Percent Word Problems and Error Analysis | Grade 7 Math | LevelKite

A percent never stands alone. It is a rate per 100: it tells how many for every 100 of some amount, called the whole. The whole is a positive amount, and the problem states which amount it is. The part is the amount you get when you take the percent of the whole. For example, $25\%$ of a whole of $80$ is a part of $20$ , because $20$ out of $80$ is the same rate as $25$ out of $100$ .

You can already solve a percent proportion. A word problem adds one job before the solving: you read the situation and decide which amount is the whole.

You will then use that same reading skill for error analysis: you study a finished solution, find the first line that goes wrong, explain why it is wrong, and correct it.

Key concept

Every percent is a percent of a whole

A percent $p\%$ is a rate of $p$ per $100$ of a whole. When the whole is a positive amount stated in the problem, the part equals that rate times the whole. Because a positive whole is not zero, you may divide both sides by it, which gives the proportion you already use: $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ .

The whole matters: $20\%$ of $60$ is $12$ , but $20\%$ of $75$ is $15$ . So before you calculate, say in words which amount is the whole. Careful arithmetic does not fix a wrong whole: you can calculate every later line correctly and still get a wrong answer.

\text{part} = \frac{p}{100} \times \text{whole}, \quad \text{whole} > 0

Use the same four moves for any percent word problem.

Name the whole. Find the positive amount the percent is of. A discount is an amount taken off a price. A sales tax and a tip are amounts added on. Each one is a percent of the amount before it is taken off or added.
Define the unknown. Choose a letter, and say what it stands for and in what unit.
Write and solve the equation. Put the amounts you know into $\text{part} = \frac{p}{100} \times \text{whole}$ .
Answer the question asked. Some problems ask for the part. Others ask for the total after the part is added or taken off, which takes one more step.

Worked example

A discount and a sale price

A backpack has a regular price of $45$ dollars. It is on sale for $20\%$ off the regular price. What is the sale price?

Name the whole. The discount is 20% of the regular price, so the regular price, 45 dollars, is the whole. It is a positive amount. The discount is the part.
Define the unknown and write the equation. Let d be the discount in dollars. The part equals the rate per 100 times the whole. Here the rate is 20 per 100, which is 0.20.
$d = \frac{20}{100} \times 45 = 0.20 \times 45$
Multiply. Two tenths of 45 is 9, so the discount is 9 dollars. A quick check agrees: 10% of 45 is 4.5, and 20% is twice that.
$d = 0.20 \times 45 = 9$
Answer the question asked. The problem wants the sale price, not the discount. A discount is taken off the regular price, so subtract it.
$45 - 9 = 36$

Answer

The sale price of the backpack is $36$ dollars.

Some problems give two amounts, a starting one and a later one, and ask for a percent.

A percent change tells how much an amount went up or down, as a rate per 100 of the amount it started from. That starting amount is the original amount. It must be positive, and it is the whole. The part is the amount of change: the larger of the two amounts minus the smaller. If the amount went up, the change is a percent increase. If it went down, it is a percent decrease.

Because the original amount is positive, it is not zero, so you can divide the amount of change by it. The result is the rate, and writing that rate per 100 gives the percent $p$ .

\frac{\text{amount of change}}{\text{original amount}} = \frac{p}{100}, \quad \text{original amount} > 0

Worked example

A percent increase

A garden club had $40$ members last year. This year it has $52$ members. By what percent did the membership increase?

Name the whole. The membership started at 40, so the original amount is 40 members. It is positive, and it is the whole. The new amount, 52, is not the whole.
Find the part. The amount of change is the larger amount minus the smaller one, so the club gained 12 members.
$52 - 40 = 12$
Write the proportion. Let p be the percent of increase, which is the number of members gained for every 100 original members. Divide the change by the original amount. You can do this because 40 is not zero.
$\frac{p}{100} = \frac{12}{40} = \frac{3}{10} = 0.30$
Write the rate per 100. Multiply both sides by 100, which keeps the two sides equal.
$p = 0.30 \times 100 = 30$
Check in the story. A 30% increase adds 30% of the original 40 members, and the result should be this year's 52.
$40 + 0.30 \times 40 = 40 + 12 = 52$

Answer

The membership increased by $30\%$ of last year's $40$ members.

A percent error works the same way as a percent change. It tells how far a measured or estimated amount is from the accepted amount, which is the value taken as correct. The accepted amount must be positive, and it is the whole. The part is the size of the gap between the two amounts: the larger minus the smaller. Dividing that gap by the positive accepted amount gives the rate.

Suppose you measure a desk as $76$ cm long, and its accepted length is $80$ cm. The gap is $80 - 76 = 4$ cm. Divide by the accepted length: $4 \div 80 = 0.05$ , which is $5$ per $100$ . The percent error is $5\%$ .

Just as a percent change never divides by the new amount, a percent error never divides by the measured amount. Here, $4 \div 76$ would compare the gap with the wrong length.

Now turn the routine around. In error analysis, someone has already written the solution, and your job is to check it. Here the word error means a mistake in someone's work. It is not a percent error, which is about a measurement.

Read the work one line at a time and use the four checks in the table below. Look for the first line that fails, because the lines built on it carry the mistake along, even when their own arithmetic is correct.

Finish with a test that works for any percent problem: put the final answer back into the story. A correct answer makes every statement in the problem true. If any statement comes out false, the answer is wrong.

Four checks for a finished percent solution

Check	Question to ask	Example of a line that fails
Whole	Is the percent taken of the positive amount that the problem says it is of?	Club: $12 \div 52$ divides by the new amount instead of the original $40$ .
Rate	Is the percent written as a rate per $100$ ?	Backpack: $20 \times 45 = 900$ uses $20$ , not $\frac{20}{100}$ . The discount would be more than the price.
Arithmetic	Is the calculation itself correct?	Backpack: $0.20 \times 45 = 0.9$ , but the product is $9$ .
Question	Does the last line answer what the problem asks?	Backpack: stopping at $9$ dollars gives the discount, not the sale price.

Worked example

Find and fix the mistake

A game is on sale for $20\%$ off its original price. The sale price is $60$ dollars. Sam finds the original price this way: $0.20 \times 60 = 12$ , then $60 + 12 = 72$ . Sam says the original price is $72$ dollars. Find Sam's mistake and correct it.

Name the whole. The discount is 20% of the original price, so the original price is the whole. Its value is unknown, but a price is a positive amount. The sale price, 60 dollars, is what is left after the discount.
Test Sam's first line with the Whole check. Sam multiplies 0.20 by 60, but 60 dollars is the sale price, not the whole. Sam calculates the product correctly, yet 12 dollars is 20% of the wrong amount, so it is not the discount. Sam's second line adds correctly but builds on that wrong 12.
$0.20 \times 60 = 12 \quad \text{(60 is the sale price, not the whole)}$
Put Sam's answer back into the story. If the original price were 72 dollars, the discount would be 20% of 72, and the sale price would have to come out as 60. It comes out as 57.6, so 72 is wrong.
$72 - 0.20 \times 72 = 72 - 14.4 = 57.6 \neq 60$
Write the correct equation. Let w be the original price in dollars, with w greater than 0. The discount is the part, 0.20 times w. Taking it away from one whole w leaves 0.80 of w, and that remaining amount is the sale price.
$w - 0.20w = (1 - 0.20)w = 0.80w = 60$
Solve. Divide both sides by 0.80. Dividing both sides by the same nonzero number keeps them equal. To divide by the decimal, multiply the top and the bottom by 10, which does not change the value.
$w = \frac{60}{0.80} = \frac{600}{8} = 75$
Check in the story. Take 20% of 75 off 75 and see whether the sale price is 60. It is, so every statement in the problem is true.
$75 - 0.20 \times 75 = 75 - 15 = 60$

Answer

Sam took $20\%$ of the sale price, but the discount is $20\%$ of the original price. The original price is $75$ dollars, not $72$ dollars.

The takeaway. A percent is a rate per 100 of a stated positive whole, so every percent problem begins with one question: of what? Name the whole, define the unknown with its unit, use $\text{part} = \frac{p}{100} \times \text{whole}$ , and answer the question that was asked. A percent change divides by the positive original amount, and a percent error divides by the positive accepted amount, never by the new or measured amount. To analyze a finished solution, test each line for the right whole, the right rate, correct arithmetic and the right question, then put the answer back into the story.