Percent Error and Checking Percent-of-Change Work | Grade 7 Math | LevelKite

You already know how to find a percent of change: you compare the amount of change with the amount you started from. In this lesson you use that idea in two new ways. First, you find how far off a measurement or an estimate is. Second, you check finished percent work, find the mistake, and fix it.

Begin with two measurements that each miss by 2 centimeters. One is the length of a pencil that is really 10 centimeters long. The other is the height of a door that is really 200 centimeters tall. The misses are the same size, but they are not equally serious. Missing by 2 out of 10 is a large share of the pencil. Missing by 2 out of 200 is a tiny share of the door. To judge a miss fairly, compare it with the true size of the thing you measured.

\text{pencil: } \frac{2}{10} = \frac{20}{100} = 20\% \qquad \text{door: } \frac{2}{200} = \frac{1}{100} = 1\%

Both fractions were rewritten with a denominator of 100: the first by multiplying the top and bottom by 10, the second by dividing the top and bottom by 2. Neither move changes the fraction's value, so each one now reads as a rate per 100. The pencil measurement is off by 20 centimeters for every 100 centimeters of true length. The door measurement is off by only 1 centimeter for every 100. Now the two misses are easy to compare.

A rate like this is called a percent error. It tells you how large a miss is for every 100 units of the true amount. A percent is a rate per 100 of a stated positive whole, which is the amount everything else is compared with. Here the whole is the true amount. So the true amount must be positive, and it is the number you divide by. You never divide by the measurement itself.

Key concept

Percent error is a rate per 100 of the accepted value

Let $A$ be the accepted value and $M$ be the measured value, both in the same unit, with $A > 0$ .

The error is $|M - A|$ . The bars mean absolute value: subtract, then keep the size of the result without any negative sign. So the error is never negative.

Percent error is that error written as a rate per 100 of the stated positive whole, $A$ . Divide by the accepted value $A$ , never by the measured value $M$ .

Percent of change follows the same pattern. Its whole is the positive original amount, so you divide by the original amount, never by the new amount.

\text{percent error} = \frac{|M - A|}{A} \times 100\%, \quad A > 0

Worked example

A scale reading that is too low

A metal block has a known mass of exactly $50$ grams. Your scale reads $47$ grams. What is the percent error of your reading?

Name the two values. The known mass, 50 grams, is the accepted value. The scale reading, 47 grams, is the measured value. Both are in grams, and the accepted value is positive, so it can be the whole.
$A = 50, \quad M = 47$
Find the error. Subtracting gives negative 3. The negative sign only tells you the reading is too low. The error is the size of that difference, so the reading is off by 3 grams.
$|M - A| = |47 - 50| = |-3| = 3$
Divide the error by the accepted value, because percent error is a rate per 100 of the positive accepted value, never of the measured value. Multiply the top and bottom by 2 to get a denominator of 100. Multiplying both by the same nonzero number keeps the fraction's value the same. Grams divided by grams leaves a plain number.
$\frac{3}{50} = \frac{3 \times 2}{50 \times 2} = \frac{6}{100} = 0.06$
Write the result as a percent. Multiplying by 100% is multiplying by 1, because 100% means 100 per 100. The value stays the same and only its form changes.
$0.06 \times 100\% = 6\%$
Check by multiplying back. Percent error is a rate per 100 of the positive accepted value, so 6% of the accepted 50 grams should give back the error of 3 grams. It does.
$0.06 \times 50 = 3$

Answer

The percent error of the scale reading is $6\%$ . The reading is off by $6$ grams for every $100$ grams of the accepted mass.

Worked example

Does the tracker pass the test?

The park office confirms that a trail is $12.5$ kilometers long. Your fitness tracker records $13.1$ kilometers for the same trail. A hiking club trusts a tracker only if its percent error is less than $5\%$ . Find the percent error and decide whether your tracker passes.

Decide which value is accepted. The confirmed length, 12.5 kilometers, is the accepted value, and it is positive. The tracker's 13.1 kilometers is the measured value, because it is the reading being judged.
$A = 12.5, \quad M = 13.1$
Find the error. The tracker reads high, so the difference is already positive. The reading is off by 0.6 kilometers.
$|M - A| = |13.1 - 12.5| = |0.6| = 0.6$
Divide the error by the positive accepted value, never by the tracker reading. Multiply the top and bottom by 8, which keeps the value the same and gives a denominator of 100. Then read the result as a rate per 100.
$\frac{0.6}{12.5} = \frac{0.6 \times 8}{12.5 \times 8} = \frac{4.8}{100} = 4.8\%$
Check by multiplying back. As a decimal, 4.8% is 0.048. That rate times the positive accepted value should give back the error of 0.6 kilometers, and it does.
$0.048 \times 12.5 = 0.6$
Compare with the club's limit. Both percents are rates per 100 of the same positive whole, the accepted length, so you can compare them directly. Since 4.8 is less than 5, the tracker is inside the limit.
$4.8\% < 5\%$

Answer

The tracker's percent error is $4.8\%$ , which is less than $5\%$ , so the tracker passes the club's test.

Now use the same idea to check someone else's work. Error analysis, also called auditing a solution, means reading finished work, finding the exact step where it goes wrong, explaining why that step is wrong, and writing the correct result.

A fixed order helps you look at every part of the work. The five checks in the next table fit percent of change and percent error alike, because both are rates per 100 of a stated positive whole: the original amount for percent of change, and the accepted value for percent error. In both, you divide by that whole, never by the new or measured amount. Each check uses only the amounts and operations written in the problem and the solution, so you can audit the work line by line.

Five checks for auditing percent work

Check	Question to ask	What correct work shows
1. The whole	Which amount is the original or accepted one? Is it positive?	The positive original or accepted amount is named as the whole
2. The difference	Were the right two amounts compared?	The difference between new and original amounts, or between measured and accepted values
3. The division	Was the difference divided by the whole?	Division by the positive original or accepted amount, never by the new or measured one
4. The percent and label	Is the division result written as a rate per 100, with the right label?	A percent; a change is an increase if the new amount is greater than the original, a decrease if less
5. Multiply back	Does that percent of the whole give back the difference?	The rate times the positive whole equals the size of the difference

Worked example

Audit a percent of change solution

A jacket's price drops from $80$ dollars to $50$ dollars. Sam's work has three lines. Line 1: the change is $80 - 50 = 30$ dollars. Line 2: $30 \div 50 = 0.6$ . Line 3: $0.6 = 60\%$ , so the price decreased by $60\%$ . Find Sam's mistake, explain why it is a mistake, and correct it.

Check 1, the whole. The price started at 80 dollars, so 80 dollars is the original amount. It is positive, so it is the whole for this percent of change. The new amount is 50 dollars.
$\text{original} = 80, \quad \text{new} = 50$
Check 2, the difference. Line 1 subtracts the new price from the original price and gets 30 dollars. That is the correct size of the change. The new price is less than the original, so calling it a decrease is also correct.
$80 - 50 = 30$
Check 3, the division. Line 2 divides by 50, the new price. This is the mistake. A percent of change is a rate per 100 of the positive original amount, so the change must be divided by 80, never by the new amount. Line 3 converts 0.6 to 60% correctly, but it starts from the wrong division result.
$\text{Line 2: } 30 \div 50 \quad \text{should be} \quad 30 \div 80$
Correct the division by using the positive original amount, 80 dollars. Then apply Check 4: write the result as a percent and keep the label decrease. Multiplying by 100% is multiplying by 1, so the value stays the same.
$\frac{30}{80} = \frac{3}{8} = 0.375, \qquad 0.375 \times 100\% = 37.5\%$
Check 5, multiply back. Because 80 dollars is the positive whole, 37.5% of 80 should give back the change of 30 dollars, and 80 minus 30 should be the new price, 50. Both work.
$0.375 \times 80 = 30, \qquad 80 - 30 = 50$
Run the same check on Sam's answer to see it fail. A percent of change is a rate per 100 of the positive original amount, so a 60% decrease would remove 60% of 80 dollars. That is 48 dollars, which leaves 32 dollars, not the real new price of 50 dollars.
$0.6 \times 80 = 48, \qquad 80 - 48 = 32 \ne 50$

Answer

Sam's mistake is in Line 2: Sam divided by the new price, $50$ dollars, instead of the original price, $80$ dollars. The price actually decreased by $37.5\%$ .

Four more solutions, audited with the five checks

Work being checked	Check it fails	Why	Corrected result
From $40$ to $46$ : $6 \div 40 = 0.15$ , so the increase is $0.15\%$ .	Check 4	$0.15$ means $15$ per $100$ of the positive original, $40$ . That is $15\%$ , not $0.15\%$ .	$0.15 \times 100\% = 15\%$ increase
A price is $50$ dollars after a $25\%$ increase. Original: $50 - 0.25 \times 50 = 37.50$ dollars.	Check 1	The $25\%$ is per 100 of the positive original price, not the new one. So $50$ is $125\%$ of the original.	$50 \div 1.25 = 40$ dollars; check: $1.25 \times 40 = 50$
From $90$ to $72$ : $18 \div 90 = 0.2$ , so this is a $20\%$ increase.	Check 4	The new amount, $72$ , is less than the positive original, $90$ , so the amount went down.	$20\%$ decrease
Accepted $40$ , measured $43$ : $3 \div 40 = 0.075$ , so the percent error is $7.5\%$ .	None	It divides by the positive accepted value, $40$ , and $0.075 \times 40 = 3$ gives back the error.	Already correct: $7.5\%$

Takeaway. Percent error and percent of change ask the same kind of question: how large is a difference for every 100 of the whole? The whole is a stated positive amount. For percent error it is the accepted value, and for percent of change it is the original amount. You divide by that whole, never by the measured or new amount.

When you audit percent work, find the whole first. Then check the difference, the division, the percent and its label, and finish by multiplying back. The first and third checks catch the mistake this lesson warned about: dividing by the wrong amount.