Explainer: Percentage Equations

In this explainer, we will learn how to find the part given its percentage and the whole, or vice versa, and how to solve problems involving percentage using proportions.

Proportions are particular ratios that compare a part, ๐‘Ž, to a whole, ๐‘. They are usually written as the fraction ๐‘Ž๐‘, meaning a out of ๐‘.

A percentage is a way to express a given proportion where the whole is 100: ๐‘% means ๐‘ out of 100 (which can be written as ๐‘100).

A given proportion can always be expressed as a percentage. It is finding ๐‘ so that ๐‘ and 100 are in the same proportions as ๐‘Ž and ๐‘. We can say that the ratios ๐‘Ž๐‘ and ๐‘100 form a proportion, which simply means that their ratios are equal; that is, ๐‘Ž๐‘=๐‘100.

In this context, the word โ€œproportionโ€ is often used for this equality of ratio, indicating that ๐‘ and 100 are in the same proportions as ๐‘Ž and ๐‘.

This equation can be used to express ๐‘Ž๐‘ as a percentage (namely, ๐‘%). However, it can be helpful to represent the proportions with a double-line diagram, showing both the real values and the corresponding percentage, to visualize and understand the equation better.

For instance, let us consider 37 and 68. What percentage of 68 is 37? This situation can be represented with a double-line diagram.

We are in a situation of two equal proportions: 37 out of 68 is the same as ๐‘ out of 100. The shaded area is 3768 of the big rectangle.

Remember that the 3768 means dividing the whole rectangle in 68 equal shares and taking 37 of these shares to get the part highlighted in orange.

Therefore, one share on the bottom number line is 10068, and 37 shares are then 10068โ‹…37. So, we have ๐‘=37โ‹…10068.

We can also think about this situation by finding how many 68s there are in 100; this is 10068. So, we know that if we multiply or divide a part and a whole by the same number, then the proportion is unchanged. Here, we are indeed finding the equivalent fraction to 3768 by multiplying both the numerator and the denominator by 10068.

We find that 3768=37โ‹…68โ‹…,๏Šง๏Šฆ๏Šฆ๏Šฌ๏Šฎ๏Šง๏Šฆ๏Šฆ๏Šฌ๏Šฎ and as 68 divided by 68 in the denominator makes 1, we get 3768=37โ‹…100=๐‘100.๏Šง๏Šฆ๏Šฆ๏Šฌ๏Šฎ

Hence, we have ๐‘=37โ‹…10068.

We see that if ๐‘Ž is ๐‘% of ๐‘, then ๐‘=100โ‹…๐‘Ž๐‘, which can be derived from ๐‘Ž๐‘=๐‘100 by multiplying both sides of the equation by 100.

As you see, there are many ways to reason and find how the value of ๐‘ is worked out. There is no need to remember all these ways that lead to the equation; simply pick the one that makes more sense to you so that you do not forget it.

Let us look now at three examples to see how we apply this equation to find what percentage of a whole a given part is.

Example 1: Finding the Percent of a Part Compared to a Whole

Benjamin sang with his choir for a benefit event in support of research against cancer, in which several concerts were held in parallel. His choir attracted 162 people, and 1,243 people attended the whole event. What percentage of those who attended the event listened to Benjaminโ€™s choir? Round your answer to the nearest percent.

Answer

We are told here that 162 people came to listen to Benjaminโ€™s choir out of 1,243 people who attended the event. We want to find the percentage that corresponds to 162 out of 1,243. So, we are looking for the number ๐‘ such that ๐‘ and 100 compare in the same way as 162 and 1,243.

It means that the ratios of one to the other in each pair are equal. Hence, ๐‘100=1621,243.

By multiplying both sides by 100, we find ๐‘=100ร—1621,243โ‰…13%.

We find that around 13% of the people who attended the whole event listened to Benjaminโ€™s choir.

Example 2: Finding the Percent of a Part Compared to a Whole

Charlotte has set herself a target of swimming 1.5 km. She is swimming in a 25-meter long swimming pool and has swum 39 lengths so far. What percent of her target has she already swum?

Answer

Here, we need first to work out the distance Charlotte has already swum. We are told she has swum 39 lengths of 25 meters, so the distance swum is 39ร—25=975.m

Her target is 1.5 km, that is, 1,500 m. We are looking for the number ๐‘ that compares to 100 in the same way as 975 compares to 1,500. It means that the ratios of one to the other in each pair are equal. Hence, ๐‘100=9751,500.

By multiplying both sides by 100, we find ๐‘=9751,500โ‹…100.=65.

Our answer is that Charlotte has already swum 65% of her target.

Example 3: Finding the Percent of a Part Compared to a Whole

At last yearโ€™s half marathon, Williamโ€™s time was 2 hours and 6 minutes. This year, he ran the half marathon in 1 hour and 59 minutes. What percent of his previous time is this? Give your answer correct to one decimal place.

Answer

Before we can work out what percentage of Williamโ€™s first running time his second time is, we need to convert the times given in hours and minutes into minutes only. Knowing that one hour is 60 minutes, we find that 2 hours and 6 minutes is 126 minutes, and 1 hour and 59 minutes is 119 minutes. We are now looking for the number ๐‘ that compares to 100 in the same way as 119 compares to 126. Hence, we have ๐‘100=119126.

By multiplying both sides by 100, we find ๐‘=100ร—119126=94.4.

Our answer is that Williamโ€™s running time of this year is 94.4% of his previous rime.

We have just seen how to compare a part to a whole by expressing the part as a percentage of the whole. Now, we are going to learn how to find a given percentage of a number.

For this, we are going first to use a 10ร—10 square to represent our whole. What percentage of the square shown is shaded?

We find that the number of squares in the shaded region is 69. Since the big square is made of 100 squares, then the shaded region is 69% of the big square.

We observe that the shaded region is made of 6 columns of 10 squares each and 9 squares in the 7th column. There are 10 columns in the big square, so each column is 110 of the square. And each little square is 1100 of the big square.

This means that 69%=6โ‹…110+9โ‹…1100.

If now we assign the number ๐‘› to the big square, we find that 69% of ๐‘› is ๏€ผ6โ‹…110+9โ‹…1100๏ˆโ‹…๐‘›, which can be rewritten using the distributive property of multiplication as 6โ‹…110โ‹…๐‘›+9โ‹…1100โ‹…๐‘›.

Since 6โ‹…110=0.6 and 9โ‹…1100=0.09, we find that 69% of ๐‘› is 0.6โ‹…๐‘›+0.09โ‹…๐‘›, which can be rewritten as 0.69โ‹…๐‘›.

We see that finding 69% of ๐‘› is simply multiplying ๐‘› by 69100=0.69.

Of course, this could have been derived as well from the equation saying that ๐‘Ž is ๐‘% of ๐‘; that is, ๐‘Ž and ๐‘ compare in the same way as ๐‘ and 100, which means that ๐‘Ž๐‘=๐‘100.

Now, we see that finding ๐‘% of ๐‘ is finding ๐‘Ž, which is given by multiplying both sides of the equation by ๐‘. We get ๐‘%๐‘=๐‘100โ‹…๐‘.of

Let us now look at a couple of examples of finding the percentage of a given number.

Example 4: Finding the Percentage of a Number

If the tax rate is 7%, what will the sales tax be for a truck that costs $16,000?

Answer

The tax rate is 7%, so we want to find 7% of $16,000. For this, we need to multiply $16,000 by 7100, or 0.07. (Note that dividing by 100 gives the value of 1% and multiplying the result by 7 then gives 7%.) We find that 0.07โ‹…16,000=1,120.

Hence, the sales tax for the truck amounts to $1,120.

Example 5: Finding the Percentage of a Number

A school produced 450 pounds of garbage in a month. If the school wants to reduce the weight of garbage produced in the following month to 79% of its weight, what should the target weight be?

Answer

We want to find 79% of 450 pounds. For this, we need to multiply 450 pounds by 79100=0.79. (Note that dividing by 100 gives the value of 1% and multiplying the result by 79 then gives 79%. This is equivalent to multiplying by 0.79.) We find that 0.79โ‹…450=355.5.

Hence, the target weight should be 355.5 pounds.

Key Points

  1. Expressing a given part-to-whole ratio ๐‘Ž๐‘ as a percentage, ๐‘%, is finding the number ๐‘ so that ๐‘ and 100 are in the same proportions as ๐‘Ž and ๐‘.
  2. When ๐‘ and 100 are in the same proportions as ๐‘Ž and ๐‘, their ratios are equal: ๐‘Ž๐‘=๐‘100.
  3. Rearranging this equation allows us to work out the value of ๐‘ if we know the part and the whole: ๐‘=๐‘Ž๐‘โ‹…100. This equation means that ๐‘Ž is ๐‘% of ๐‘.
  4. To find the part, ๐‘Ž, given that it is ๐‘% of a given whole ๐‘, we use the same equation but rearrange it so that ๐‘Ž is the subject: ๐‘Ž=๐‘100โ‹…๐‘.

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