentage was $72%$72% while Martin's accuracy was $70%$70%, so Alice was the more accurate of the two.
Before they used percentages, both of them were wrong.
Alice's only compared the total number of hits, while Martin only compared the total number of misses. Using percentages means they compare both hits and misses at the same time.
It should be noted that, in this case, $100%$100% accuracy would mean hitting the target with every shot. When represented as a fraction this would be $\frac{25}{25}$2525, which is one whole.
When finding quantities as percentages, we always want to compare our quantity to the whole that it is a part of.
When writing this as a fraction, we always put the quantity as the numerator and the whole as the denominator.
Practice question
Question 1
Which of the following shows how to calculate $15$15 as a percentage of $31$31?
$\frac{15}{31}\times10%$1531×10%
A$\frac{15}{31}\times1$1531×1
B$\frac{15}{31}\times100%$1531×100%
C$\frac{15}{100}\times31%$15100×31%
D
Finding quantities as percentages
Now that we know that finding quantities as percentages requires us to make the correct fraction, the rest of the steps become very familiar to us. This is because, once we have our fraction, all we need to do is convert our fraction into a percentage.
Worked examples
Example 1
Write $12$12 out of $40$40 as a percentage.
Think: Our quantity is $12$12 and the "whole" that it is out of is $40$40. We want to first express this as a fraction and then convert it into a percentage.
Do: Since the quantity is $12$12 and the whole is $40$40, we can write this as the fraction:
$\frac{12}{40}$1240
There is a common factor of $4$4 between the numerator and denominator, and after we cancel that factor the fraction becomes:
$\frac{3}{10}$310
Converting this into a percentage gives us:
$30%$30%
So the quantity $12$12 out of $40$40 is equal to $30%$30% of the whole.
Example 2
There are $20$20 koi carp in a pond. If $13$13 of the koi carp are orange, what percentage of the koi carp are not orange?
Think: To start by writing this as a fraction, we need to know what numbers represent the quantity and the whole. How many koi carp are there in total and how many of them are not orange?
Do: There are $20$20 koi carp in total so this will be our whole. Since $13$13 of them are orange, $7$7 of them will be "not orange". This is the quantity we are interested in, and we write it as a fraction:
$\frac{7}{20}$720
Converting this into a percentage gives us:
$35%$35%
So exactly $35%$35% of the koi carp are not orange.
Reflect: When finding quantities as percentages, we identify the numbers that will represent our quantity and our whole. After using those numbers to make a fraction, we convert the fraction into a percentage.
Practice question
Question 2
There are $11$11 boys and $14$14 girls in a class.
Find the total number of students in the class.
What percentage of the class is boys?
What percentage of the class is girls?
Quantities with different units
Sometimes when writing a quantity as a percentage, the quantity may not be in the same units as the whole we are comparing it to. In order to solve these problems, we will need to be able to convert between different units to set up the correct fraction.
Worked Example
What is $24$24 cm as a percentage of $2$2 m?
Think: We can identity the quantity to be $24$24 cm and the whole to be $2$2 m, but we can't make our fraction until these numbers have the same units.
Do: Since $1$1 m = $100$100 cm, we can convert $2$2 m into $200$200 cm. Now that the whole and the quantity have the same units, we can make our fraction with the quantity of $24$24 as the numerator and the whole of $200$200 as the denominator:
$\frac{24}{200}$24200
Converting this into a percentage gives us:
$12%$12%
So $24$24 cm is equal to $12%$12% of $2$2 m.
Reflect: When finding quantities as percentages when dealing with multiple units in the same problem, we want to make all values have the same units. After doing this, we can solve the problem using the method we have learned in this lesson.
Practice question
Question 3
What percentage of $2$2 hours is $48$48 minutes?