Percent of a Number and Finding the Whole | Grade 7 Math | LevelKite

A percent is a rate per 100: it tells you how many for every 100. So $35\%$ means 35 for every 100, which you can also write as $\frac{35}{100}$ or $0.35$ .

A percent is always a percent of some amount, called the whole. The whole is a positive number, and the problem always states which amount it is. The amount you get by taking the percent of the whole is called the part.

That makes three quantities: the percent, the whole and the part. In this lesson you will know two of them and find the third. First you will find the part, and then you will find the whole.

To see what taking a percent of a whole means, read the table below. It finds $35\%$ of several wholes by counting the hundreds in each whole, because $35\%$ gives 35 for each hundred.

Finding 35% of different positive wholes

Whole	Hundreds in the whole	35% of the whole
$100$	$100 \div 100 = 1$	$35 \times 1 = 35$
$200$	$200 \div 100 = 2$	$35 \times 2 = 70$
$300$	$300 \div 100 = 3$	$35 \times 3 = 105$
$50$	$50 \div 100 = 0.5$	$35 \times 0.5 = 17.5$
$80$	$80 \div 100 = 0.8$	$35 \times 0.8 = 28$

Each row did the same two things: it divided the whole by $100$ to count the hundreds, and then it multiplied by $35$ . Doing both is the same as multiplying the whole by $\frac{35}{100}$ , because $35 \times \frac{\text{whole}}{100}$ and $\frac{35}{100} \times \text{whole}$ both equal $\frac{35 \times \text{whole}}{100}$ .

So for any positive whole, $35\%$ of the whole is $\frac{35}{100} \times \text{whole}$ , which is $0.35 \times \text{whole}$ . Taking a percent of a positive whole means multiplying the whole by the percent written as a rate per 100.

Nothing in this reasoning depends on the number $35$ , so it works for any percent of a positive whole.

Key concept

The percent equation

A percent $p\%$ is a rate of $p$ per 100, where $p$ is the number in front of the percent sign. When the percent is taken of a stated positive whole, the part equals the rate $\frac{p}{100}$ times the whole. For $35\%$ , $p = 35$ and the rate is $\frac{35}{100} = 0.35$ .

This one equation connects all three quantities. When the part is unknown, you multiply the whole by the rate. When the whole is unknown, the same equation still holds, and you will soon use it to work backward to the whole.

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

Worked example

Finding a percent of a number

What is $35\%$ of $80$ ?

Name the three quantities. The percent is 35%. The whole is 80, because 80 is the amount the percent is taken of, and it is a stated positive number. The part is the unknown.
$\text{part} = 35\% \text{ of } 80$
Write the percent as a rate per 100. Percent means per 100, so 35% is 35 hundredths, which is the decimal 0.35.
$35\% = \frac{35}{100} = 0.35$
Use the percent equation: the part equals the rate times the whole. The equation applies because the whole, 80, is a stated positive number.
$\text{part} = 0.35 \times 80$
Multiply. Because 0.35 is 35 hundredths, multiply 35 by 80 to get 2800, and then divide by 100.
$0.35 \times 80 = \frac{35 \times 80}{100} = \frac{2800}{100} = 28$
Check the size. The whole is positive and 35% is less than 50%, so the part must be less than half of 80, which is 40. The result 28 is less than 40, and it matches the last row of the table above.
$28 < 40$

Answer

$35\%$ of $80$ is $28$ .

Now turn the question around. Sometimes you know the part and the percent, and the whole is the missing quantity. The problem still states which amount is the whole, and the whole is still positive. You just do not know its value yet.

Start with a familiar case: $10$ is $50\%$ of what number? Since $50\%$ is one half, half of the whole is $10$ , so the whole is $20$ .

Here is the same reasoning in symbols. The percent equation says $10 = 0.50 \times \text{whole}$ . The whole was multiplied by $0.50$ to make the part, so dividing the part by $0.50$ gives the whole back: $10 \div 0.50 = 20$ . Division undoes the multiplication.

The same step works for any percent $p\%$ of a positive whole, as long as the percent is not $0\%$ . Divide both sides of the percent equation by the rate $\frac{p}{100}$ . Dividing both sides of an equation by the same nonzero number keeps the two sides equal. The rate cannot be $0$ , because you cannot divide by $0$ . The result is this rule:

\text{whole} = \text{part} \div \frac{p}{100} \quad \text{when } p \neq 0

Worked example

Finding the whole

$12$ is $30\%$ of what number?

Name the three quantities. The percent is 30%. The number 12 is the part, because it is what you get when you take 30% of something. The whole is the unknown, so let w stand for the whole. A whole is positive, so w is positive.
$12 = 30\% \text{ of } w$
Write the percent equation with these quantities. 30% is 30 hundredths, or 0.30, and the part equals the rate times the whole.
$12 = 0.30 \times w$
Divide both sides by 0.30. Dividing both sides of an equation by the same nonzero number keeps the two sides equal, and 0.30 is not 0. On the right side, dividing by 0.30 undoes multiplying by 0.30, so w is left alone.
$\frac{12}{0.30} = \frac{0.30 \times w}{0.30} \quad \text{so} \quad w = \frac{12}{0.30}$
Divide. First multiply both 12 and 0.30 by 10 to clear the decimal. Multiplying both numbers in a division by the same nonzero number does not change the result.
$w = \frac{12 \times 10}{0.30 \times 10} = \frac{120}{3} = 40$
Check by finding 30% of 40. You get 12, which is the part you started with. The whole, 40, is also positive, as it must be.
$0.30 \times 40 = \frac{30 \times 40}{100} = \frac{1200}{100} = 12$

Answer

$12$ is $30\%$ of $40$ , so the whole is $40$ .

One fact about a positive whole, 20% of 45 is 9, asked two ways

Unknown	Question	Equation	What you do
The part	What is $20\%$ of $45$ ?	$\text{part} = 0.20 \times 45$	Multiply: $0.20 \times 45 = \frac{900}{100} = 9$
The whole	$9$ is $20\%$ of what number?	$9 = 0.20 \times \text{whole}$	Divide: $9 \div 0.20 = 90 \div 2 = 45$

Worked example

Finding the whole in a real situation

Maya has read $54$ pages of a book. That is $45\%$ of all the pages in the book. How many pages does the book have?

Name the quantities with their units. The part is 54 pages and the percent is 45%. The whole is the number of pages in the book. It is unknown, so let w stand for that number of pages. A book has a positive number of pages, so w is positive.
Write the percent equation. 45% is 0.45, and the part equals the rate times the whole.
$54 = 0.45 \times w$
Divide both sides by 0.45. It is not 0, so the two sides stay equal and w is left alone.
$w = \frac{54}{0.45}$
Divide. Multiply both numbers by 100 to clear the decimal; multiplying both by the same nonzero number does not change the result. Then divide 5400 by 45: since 45 times 100 is 4500 and 45 times 20 is 900, 45 times 120 is 5400.
$w = \frac{54 \times 100}{0.45 \times 100} = \frac{5400}{45} = 120$
Check by finding 45% of 120 pages. You get 54 pages, the part Maya has read.
$0.45 \times 120 = \frac{45 \times 120}{100} = \frac{5400}{100} = 54$

Answer

The book has $120$ pages.

Takeaway. A percent is a rate per 100 of a stated positive whole, and one equation connects the three quantities: the part equals the rate (the percent written as hundredths) times the whole. When the part is unknown, multiply the whole by the rate. When the whole is unknown and the percent is not $0\%$ , divide the part by the rate, and then check by taking the percent of the whole you found.

Next, you will find what percent one number is of another and solve percent problems using proportions.