Finding the Percent and Using Percent Proportions | Grade 7 Math | LevelKite

You can already find a percent of a number, and you can find a whole from a part and its percent. Now you will answer the third kind of question: one number is what percent of another?

Start with what a percent means. A percent is a rate per 100. It tells you how much there is for every 100 units of a stated positive whole. The whole is the amount you measure the percent against. The part is the amount you compare to it.

So " $12$ is what percent of $48$ ?" asks this: if you counted the whole, $48$ , as $100$ , what would the part, $12$ , count as? In every question in this lesson, the words percent of or % of point to the whole: it is the amount that comes right after them.

Here is how to find that percent. Let $p$ be the percent number, so you write the answer as $p\%$ .

Divide the part by the whole. This tells you how much of the part there is for each $1$ unit of the whole. You can divide because the whole is positive, so it is not zero.
Multiply by 100. This tells you how much there is for every $100$ units of the whole, and that is what a percent means.

For $12$ and $48$ : $12 \div 48 = 0.25$ , and $0.25 \times 100 = 25$ . So $12$ is $25\%$ of $48$ : when the whole counts as $100$ , the part counts as $25$ .

This rule works when the part and the whole are positive amounts in the same unit.

p = \frac{\text{part}}{\text{whole}} \times 100 \qquad (\text{whole} > 0)

Worked example

Divide, then multiply by 100

What percent of $40$ is $14$ ?

Name the quantities. You measure the percent against $40$ , so $40$ is the whole. You compare $14$ to it, so $14$ is the part. Neither number has a unit, so they already match. Let $p$ be the percent number.
$\text{part} = 14, \quad \text{whole} = 40$
Divide the part by the whole to find how much of the part there is for each $1$ unit of the whole. The whole, $40$ , is positive, so you are not dividing by zero.
$14 \div 40 = 0.35$
Multiply by $100$ to change the amount for each $1$ unit into the amount for every $100$ units. That amount is the percent number.
$p = 0.35 \times 100 = 35$
Check by going the other way. Finding $35\%$ of the whole should give the part back.
$0.35 \times 40 = 14$

Answer

$14$ is $35\%$ of $40$ .

Key concept

The percent proportion

A ratio compares two amounts by division. A proportion is an equation that says two ratios are equal.

A statement that a part is $p\%$ of a whole is a proportion. It means the part compares to the whole the same way $p$ compares to $100$ .

This is true when the whole is a stated positive amount and the part is a positive amount in the same unit. Keep matching quantities in matching positions: the part goes across from $p$ , and the whole goes across from $100$ .

\frac{\text{part}}{\text{whole}} = \frac{p}{100}

The proportion and the divide-then-multiply rule say the same thing. Start with the proportion and multiply both sides by $100$ . Multiplying both sides of an equation by the same number that is not zero keeps the two sides equal, so you get $\frac{\text{part}}{\text{whole}} \times 100 = p$ .

The proportion can do more, though. Any one of its three values can be the unknown, the value you need to find: the part, the whole, or the percent number. You write the same proportion each time, put a letter in the unknown's position, and solve.

The table shows one fact asked three ways. Each quantity keeps its position in every row. Only the letter moves.

One fact asked three ways: 24 is 30% of 80

Question	Unknown	Proportion	Value of the unknown
What is $30\%$ of $80$ ?	the part, $a$	$\frac{a}{80} = \frac{30}{100}$	$a = 24$
$24$ is $30\%$ of what number?	the whole, $w$	$\frac{24}{w} = \frac{30}{100}$	$w = 80$
$24$ is what percent of $80$ ?	the percent number, $p$	$\frac{24}{80} = \frac{p}{100}$	$p = 30$

Worked example

Find the percent with a proportion

A bottle holds $250$ mL of a fruit drink. Of that drink, $45$ mL is real juice. What percent of the drink is real juice?

Name the quantities with their units. You measure the percent against the drink, so the whole is $250$ mL. The part is the $45$ mL of juice. Both are positive amounts in milliliters, so the units match. Let $p$ be the percent number.
$\text{part} = 45 \text{ mL}, \quad \text{whole} = 250 \text{ mL}$
Write the percent proportion, because the part compares to the whole the same way $p$ compares to $100$ . Keep matching positions: the part across from $p$ , the whole across from $100$ .
$\frac{45}{250} = \frac{p}{100}$
Multiply both sides by $100$ . It is not zero, so the two sides stay equal.
$\frac{45}{250} \times 100 = \frac{p}{100} \times 100$
Simplify. On the right, multiplying by $100$ undoes dividing by $100$ , so only $p$ is left. On the left, $45 \times 100 = 4500$ . Dividing the top and bottom by $10$ keeps the fraction's value the same and makes the division easier.
$p = \frac{4500}{250} = \frac{450}{25} = 18$
Check: $18\%$ of the whole should give the part, $45$ mL, back.
$0.18 \times 250 = 45$

Answer

Real juice makes up $18\%$ of the drink.

Worked example

Find the whole with a proportion

A class has collected $84$ cans for a food drive. That is $35\%$ of its goal. How many cans is the goal?

Name the quantities. You measure the percent against the goal, so the goal is the whole, and it is the unknown. Let $w$ be the goal in cans, a positive number. The part is $84$ cans, and the percent number is $35$ . Both amounts are in cans, so the units match.
$\text{part} = 84 \text{ cans}, \quad \text{whole} = w \text{ cans}$
Write the percent proportion with each quantity in its matching position: the part across from $35$ , the whole across from $100$ .
$\frac{84}{w} = \frac{35}{100}$
Multiply both sides by $w$ and by $100$ . Neither is zero, so the two sides stay equal.
$\frac{84}{w} \times w \times 100 = \frac{35}{100} \times w \times 100$
Simplify each side. On the left, multiplying by $w$ undoes dividing by $w$ . On the right, multiplying by $100$ undoes dividing by $100$ .
$84 \times 100 = 35 \times w$
Work out $84 \times 100 = 8400$ . Then divide both sides by $35$ , which is not zero, to leave $w$ alone. Dividing the top and bottom by $5$ keeps the value the same and makes the division easier.
$w = \frac{8400}{35} = \frac{1680}{7} = 240$
Check: $35\%$ of the goal should give the part, $84$ cans, back.
$0.35 \times 240 = 84$

Answer

The class's goal is $240$ cans.

Two quick checks catch most slips. Both assume the part and the whole are positive amounts in the same unit.

Put it back. Find your percent of the whole. You should get the part.
Compare with 100. If the part is less than the whole, the percent is less than $100$ . If the part is greater than the whole, the percent is greater than $100$ .

Also look at your proportion before you solve it. In the form you use in this lesson, the part sits across from the percent number, and the whole sits across from $100$ .

Takeaway. A percent is a rate per 100 of a stated positive whole. To find what percent a part is of that whole, divide the part by the whole and multiply by $100$ . The percent proportion $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ holds the same idea in one equation, and you can solve it for whichever value is missing. Both tools work when the part and the whole are positive amounts in the same unit and each quantity stays in its matching position.

Next, you will use these tools to solve percent word problems and to find and fix errors in percent equations.