Percents · Grades 7
Percent Word Problems: Which of the Three Questions Is It?
Quick answer
Every percent problem gives you two of three quantities — the part, the whole and the percent — and asks for the third. Know the whole and the percent, multiply. Know the part and the whole, divide part by whole. Know the part and the percent, divide part by the decimal. Naming which one you have is most of the work.
What you'll learn
- Identify which of the three percent questions a problem is asking
- Solve multi-step percent problems involving money
- Check that an answer is a sensible size before reporting it
Why every percent problem is one of three
Every percent problem involves the same three things:
- the whole — what everything is measured against
- the part — some amount of it
- the percent — how big the part is compared with the whole
A problem always hands you two and asks for the third. Working out which one is missing is most of the job.
| Missing | You do | Example |
|---|---|---|
| the part | multiply | of is |
| the percent | divide part by whole | out of is |
| the whole | divide part by the decimal | is of |
All three lines describe the same situation. They differ only in which number was withheld.
Finding the whole
The whole is the amount everything else is compared against. In practice:
| Problem | The whole |
|---|---|
| a discount on a price | the original price |
| a test score | the total marks available |
| a population change | the population before |
| a tip on a bill | the bill before the tip |
The word “of” points at it almost every time. So does “out of”.
The size check
Before writing an answer down, ask whether it is the right size.
| The percent is | The answer should be |
|---|---|
| under | smaller than the whole |
| exactly | the same |
| over | bigger |
This one check catches misplaced decimal points, upside-down divisions and multiplying where you meant to divide — which between them account for nearly every wrong answer on this topic.
Two-step problems
Longer problems are two short ones. The difficulty is not the arithmetic but keeping track of what each intermediate number means.
Write it down. “$160 is the price after the discount” takes three seconds and prevents feeding it into the wrong next step.
Worked examples
Common mistakes
Practice problems
-
A $70 pair of shoes has sales tax. How much is the tax?
Answer
$5.60
Full solution
The whole and the percent are known, so multiply: .
-
A student scores out of . What percent is that?
Answer
Full solution
, so .
-
is of what number?
Answer
Full solution
. Checking: ✓
-
A $45 shirt is off. What do you pay?
Answer
$36
Full solution
You pay : .
-
A phone bill rises from $40 to $46. What is the percent increase?
Answer
Full solution
Change is , and , so a increase.
The original $40 is the whole, not the new $46.
-
A $150 item is off, then tax is added. What is the total?
Hint
Two steps. Tax goes on the sale price.
Answer
$110.25
Full solution
, then .
-
After a discount, a bag costs $52. What was the original price?
Hint
$52 is of it.
Answer
$65
Full solution
. Checking: ✓
-
In a class of , there are girls. What percent are boys?
Hint
Two routes. Do both and check they agree.
Answer
Full solution
There are boys, and , so .
Or: girls are , and the rest is .
-
A $400 investment earns simple interest for years. What is it worth at the end?
Answer
$448
Full solution
, and the total is .
The question asked what it is worth, so the interest alone is not the answer.
-
A shop reports that of its customers are local, and of local customers buy something. What percent of all customers buy something?
Hint
The is a percent of the locals, not of everyone.
Answer
Full solution
, so .
Checking with customers: are local, and of is — which is of ✓
Adding the percents, or taking of everyone, would both overstate it, because the second percent was never measured against the whole.
Frequently asked questions
How do I know whether to multiply or divide?
If you know the whole and want a part, multiply. If you know a part and want to express it as a percent, divide the part by the whole. If you know a part and its percent and want the whole, divide the part by the decimal.
How do I find which number is the whole?
It is the amount everything else is measured against — usually the original price, the full class, the starting population. It normally follows the words of or out of.
The answer came out bigger than the number I started with. Is that wrong?
Only if the percent was under 100. Below 100% a part must be smaller than its whole, so a larger answer means a step went the wrong way. Above 100% a larger answer is correct.
What do I do with two-step problems?
Do one step at a time and write down what each answer means before starting the next. Most errors in multi-step problems come from feeding the right number into the wrong step, not from the arithmetic.
Do I round as I go?
No. Carry the full figure and round once at the end. Rounding partway compounds the error, and with money the difference can show up in the final cents.
Formulas on this page
Standards alignment
This lesson covers the following Common Core State Standards for Mathematics.
- CCSS.MATH.CONTENT.7.RP.A.3Ratios and Proportional RelationshipsUse proportional relationships to solve multistep ratio and percent problems.
- CCSS.MATH.CONTENT.6.RP.A.3cRatios and Proportional RelationshipsFind a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.