Find the Whole and Solve Percent Problems | Grade 6 Math | LevelKite

Picture a download bar on a tablet. It says 20% complete, and it shows that $8$ megabytes (MB) of the file have arrived. A megabyte is a unit of file size. How big is the whole file?

You already know the three roles in a percent problem: the part, the whole, and the percent. Here you know the part, $8$ MB, and the percent, $20\%$ . The whole is missing. In this lesson you will learn to find a missing whole. Then you will use that skill to solve percent word problems.

Start with what the numbers mean. A percent is a rate per $100$ . The whole is the full amount, so it is the $100\%$ amount. The part is a share of that whole.

The download bar tells you that $20\%$ of the file is $8$ MB. Picture the file cut into $100$ equal slices. Then $20$ of those slices add up to $8$ MB.

Percents and amounts of the same whole scale together. If you divide the percent by a number that is not zero, the amount is divided by that same number. If you multiply the percent by a number, the amount is multiplied by that same number. So you can step down from $20\%$ to $1\%$ , and then up to $100\%$ .

Stepping from 20% of the file to the whole file

Percent of the file	Megabytes (MB)	How to get it
$20\%$	$8$	Read it from the download bar.
$1\%$	$8 \div 20 = 0.4$	Divide both numbers in the first row by $20$ .
$100\%$	$0.4 \times 100 = 40$	Multiply both numbers in the second row by $100$ .

The table shows that the whole file is $40$ MB. Look at the middle row: $1\%$ of the file is $0.4$ MB. A file size can be split into smaller amounts, so a fractional share (a share that is not a whole number) like $0.4$ MB is a fine result.

Now write the same reasoning as an equation. Let $w$ stand for the size of the whole file, in megabytes. The phrase 20% of the whole means $\frac{20}{100}$ of it, which is $0.20 \times w$ . The download bar says this amount is $8$ MB, so $0.20 \times w = 8$ .

To find $w$ , use the inverse operation, the operation that undoes another one. Dividing by $0.20$ undoes multiplying by $0.20$ . Divide both sides of the equation by $0.20$ . The sides stay equal because two equal amounts are divided by the same number, and that number is not zero.

To divide by $0.20$ , first multiply the dividend (the number being divided) and the divisor (the number you divide by) by $100$ . That turns the divisor into the whole number $20$ . Multiplying both by the same nonzero number does not change the quotient, which is the result of the division. The equation gives the same whole as the table:

w = 8 \div 0.20 = 800 \div 20 = 40

Key concept

Find the whole: divide the part by the percent as a decimal

The whole is the $100\%$ amount. A part equals its percent, written as a decimal, times the whole: $\text{part} = \text{percent as a decimal} \times \text{whole}$ .

When you know the part and the percent, undo that multiplication: divide the part by the percent written as a decimal. This does both steps of the table at once. Dividing by $0.20$ is the same as dividing by $20$ and then multiplying by $100$ , because $0.20 = \frac{20}{100}$ .

The rule works whenever the percent is not $0\%$ . A divisor can never be zero. Also, $0\%$ of every whole is $0$ , so a $0\%$ part cannot tell you what the whole is.

\text{whole} = \text{part} \div \text{percent as a decimal}

Worked example

Find the whole from a part and a percent

$36$ is $45\%$ of what number?

Sort the roles. 36 is the part, because it is 45% of something. 45% is the percent. The number you are looking for is the whole. Let w stand for the whole.
Write the percent as a decimal. Percent means per 100, so divide the percent number by 100.
$45\% = \frac{45}{100} = 0.45$
Write the relationship as an equation. 45% of the whole is the part, so the percent as a decimal times w equals 36.
$0.45 \times w = 36$
Divide both sides by 0.45. This undoes the multiplication and leaves w alone. The two sides stay equal because both are divided by the same number, and 0.45 is not zero.
$w = 36 \div 0.45$
Multiply the dividend and the divisor by 100 to make the divisor a whole number. The quotient stays the same because both numbers are multiplied by the same nonzero number. Then divide: 45 times 80 is 3600, so 3600 divided by 45 is exactly 80.
$36 \div 0.45 = 3600 \div 45 = 80$
Check by multiplying back. The quotient 80 is exact, with no rounding, so the product must equal the part exactly. 45 times 80 is 3600, and 3600 divided by 100 is 36.
$0.45 \times 80 = 36$

Answer

$36$ is $45\%$ of $80$ , so the whole is $80$ .

Check every whole you find. Multiply it by your divisor, the percent as a decimal, which is not zero.

Exact quotient (no rounding): the product equals the part exactly.
Rounded quotient: the check is only approximate. If $30\%$ of a whole is $20$ , then $20 \div 0.30$ rounds to $66.67$ , and $0.30 \times 66.67 = 20.001$ , close to $20$ but not equal.

Some divisions share objects that cannot be split. $50$ marbles among $8$ friends is $6$ each with $2$ left over. The $2$ is the remainder. Check with divisor $\times$ whole-number quotient $+$ remainder: $8 \times 6 + 2 = 50$ .

Keep that apart from two other results. $48 \div 8 = 6$ has no remainder. $50$ liters poured equally into $8$ jugs is $6.25$ liters each: a fractional share, with nothing left over.

Now put the rule to work in word problems. Each percent word problem in this lesson is a story about a part, a whole, and a percent. The story gives you two of them. The third one is missing, and the question depends on it. Use four moves:

Sort the roles. Decide which amount is the whole, the full $100\%$ amount. Decide which amount is the part, a share of that whole. The percent links the two. Notice which of the three is missing.
Name the unknown. Pick a letter. Say what it stands for and what unit it is measured in.
Write the relationship and solve. Start from $\text{part} = \text{percent as a decimal} \times \text{whole}$ . The table below shows how to find each missing piece.
Check, then answer the question asked. Reread the question. Some problems need one more step after you find the missing piece.

Finding each missing piece

You know	You need	How to find it
The part and the percent	The whole	Divide the part by the percent as a decimal. The percent must not be $0\%$ .
The part and the whole	The percent	Divide the part by the whole. Write the quotient as a percent. The whole must not be $0$ .
The percent and the whole	The part	Multiply the percent as a decimal by the whole.

Worked example

A word problem with a missing whole

Maya has read $84$ pages of a novel. That is $35\%$ of the pages in the novel. How many pages does the novel have?

Sort the roles. The 35% is 35% of the pages in the novel, so the whole is the number of pages in the novel. The 84 pages Maya has read are the part. Let w stand for the number of pages in the whole novel.
Write the relationship. As a decimal, 35% is 0.35. 35% of the whole is the part, so 0.35 times w equals 84.
$0.35 \times w = 84$
Divide both sides by 0.35, which is not zero, to undo the multiplication. Multiply the dividend and the divisor by 100 so the divisor is a whole number. Then divide: 35 times 240 is 8400, so 8400 divided by 35 is exactly 240.
$w = 84 \div 0.35 = 8400 \div 35 = 240$
Check. The quotient is exact, so multiplying back must give the part exactly. 35 times 240 is 8400, and 8400 divided by 100 is 84. The size makes sense too: 84 pages is only 35% of the novel, so the novel must have more than 84 pages.
$0.35 \times 240 = 84$

Answer

The novel has $240$ pages.

Worked example

A two-step percent word problem

A rain barrel is $60\%$ full. It has $45$ liters of water in it. How many more liters of water will fill the barrel?

Sort the roles. The percent is 60% of a full barrel, so the whole is the number of liters in a full barrel. The 45 liters in the barrel now are the part. Let w stand for the number of liters in a full barrel. The question asks for the liters still needed, so find w first.
Write the relationship. 60% of the whole is the part, so 0.60 times w equals 45.
$0.60 \times w = 45$
Divide both sides by 0.60, which is not zero, to undo the multiplication. Multiply the dividend and the divisor by 100. 60 times 75 is 4500, so 4500 divided by 60 is exactly 75.
$w = 45 \div 0.60 = 4500 \div 60 = 75$
Check the whole. The quotient 75 is exact, so multiplying back must give the part exactly. 60 times 75 is 4500, and 4500 divided by 100 is 45.
$0.60 \times 75 = 45$
Answer the question that was asked. A full barrel holds 75 liters, and 45 liters are already in it. Subtract to find the liters still needed.
$75 - 45 = 30$
Check the answer a second way. The barrel is 60% full, so 100% minus 60%, which is 40%, is still empty. 40% of 75 liters is 0.40 times 75. 40 times 75 is 3000, and 3000 divided by 100 is 30. The two answers match.
$0.40 \times 75 = 30$

Answer

The barrel needs $30$ more liters of water to be full.

Takeaway. The whole is the $100\%$ amount, and $\text{part} = \text{percent as a decimal} \times \text{whole}$ . When you know a part and its percent, divide the part by the percent as a decimal to find the whole. This works whenever the percent is not $0\%$ , because a divisor can never be zero. In a word problem, sort the roles, name the unknown with its unit, write the relationship, solve, check, and then answer the question that was asked.

Next, you will reason about percents greater than $100\%$ and percents less than $1\%$ .