4.03 Percentages in context

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\% is a whole, 50\% is a half and 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.

For example if we want to find 50\% of 30, we can just find half of 30 which is 15.

A percentage might not be one of those in the table, but it might be a multiple.

When doing research on the popularity of different public transports in Australia, Claris found a study that claims that 65\% of people drive to work. Out of the 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\% of 60.

While 65\% does not relate to any of our simple fractions, we can notice that 65\% is equal to thirteen lots of 5\%. If we can find 5\% by converting it to a fraction, we can then just multiply the answer by 13.

We can find 5\% of 60 by dividing it by 20:

5\% \text{ of } 60=\dfrac{1}{20}\times 60 = 3

Then we multiply this answer by 13:

65\% \text{ of } 60=13\times3 = 39

So Claris can expect 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.

Examples

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\% of 60 as \dfrac{48}{100}\times60 or 0.48 \times 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 to find 50\% of a quantity. However, another way to divide by 2 is to multiply by \dfrac{1}{2} or 0.5. Notice that both \dfrac{1}{2} and are equivalent to 50\%.

We can find percentages of quantities directly by multiplying the quantity by either the fraction or decimal equivalent to the percentage.

Examples

So far, the percentages that we have found have always been below 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\% 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.

When finding a percentage of a quantity, swapping the values of the percentage and the quantity does not affect the final solution.

Examples