# Lesson Explainer: Percentage Equations Mathematics • 7th Grade

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, , to a whole, . They are usually written as the fraction , meaning a out of .

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

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

In this context, the word “proportion” is often used for this equality of ratio, indicating that and 100 are in the same proportions as and .

This equation can be used to express as a percentage (namely, ). 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 out of 100. The shaded area is of the big rectangle.

Remember that the 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 , and 37 shares are then . So, we have

We can also think about this situation by finding how many 68s there are in 100; this is . 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 by multiplying both the numerator and the denominator by .

We find that and as 68 divided by 68 in the denominator makes 1, we get

Hence, we have

We see that if is of , then which can be derived from by multiplying both sides of the equation by 100.

As you see, there are many ways to reason and find how the value of 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

Nader 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 Nader’s choir? Round your answer to the nearest percent.

### Answer

We are told here that 162 people came to listen to Nader’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 such that 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,

By multiplying both sides by 100, we find

We find that around of the people who attended the whole event listened to Nader’s choir.

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

Mona 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 Mona has already swum. We are told she has swum 39 lengths of 25 metres, so the distance swum is

Her target is 1.5 km, that is, 1‎ ‎500 m. We are looking for the number 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,

By multiplying both sides by 100, we find

Our answer is that Mona has already swum of her target.

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

At last year’s half marathon, Maged’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 Maged’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 that compares to 100 in the same way as 119 compares to 126. Hence, we have

By multiplying both sides by 100, we find

Our answer is that Maged’s running time of this year is 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 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 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 of the square. And each little square is of the big square.

This means that

If now we assign the number to the big square, we find that of is which can be rewritten using the distributive property of multiplication as

Since and , we find that of is which can be rewritten as

We see that finding of is simply multiplying by .

Of course, this could have been derived as well from the equation saying that is of ; that is, and compare in the same way as and 100, which means that

Now, we see that finding of is finding , which is given by multiplying both sides of the equation by . We get

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 , what will the sales tax be for a truck that costs \$16‎ ‎000?

### Answer

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

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 of its weight, what should the target weight be?

### Answer

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

Hence, the target weight should be 355.5 pounds.

### Key Points

• Expressing a given part-to-whole ratio as a percentage, , is finding the number so that and 100 are in the same proportions as and .
• When and 100 are in the same proportions as and , their ratios are equal:
• Rearranging this equation allows us to work out the value of if we know the part and the whole: This equation means that is of .
• To find the part, , given that it is of a given whole , we use the same equation but rearrange it so that is the subject:

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