Why do I divide by the original amount instead of the new amount?

Because percent change answers the question "how big was this change relative to where we started?" The original amount is the reference point, the whole that the change is being compared to. Dividing by the new amount answers a different question and gives a different percent. Ask which value came first in time; that one is the denominator.

How do I know whether to multiply by 1.25 or by 0.25?

Read whether the problem gives you a whole or a part. "Increased by 25 percent" means you keep all of the original and add a quarter more, so multiply by 1.25. "25 percent of" means you want only the part, so multiply by 0.25. And "decreased by 25 percent" means 75 percent remains, so multiply by 0.75.

Can a percent increase be more than 100 percent?

Yes. If a quantity more than doubles, the increase is more than 100 percent — going from 20 to 70 is 50/20 = 250\%. A percent decrease, though, can never exceed 100 percent, because a 100 percent decrease already brings the quantity to zero.

Why does a 30 percent increase followed by a 30 percent decrease not bring me back to the start?

The two percents are taken from different amounts. Starting at 100, a 30 percent increase gives 130; then 30 percent of 130 is 39, so you land at 91. The decrease is calculated from the larger number, so more comes off than went on. Multiplying multipliers shows the net effect: 1.3 × 0.7 = 0.91, a 9 percent decrease overall.

M7MATH-4.2

Percent Increase & Decrease

Learn to compute percent increase and decrease by comparing the change to the original amount, and use a single multiplier like 1.15 or 0.85 to find the new amount fast.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Percent Increase & Decrease, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Percent Increase & Decrease free

What this lesson covers

A sweatshirt that used to cost 40 dollars now costs 46 dollars. A town's population dropped from 12,000 to 10,800. In both cases something changed, and the useful question is not just "how much?" but "how much compared to what it started at?" That comparison is percent change, and it is the reason a 5 dollar raise means something completely different to someone earning 20 dollars an hour than to someone earning 200 dollars an hour.

In this lesson you will learn the one formula behind every percent increase and percent decrease problem, why the denominator must always be the original amount, and how to skip the two-step "find the change, then add it" routine by multiplying once by a number like

1.15

0.88

. That single-multiplier idea is the tool you will lean on for the rest of the unit and again in algebra.

The Percent Change Formula

Percent change measures a change as a fraction of where you started:

\text{percent change} = \frac{\text{amount of change}}{\text{original amount}} \times 100\%

The amount of change is the new amount minus the original amount. If that difference is positive, you have a percent increase. If it is negative, you have a percent decrease, and you report the size of the drop as a positive percent ("a 10 percent decrease").

Work it in three steps. First, subtract to find the change. Second, divide by the original. Third, convert the decimal to a percent by multiplying by 100.

Example: a plant grows from 24 cm to 30 cm. The change is

30 - 24 = 6

cm. Then

\frac{6}{24} = 0.25 = 25\%

, so the height increased by 25 percent.

Example: attendance falls from 250 students to 225. The change is

225 - 250 = -25

. Then

\frac{25}{250} = 0.1 = 10\%

, a 10 percent decrease.

The single most common mistake is dividing by the wrong number. In the attendance problem,

\frac{25}{225} \approx 11.1\%

is wrong because 225 is not where the situation started. Before you divide, ask yourself out loud: which number came first in time? That number goes on the bottom. Notice also what percent change is not: it is not the difference between two percents. Going from 20 percent to 25 percent battery is a 5 percentage-point difference but a 25 percent increase.

One Multiplier Instead of Two Steps

Suppose a 60 dollar jacket price increases by 15 percent. The slow way: find 15 percent of 60, which is 9 dollars, then add to get 69 dollars. The fast way uses the fact that the new price is 100 percent of the original plus 15 percent of the original, or 115 percent of it:

\text{new} = \text{original} \times (1 + r)

60 \times 1.15 = 69

dollars in one keystroke sequence. For a decrease you keep what is left over: a 15 percent drop leaves 85 percent, so

\text{new} = \text{original} \times (1 - r)

and

60 \times 0.85 = 51

dollars.

Change	Percent kept	Multiplier
Up 8%	108%	$1.08$
Up 40%	140%	$1.40$
Up 7.5%	107.5%	$1.075$
Down 8%	92%	$0.92$
Down 40%	60%	$0.60$
Down 5%	95%	$0.95$

Here

r

is the percent written as a decimal, so 8 percent becomes

0.08

, not

0.8

. Writing

1 + 0.8 = 1.8

turns an 8 percent raise into an 80 percent raise, which is the error to watch for.

A good habit: before computing, predict whether your answer should be bigger or smaller than the original. An increase multiplier is always greater than 1; a decrease multiplier is always between 0 and 1. If you multiply by

0.85

and get a bigger number, something went wrong.

Working Backward to the Original

Many real problems hand you the new amount and ask for the original. A sale price is 51 dollars after a 15 percent discount — what was the regular price? Students often take 15 percent of 51 and add it back, getting about 58.65 dollars. That is wrong, because the 15 percent was taken from the larger original price, not from 51.

Set it up as an equation with the multiplier:

0.85x = 51

Divide both sides by

0.85

x = \frac{51}{0.85} = 60

dollars. Check by going forward:

60 \times 0.85 = 51

. That check catches almost every error in this kind of problem.

The same idea handles increases. If a population grew 20 percent and is now 8,400, then

1.2x = 8400

, so

x = \frac{8400}{1.2} = 7000

.

Related trap: percent changes do not simply add. A price goes up 10 percent, then down 10 percent. Starting at 200 dollars,

200 \times 1.1 = 220

, and

220 \times 0.9 = 198

. You end 2 dollars below where you started, not back at 200, because the 10 percent decrease was taken from the larger 220. To combine two changes, multiply the multipliers:

1.1 \times 0.9 = 0.99

, a net 1 percent decrease. Adding and subtracting percents of different wholes is the deepest misconception in this topic, and multipliers are what protect you from it.

Reading the Words Carefully

Percent problems live or die on wording. "Increased by 30 percent" and "increased to 30 percent of the original" describe opposite things: the first multiplies by

1.30

, the second by

0.30

. Circle the words by, to, of, more than, and less than before you touch the numbers.

Phrase	Meaning	Computation
25% more than 80	increase	$80 \times 1.25 = 100$
25% of 80	part of it	$80 \times 0.25 = 20$
25% less than 80	decrease	$80 \times 0.75 = 60$
80 is what percent of 100	comparison	$\frac{80}{100} = 80\%$
80 increased to 100	percent increase	$\frac{20}{80} = 25\%$

Notice the last two rows use the same pair of numbers and give different answers, because they ask different questions. "What percent of" puts the whole in the denominator; "percent increase" puts the original in the denominator and only the change on top.

Rounding matters too. If a problem gives money, round to the nearest cent unless told otherwise, and round percents to the nearest tenth if the division does not come out even. For example, going from 45 to 52 is

\frac{7}{45} \approx 0.1556 \approx 15.6\%

. Do not round in the middle of a multi-step problem; carry the full decimal and round only your final answer, or your result can drift.

Finally, always label your answer as an increase or a decrease. "12 percent" alone does not tell your reader which direction the quantity moved.

Choosing a Strategy Fast

Once you recognize the shape of a percent problem, the method follows automatically. Sort every problem into one of three types by asking what is missing.

What you know	What is missing	Method
Original and new amounts	The percent	$\frac{\text{change}}{\text{original}} \times 100\%$
Original and the percent	The new amount	Multiply by $1 + r$ or $1 - r$
New amount and the percent	The original	Solve $(1 \pm r)x = \text{new}$

Estimation is your safety net for all three. Ten percent of a number is easy — move the decimal point one place left — and you can build from there. Ten percent of 340 is 34, so 5 percent is 17 and 15 percent is 51. If a problem says 340 increased by 15 percent and your answer is 3,910, you know instantly it is off, because the answer should land near 391.

Benchmarks help you sanity-check percent change too. If a quantity roughly doubles, that is about a 100 percent increase. If it grows by half, about 50 percent. If it drops to about three quarters of what it was, that is about a 25 percent decrease. A quantity can increase by more than 100 percent (tripling is a 200 percent increase), but it can never decrease by more than 100 percent, since a 100 percent decrease already means the quantity is gone. Getting a percent decrease larger than 100 is a sure sign you divided by the wrong amount.

Key terms

Percent change.: The amount of change expressed as a percent of the original amount: $\frac{\text{change}}{\text{original}} \times 100\%$ .
Percent increase.: A percent change in which the new amount is larger than the original amount.
Percent decrease.: A percent change in which the new amount is smaller than the original amount, reported as a positive percent.
Original amount.: The starting value in a comparison. It is always the denominator when computing percent change.
Amount of change.: New amount minus original amount. Its size, ignoring sign, goes in the numerator of the percent change formula.
Multiplier (scale factor).: The single number you multiply by to get the new amount: $1 + r$ for an increase, $1 - r$ for a decrease, where $r$ is the percent as a decimal.
Percentage point.: A unit for the plain difference between two percents. Going from 20 percent to 25 percent is 5 percentage points but a 25 percent increase.

Worked example

A bike shop raises the price of a helmet from 32 dollars to 38 dollars. (a) Find the percent increase, rounded to the nearest tenth of a percent. (b) Later the shop marks the 38 dollar price down by 25 percent. Find the sale price using a single multiplier. (c) Compared with the original 32 dollars, is the sale price an increase or a decrease, and by what percent?

Part (a). Find the amount of change first:

38 - 32 = 6

dollars. The original amount is 32, since that is where the price started. Divide:

\frac{6}{32} = 0.1875

. Multiply by 100 to get a percent:

18.75\%

, which rounds to

18.8\%

. So the price increased by about 18.8 percent. A quick check: 10 percent of 32 is 3.2 and 20 percent is 6.4, so a 6 dollar jump should be a bit under 20 percent. It is.

Part (b). A 25 percent markdown means the shopper still pays

100\% - 25\% = 75\%

of the price, so the multiplier is

0.75

. The markdown is taken from the current price of 38 dollars, not from the old 32:

38 \times 0.75 = 28.50

. The sale price is 28 dollars and 50 cents.

Part (c). Now compare 28.50 with the original 32. The change is

28.50 - 32 = -3.50

, which is negative, so this is a decrease. Divide the size of the change by the original:

\frac{3.50}{32} = 0.109375

, or about

10.9\%

. The sale price is roughly a 10.9 percent decrease from the original 32 dollars.

Notice that the 18.75 percent increase and the 25 percent decrease did not combine into a 6.25 percent decrease. Multiplying the multipliers shows why:

1.1875 \times 0.75 = 0.890625

, and

1 - 0.890625 = 0.109375

, matching the 10.9 percent found above.

Practice questions

A jacket originally priced at 80 dollars is reduced to 68 dollars. What is the percent decrease?

15%
17.6%
12%
85%

Answer: 15%

The amount of change is

80 - 68 = 12

dollars. Divide by the ORIGINAL price, 80:

\frac{12}{80} = 0.15 = 15\%

. The choice 17.6% comes from dividing by the new price, 68, which is the most common error here. The choice 12% mistakes the amount of change in dollars for the percent, and 85% is the percent of the original that remains, not the percent decrease.

After a 20 percent increase, a phone plan costs 42 dollars per month. Explain why adding 20 percent of 42 to 42 does not give the old price, then find the old price correctly.

Answer: The old price was 35 dollars per month. Reversing an increase requires dividing by 1.20, not adding 20 percent of the new amount.

The 20 percent increase was computed from the OLD price, so 20 percent of 42 is not the same amount of money that was added. Set up the multiplier equation: old price times

1.20

equals 42, or

1.2x = 42

. Divide both sides by

1.2

to get

x = 35

. Check forward:

35 \times 1.2 = 42

, which confirms it. By contrast, the incorrect method gives

42 + 8.40 = 50.40

, which is larger than the new price — a clear signal it cannot be the original.

A store increases a 50 dollar item by 10 percent, then decreases the new price by 10 percent. Find the final price and the overall percent change from the original.

Answer: The final price is 49 dollars and 50 cents, a 1 percent decrease overall.

Apply the multipliers in order:

50 \times 1.10 = 55

, then

55 \times 0.90 = 49.50

. The two percents do not cancel because the 10 percent decrease is taken from 55, a bigger base than 50, so more money comes off than went on. Combining multipliers gives the same result directly:

1.10 \times 0.90 = 0.99

, and

50 \times 0.99 = 49.50

. Since

0.99 = 1 - 0.01

, the net effect is a 1 percent decrease.

FAQ

Why do I divide by the original amount instead of the new amount?: Because percent change answers the question "how big was this change relative to where we started?" The original amount is the reference point, the whole that the change is being compared to. Dividing by the new amount answers a different question and gives a different percent. Ask which value came first in time; that one is the denominator.
How do I know whether to multiply by 1.25 or by 0.25?: Read whether the problem gives you a whole or a part. "Increased by 25 percent" means you keep all of the original and add a quarter more, so multiply by $1.25$ . "25 percent of" means you want only the part, so multiply by $0.25$ . And "decreased by 25 percent" means 75 percent remains, so multiply by $0.75$ .
Can a percent increase be more than 100 percent?: Yes. If a quantity more than doubles, the increase is more than 100 percent — going from 20 to 70 is $\frac{50}{20} = 250\%$ . A percent decrease, though, can never exceed 100 percent, because a 100 percent decrease already brings the quantity to zero.
Why does a 30 percent increase followed by a 30 percent decrease not bring me back to the start?: The two percents are taken from different amounts. Starting at 100, a 30 percent increase gives 130; then 30 percent of 130 is 39, so you land at 91. The decrease is calculated from the larger number, so more comes off than went on. Multiplying multipliers shows the net effect: $1.3 \times 0.7 = 0.91$ , a 9 percent decrease overall.

Learn this with a teacher, not a page

The Crimsora tutor teaches Percent Increase & Decrease live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

Try a live session free