Identify the Part, the Whole, and the Percent | Grade 6 Math | LevelKite

You already know how to find a percent of a number. For example, $25\%$ of $80$ is $20$ .

Look at that sentence again. It holds three numbers: $25\%$ , $80$ and $20$ . Each number has its own job. In this lesson you will learn the names of those jobs. Then you will find the percent when it is the missing number.

A percent statement such as " $25\%$ of $80$ is $20$ " compares two amounts. Three words name the numbers in it.

The whole is the full amount you compare against. It is the reference amount, so it counts as $100\%$ .
The part is the amount you compare with the whole.
The percent tells how large the part is compared with the whole. Percent means "per hundred", so $25\%$ means $25$ for every $100$ , or $\frac{25}{100}$ .

In that statement, the whole is $80$ , the part is $20$ , and the percent is $25\%$ . The part, $20$ , is $25$ hundredths of the whole, $80$ .

Naming the part, the whole, and the percent

Statement	Part	Whole	Percent
$20$ is $25\%$ of $80$	$20$	$80$	$25\%$
$30\%$ of $50$ is $15$	$15$	$50$	$30\%$
$60$ is $150\%$ of $40$	$60$	$40$	$150\%$
$9$ out of $36$ is what percent?	$9$	$36$	unknown
What percent of $40$ is $18$ ?	$18$	$40$	unknown

To name the part and the whole, ask two questions. Which amount is the full $100\%$ ? That is the whole. Which amount is compared with it? That is the part. The percent is the number that tells how the two compare. It may be the unknown, as in the last two rows.

Words can help. The whole often comes right after "percent of" or "out of". Still, let the meaning decide, not the word order or the size of the numbers. Look at the third row of the table. The part, $60$ , is larger than the whole, $40$ , so the percent is more than $100\%$ .

Worked example

Name the part, the whole, and the percent

A school band has $40$ members, and $12$ of them play the flute. That is $30\%$ of the band. Name the part, the whole, and the percent. Then check that the three numbers fit together.

Find the whole first. The percent describes the band, so the whole band is the full 100% amount. The band has 40 members.
$\text{whole} = 40$
Find the part. The flute players are the group you compare with the whole band. There are 12 of them.
$\text{part} = 12$
Find the percent. It is the number with the percent sign. It tells how large the part is compared with the whole: 30 for every 100.
$\text{percent} = 30\%$
Check with what you already know. Taking 30% of the whole should give the part. The 30% here is exact, not rounded, so the product must equal the part exactly.
$\frac{30}{100} \times 40 = 0.30 \times 40 = 12$

Answer

The part is $12$ flute players, the whole is $40$ band members, and the percent is $30\%$ .

Now suppose the percent is the missing number. You know that $7$ is some percent of $20$ . The part is $7$ and the whole is $20$ .

A percent tells what the part would be if the whole were $100$ . So scale the whole up to $100$ . Since $20 \times 5 = 100$ , multiply both numbers by $5$ . Multiplying the part and the whole by the same nonzero number does not change how they compare. The part becomes $7 \times 5 = 35$ . So $7$ out of $20$ matches $35$ out of $100$ , and $7$ is $35\%$ of $20$ .

Scaling is quick when the whole fits evenly into $100$ . Division works for any whole that is not zero. Here $7 \div 20 = 0.35$ , which says the part is $0.35$ of one whole. Then $0.35$ is $35$ hundredths, and $35$ hundredths is $35\%$ .

Here is why dividing works. Let $p$ stand for the number of percent, so the percent you want is $p\%$ . You already know that taking $p\%$ of the whole gives the part. For this problem, that is the equation $\frac{p}{100} \times 20 = 7$ .

To get $p$ alone, undo each operation with its inverse, the operation that reverses it. Dividing by $20$ undoes multiplying by $20$ . You can divide by $20$ because it is not zero. Multiplying by $100$ undoes dividing by $100$ . Do the same thing to both sides of the equation each time, so the two sides stay equal.

The steps below end with $p = 35$ . So the percent is $35\%$ , the same answer as before.

\begin{aligned} \frac{p}{100} \times 20 &= 7 \\ \frac{p}{100} &= 7 \div 20 = 0.35 \\ p &= 0.35 \times 100 = 35 \end{aligned}

Key concept

A percent compares a part with a whole

The whole is the reference amount, the full $100\%$ . The part is the amount you compare with it. The percent tells how many hundredths of the whole the part is.

To find what percent the part is of the whole, divide the part by the whole, then multiply by $100$ . In the formula, $p$ is the number of percent, so you write the answer as $p\%$ .

The whole is the number you divide by, so it must not be zero.

p = \frac{\text{part}}{\text{whole}} \times 100 \quad (\text{whole} \neq 0)

Worked example

Find a missing percent

What percent of $40$ is $18$ ?

Name the part and the whole. The whole is 40, the full amount that 18 is compared with. The part is 18. The percent is unknown, so call it p percent. The whole is not zero, so you can divide by it.
$\text{part} = 18, \quad \text{whole} = 40$
Divide the part by the whole. The quotient, the result of a division, tells how much of one whole the part is. Write the division as a fraction, divide both numbers by 2, then multiply both by 5 to reach hundredths. The value stays the same because you divide or multiply the top and bottom by the same nonzero number.
$18 \div 40 = \frac{18}{40} = \frac{9}{20} = \frac{45}{100} = 0.45$
Multiply the quotient by 100. Percent means per hundred, so this counts how many hundredths of the whole the part is.
$p = 0.45 \times 100 = 45$
Check. The quotient 0.45 is exact, not rounded, so the quotient times the divisor must equal the dividend exactly. The divisor is the number you divide by, here the whole, 40. The dividend is the number you divide, here the part, 18.
$0.45 \times 40 = 18$

Answer

$18$ is $45\%$ of $40$ .

Worked example

Find a percent that needs rounding

A team wins $5$ of its $6$ games. What percent of its games does the team win? Round to the nearest whole percent.

Name the part and the whole. The whole is all 6 games, the full 100% amount. The part is the 5 wins. The whole is not zero, so you can divide by it.
$\text{part} = 5, \quad \text{whole} = 6$
Divide the part by the whole. Keep dividing into decimals instead of stopping with a remainder, because a percent measures a fraction of the whole. Think of 5 as 50 tenths. Dividing by 6 gives 8 tenths with 2 tenths left. Those 2 tenths are 20 hundredths. Dividing by 6 gives 3 hundredths with 2 hundredths left. A 2 is left at every later place too, so the digit 3 repeats without end.
$5 \div 6 = 0.8333\ldots$
Multiply by 100 to count hundredths. Then round to the nearest whole number. The first digit after the decimal point is 3, which is less than 5, so round down to 83.
$0.8333\ldots \times 100 = 83.33\ldots \approx 83$
Check. Write 83% as the decimal 0.83, which is a rounded quotient. Because it is rounded, its product with the whole should be close to the part, 5, but not exactly equal to it.
$0.83 \times 6 = 4.98 \approx 5$

Answer

The team wins about $83\%$ of its games.

The big idea: a percent compares a part with a whole. The whole is the full $100\%$ amount. The part is the amount you compare with it. The percent tells how many hundredths of the whole the part is.

To find a missing percent, divide the part by the whole, then multiply by $100$ . This works whenever the whole is not zero. To check, multiply the quotient by the whole. An exact quotient gives back the part exactly. A rounded quotient gives a number close to the part.

Next, you will work in the other direction. You will find the whole when you know a part and a percent, and you will solve percent word problems.