Solve percent part-whole problems

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Lesson progressPractice problems 0/3

Difficulty: Intermediate
Estimated time: 28 minutes
Domains: Problem Solving and Data Analysis
Techniques: Part-wholeReference-wholeComplementsNested-percentagesLiteral-percent-entry

What you’ll learn

Tell which amount is the part, which is the percent and which is the whole.
Type percent relationships with numbers into Desmos as written, with the whole after “of.”
Find any one of the three amounts from the other two.
Handle leftover groups and percents of subgroups.
Choose between Desmos, mental math and algebra.

Prerequisites

Reason with ratios and proportions0%
Recognize part-to-whole relationships and equivalent ratios.

Why this matters on the SAT

Know what the percent is taken of

SAT percent questions show up in surveys, inventories, populations and mixtures. The arithmetic is usually short. The real decision is which amount is the whole, the amount that counts as $100\%$ . Once you know that, one relationship finds a missing part, percent or total.

Solution to the example

The whole is all $640$ plants, so $640$ goes after of. Type the relationship the way it reads:

37.5\%\text{ of }640=240.

The answer is C. The question asks for some of the plants, so the answer is a number of plants, not a percent. Choice D, $400$ , is the plants that aren’t succulents.

SAT example

A greenhouse contains $640$ plants. Of these plants, $37.5\%$ are succulents. How many of the plants are succulents?

A
$24$
B
$171$
C
$240$
D
$400$

Use one relationship for three roles

You’ve already compared a group with its total in part-to-whole relationships. A percent makes the same comparison, counted out of $100$ . Take “ $30\%$ of $80$ students”:

30\%\text{ of }80=24\text{ students}.

Each number plays a role:

The $80$ students are the whole, $W$ , the amount that counts as $100\%$ .
The $24$ students are the part, $P$ , the piece of the whole you’re describing.
The $30\%$ is the percent. It says the part is $30$ hundredths of the whole.

Those three roles always fit one relationship:

\boxed{\text{part}=\text{percent of whole}}.

The word of often points to the whole, and it also tells you what to type. When the question has numbers, type the relationship into Desmos the way it reads: keep the percent sign, put the whole after of, and let Desmos do the arithmetic.

The part and the whole always share a unit. If the whole is in liters, so is the part. The percent has no unit, because it compares two amounts measured the same way.

The whole doesn’t have to be everyone in the problem. In “ $40\%$ of the $24$ orchestra students,” the whole is the orchestra, not the entire school.

Percents as decimals

Percent means “per hundred,” so

p\%=\frac{p}{100}.

Divide by $100$ to get the decimal:

\begin{aligned} 8\%&=0.08,\\[1.4em] 37.5\%&=0.375,\\[1.4em] 125\%&=1.25. \end{aligned}

For algebra, write the relationship with the decimal, $r$ , in place of the percent:

P=rW.

Desmos reads the percent sign as “divide by $100$ ,” so you don’t need this step there.

A percent can also be more than $100\%$ . Then the part comes out larger than its whole, and it’s still a part-whole question, not a percent-change one.

Check your understanding:

$90$ is $120\%$ of what whole? Write a part-whole equation and find the whole.

Check your understanding:

Take the statement “ $18$ liters is $12\%$ of a tank’s capacity.” Which amount is the part, what is the percent as a decimal, which amount is the whole, and what unit is the unknown in?

Common mistake:

If you type 12 of x, you’ve dropped the percent sign, and the relationship comes out one hundred times too big. Keep 12% when you type numbers into Desmos, or use $0.12$ when you write algebra. To check, remember that $12\%$ of a positive whole has to be smaller than the whole.

Find the missing role

Any one of the three amounts can be the one you’re asked for. Label the part and the whole, then put $x$ in the missing spot. Desmos solves the equation whichever one it is.

What’s missing	What to type	The question sounds like
Part	`percent% of whole`	“What is this percent of the whole?”
Percent	`x% of whole = part`	“The part is what percent of the whole?”
Whole	`percent% of x = part`	“The part is this percent of what total?”

Find the part by multiplying

If $32\%$ of $275$ survey responses chose option A, then

32\%\text{ of }275=88.

So $88$ responses chose option A.

Find the percent by dividing

Say $54$ of $240$ seedlings are basil. The part is $54$ and the whole is $240$ , so type

x% of 240 = 54

Desmos gives $x=22.5$ , so $22.5\%$ of the seedlings are basil. Underneath, that’s a division, part over whole: $\frac{54}{240}\times100\%=22.5\%$ .

Find the whole by dividing

Say $84$ tickets are $35\%$ of all the tickets. Type

35% of x = 84

The vertical solution line is at $x=240$ , so there are $240$ tickets in all. This time you divide the part by the decimal: $84=0.35W$ , so $W=\frac{84}{0.35}=240$ .

Check your understanding:

Try all three with the same numbers: (1) What is $15\%$ of $360$ ? (2) $54$ is what percent of $360$ ? (3) $54$ is $15\%$ of what number?

Example: Find the whole from what’s left

Worked example

In a museum survey, $35\%$ of the visitors chose a guided tour. The remaining $390$ visitors chose a self-guided tour. How many visitors were surveyed in all?

A
$136.5$
B
$253.5$
C
$600$
D
$780$

Step 1

What percent are the 390 visitors?

If $35\%$ chose a guided tour, everyone else chose a self-guided tour:

100\%-35\%=65\%.

So the $390$ visitors are the part that makes up $65\%$ of the whole. This leftover share is called the complement.

Step 2

Write the part-whole equation

Let $W$ be the total number of visitors. The $390$ visitors are $65\%$ of that total, so $W$ goes after of:

65\%\text{ of }W=390.

Step 3

Solve, then check the total

Type

65% of x = 390

Desmos shows the vertical solution line at $x=600$ . To check, the guided-tour group is 35% of 600, or $210$ , and $210+390=600$ . The two groups add back to the whole, so the answer is C.

Common mistake:

It’s easy to type 65% of 390, but that treats $390$ as the whole. The question says $390$ is only the part that’s left, and that entry leads to choice B, $253.5$ . The total is what you don’t know, so it’s the $x$ : 65% of x = 390. Then check that the two groups add back to the total.

How to choose a method

The three methods below all use the same relationship, and in each one you decide which amount is the whole. After that, pick the quickest method you trust.

Type the percent into Desmos as written

When the question has numbers, start here. This is literal percent entry, and it pays off most with an awkward percent. Say $63$ lab samples are $18.75\%$ of all the samples. The whole is unknown, so it’s $x$ :

18.75% of x = 63

The vertical solution line is at $x=336$ . By hand, you’d turn $18.75\%$ into $0.1875$ and then divide $63$ by it, which gives you two places to slip. Typing the percent as written skips both.

With a friendly percent, your head can be faster. $25\%$ is one fourth, so $25\%$ of $68$ is $17$ .

Write an equation when the answer has letters

Some answers are expressions, not numbers. If $40\%$ of $s$ shirts are on sale, the sale shirts are $0.40s$ . That’s $P=rW$ again, with $r=\frac{p}{100}$ . Writing it out can also make the roles clearer when a question is hard to untangle.

Set up a proportion

You can also write

\frac{P}{W}=\frac{p}{100}.

It says the same thing as $P=rW$ and keeps the comparison in view, as long as the roles line up: part over whole equals percent over $100$ . It takes more writing, so use it when the fraction helps you more than it slows you down.

Related: Solve percent problems in Desmos has more practice turning percent sentences into Desmos equations, plus the percent changes that come next.

Check your understanding:

Which method would you start with? (1) Find $25\%$ of $68$ . (2) $63$ is $18.75\%$ of what total? (3) Write a relationship that says $q$ is $p\%$ of $n$ .

Common mistake:

Typing 18.75% of 63 = x makes $63$ the whole, so you get a smaller part instead of the total. Desmos solves whatever you type. It can’t tell which amount is the whole. Put the unknown total after of, then check: $63$ is only $18.75\%$ of a positive whole, so the total has to be bigger than $63$ .

Calculator loads as you approach

x is the unknown total. The vertical solution line shows x equals 336.

Keep part-whole and percent change separate

Some percent questions only look like part-whole questions. To tell them apart, ask whether an amount changes.

In a part-whole question, nothing grows or shrinks. One amount is a share of another, like the succulents out of all $640$ plants, so you can write part $=$ percent of whole directly. Leftover groups and groups inside groups count too, such as “ $40\%$ of the students who play a sport.”

If a town grows from $500$ to $600$ people, you know the amount before and after. If a $40 price goes up $10\%$ , you know the start and one change. Both are percent change questions.

If you know a price only after a $15\%$ increase and need the price before, or a price drops twice in a row, that’s where reverse and combined percent changes come in. Before you can write a part-whole equation, you turn each change into a change multiplier, the number you multiply the old amount by. After a $15\%$ increase, the new price is $115\%$ of the old one, so the multiplier is $1.15$ .

Check your understanding:

Which kind is each one? (1) $48$ is $30\%$ of what total? (2) A population rises from $800$ to $920$ . What is the percent increase? (3) After a $20\%$ discount, a price is $64. What was the original price? (4) $60\%$ of students take art, and $25\%$ of those students take choir. What percent of all students take both?

Practice problems

For each one, name the whole first, and make sure your answer is what the question asks for, in the right unit.

Find a part of the total

Practice problem

A research station processed $375$ soil and water samples. Of these samples, $28\%$ were soil samples. How many soil samples did the station process?

Calculator loads as you approach

Name the whole, then type the percent of it.

Find what percent of the tank is full

Practice problem

A water tank has a total capacity of $2{,}400$ liters. The tank currently contains $1{,}620$ liters of water.

The amount of water currently in the tank is what percent of the tank’s total capacity?

Your answer

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Let x stand for the unknown percent.

Find the whole through groups inside groups

Practice problem

At a science camp, $60\%$ of the participants chose field biology. Of the field-biology participants, $35\%$ chose a wetlands project. Of the field-biology participants who did not choose wetlands, $25\%$ chose a forestry project. The remaining $234$ field-biology participants chose neither project.

How many participants attended the science camp?

Calculator loads as you approach

Label each group that’s left, then type one percent equation.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Find the whole first. It’s the amount that counts as $100\%$ in that statement.
Part $=$ percent of whole, with the whole after of.
With numbers, type the percent into Desmos as written, percent sign and all. For an answer in letters, write $P=rW$ .
The part and whole share a unit, and a subgroup can be the whole for the next percent.
A friendly percent like $25\%$ is often faster in your head.

Related lesson

When the original is unknown after a change, or several changes happen in a row, use Reverse and combine percent changes.

Next lesson

Model percent increase and decrease

Compare a new quantity with its original and apply a one-step percent change.

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Practice

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275 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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