Percent of Change: Compare with the Original Amount | Grade 8 Math | LevelKite

Two plants each grow $5$ cm in a month. A bean plant starts at $10$ cm tall, and a young tree starts at $100$ cm tall. Did they change by the same amount?

In centimeters, yes. But the bean plant added half of its starting height, while the tree added only a small part of its starting height. To describe a change fairly, you compare it with where the amount started.

Three terms make this precise:

The original amount is the amount before the change. It is the reference, the whole that you compare the change with.
The new amount is the amount after the change.
The amount of change is the new amount minus the original amount.

The percent of change tells you how large the amount of change is as a percent of the original amount. In this lesson, the problem always states the original amount, the original amount is positive, and both amounts use the same unit.

Same amount of change, different original amounts

Plant	Original height	New height	Amount of change	Change divided by original	Percent of change
Bean plant	$10$ cm	$15$ cm	$15 - 10 = 5$ cm	$\frac{5 \text{ cm}}{10 \text{ cm}} = 0.5$	$50\%$
Young tree	$100$ cm	$105$ cm	$105 - 100 = 5$ cm	$\frac{5 \text{ cm}}{100 \text{ cm}} = 0.05$	$5\%$

Read across each row of the table. Both plants gained $5$ cm, but $5$ cm is $50\%$ of the bean plant's original height and only $5\%$ of the tree's original height. The amounts of change match. The percents differ because the original amounts differ.

This is the percent equation you already know, $\text{part} = \text{percent} \times \text{whole}$ , with two fixed roles: the whole is the original amount, and the part is the amount of change.

Now use symbols. Let $a$ be the original amount that the problem states, where $a > 0$ . Let $b$ be the new amount, measured in the same unit as $a$ . Let $p$ be the percent of change written as a decimal. With these symbols, the percent equation reads $b - a = p \times a$ .

Because $a > 0$ , you know $a$ is not $0$ , so you can divide both sides by $a$ . Dividing both sides of an equation by the same number that is not $0$ keeps the two sides equal. On the right side, $p \times a$ divided by $a$ is $p$ , because $a \div a = 1$ . That leaves $p$ by itself:

p = \frac{b - a}{a}, \quad \text{where } a > 0

Key concept

Percent of change compares the change with the original

Let $a$ be the stated original amount, with $a > 0$ , and let $b$ be the new amount in the same unit. To find the percent of change, divide the amount of change, $b - a$ , by the original amount $a$ and write the result as a percent. The matching units divide out, so the result has no unit. Multiplying by $100\%$ only rewrites the number as a percent, because $100\% = 1$ .

You divide by a positive number, so the result has the same sign as $b - a$ :

Positive: the amount went up. This is a percent increase.
Negative: the amount went down. This is a percent decrease. You state its size, so $-15\%$ is a $15\%$ decrease.
Zero: the amount did not change.

\text{percent of change} = \frac{b - a}{a} \times 100\%, \quad a > 0

Worked example

A percent increase

A robotics club had $40$ members last year. This year it has $52$ members. What is the percent of change in the number of members?

Name the two amounts. The original amount is the number of members before the change, 40. The problem states it and it is positive, so you can use it as the reference. The new amount is 52, counted in the same unit, members.
$a = 40, \quad b = 52$
Find the amount of change by subtracting the original amount from the new amount. The result is positive, so the number of members went up.
$b - a = 52 - 40 = 12$
Compare the change with the original amount by dividing. The unit, members, divides out. Then simplify the fraction by dividing the top and bottom by 4, and write it as a decimal.
$\frac{12 \text{ members}}{40 \text{ members}} = \frac{12}{40} = \frac{3}{10} = 0.3$
Write the decimal as a percent by multiplying by 100%. The value does not change, because 100% equals 1.
$0.3 \times 100\% = 30\%$
Check with the percent equation. 30% of the original 40 should equal the amount of change, and adding that change to 40 should give the new amount.
$0.3 \times 40 = 12, \quad 40 + 12 = 52$

Answer

The percent of change is $30\%$ , so the club had a $30\%$ increase from its original $40$ members.

Worked example

A percent decrease

A runner's time for one lap drops from $80$ seconds to $68$ seconds. What is the percent of change in the lap time?

The original amount is the time before the change, 80 seconds. The problem states it and it is positive. The new amount is 68 seconds. Both times are in seconds, so the units match.
$a = 80, \quad b = 68$
Subtract the original amount from the new amount. The result is negative, which tells you the time went down.
$b - a = 68 - 80 = -12$
Divide the change by the original amount, then simplify by dividing the top and bottom by 4. A negative number divided by a positive number is negative, so the sign still shows a decrease.
$\frac{-12}{80} = -\frac{3}{20} = -0.15$
Multiply by 100% to write the decimal as a percent. The value stays the same because 100% equals 1.
$-0.15 \times 100\% = -15\%$
Check: 15% of the original 80 should equal the size of the change, and subtracting it from 80 should give the new amount.
$0.15 \times 80 = 12, \quad 80 - 12 = 68$

Answer

The percent of change is $-15\%$ , so the lap time had a $15\%$ decrease from its original $80$ seconds.

Worked example

Same change, different original

The robotics club from the first example has $52$ members. The next year it falls back to $40$ members. What is the percent of change, to the nearest tenth of a percent?

Now 52 is the number of members before the change, so 52 is the stated original amount, and it is positive. The new amount is 40. The starting point switched, so the reference switched too.
$a = 52, \quad b = 40$
Subtract the original amount from the new amount. The negative result shows a decrease.
$b - a = 40 - 52 = -12$
Divide the change by the original amount, 52, and simplify by dividing the top and bottom by 4. This fraction is the exact value.
$\frac{-12}{52} = -\frac{3}{13}$
Write the fraction as a decimal by dividing 3 by 13. The digits repeat without ending, and the three dots show that the decimal continues.
$-\frac{3}{13} = -0.230769\ldots$
Multiply by 100% to write the decimal as a percent, then round to the nearest tenth of a percent. Use the approximately-equal sign here, because rounding changed the value slightly: the rounded percent is close to the exact value but not equal to it.
$-0.230769\ldots \times 100\% = -23.0769\ldots\% \approx -23.1\%$
Check with the rounded percent: 23.1% of the original 52 should be close to 12, the size of the change. It is close but not exactly 12, because the percent was rounded.
$0.231 \times 52 = 12.012 \approx 12$

Answer

The percent of change is about $-23.1\%$ , so the club had a decrease of about $23.1\%$ from its original $52$ members. The exact change is $-\frac{3}{13}$ of the original amount.

The same 12 members compared with two different original amounts

Change in members	Original amount	Amount of change	Change divided by original	Percent of change
From $40$ to $52$	$40$	$52 - 40 = 12$	$\frac{12}{40} = 0.3$	$30\%$ increase
From $52$ to $40$	$52$	$40 - 52 = -12$	$\frac{-12}{52} \approx -0.2308$	about $23.1\%$ decrease

The club gained $12$ members and later lost $12$ members, but the two percents are not opposites. The $12$ members are $30\%$ of the original $40$ and only about $23.1\%$ of the original $52$ . Each percent refers to its own whole, the original amount for that change. So before you divide, ask yourself: which amount came first?

A percent of change compares the amount of change with the original amount. Subtract the original amount from the new amount, divide by the original amount, and write the result as a percent. This works when the problem states the original amount, that amount is positive, and both amounts use the same unit. Then a positive result is a percent increase, and a negative result is a percent decrease. If rounding changes a percent, say about or write $\approx$ to show that the value is close but not exact.

Next, you will work on percent word problems and learn to find and correct errors in percent problems.