4.02 Percentages of quantities
Introduction
Percentages are useful because 1\% always represents 1 part out of 100. This helps us compare different quantities out of different wholes because, after we convert them to percentages, we can easily compare them.
Find 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.
Alice and Martin both take part in an archery competition.
Alice takes 25 shots and hits the target 18 times.
Martin takes 10 shots and hits the target 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 times out of 25, her accuracy can be expressed as the fraction \dfrac{18}{25}.
Similarly, Martin's accuracy can be expressed as the fraction \dfrac{7}{10}.
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\% while Martin's accuracy was 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\% accuracy would mean hitting the target with every shot. When represented as a fraction this would be \dfrac{25}{25}, 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.
Examples
Example 1
Which of the following shows how to calculate 15 as a percentage of 31?
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.
Find 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.
Examples
Example 2
There are 2 boys and 7 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?
To find a quantity as a percentage, first write it as a fraction with the quantity as the numerator and the whole as the denominator, then multiply the fraction by 100\%.
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.
Examples
Example 3
What percentage of 4 hours is 36 minutes?
To compare quantities with different units, we need to convert one of the quantities to the same units as the other.