Model percent increase and decrease

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Difficulty: Intermediate
Estimated time: 25 minutes
Domains: Problem Solving and Data Analysis
Techniques: Percent-changePercent-multipliersOriginal-valuePercent-of-versus-greater

What you’ll learn

Find the original and new amounts in a one-step change.
Tell absolute change apart from percent increase or decrease.
Apply an increase or decrease with one multiplier.
Turn a multiplier into a percent change and back, including changes above $100\%$ .
Tell “ $p\%$ of” apart from “ $p\%$ greater than.”
Use Desmos for awkward arithmetic and round only the final percent.

Prerequisites

Solve percent part-whole problems0%
Find a percent, part, or whole from the other two quantities.

Why this matters on the SAT

Measure change from the right starting point

SAT percent-change questions are often about everyday things like prices, populations, production and measurements, and a small change in wording can change what the numbers mean. Sometimes you’re given the percent and need the new amount. Sometimes you’re given two amounts and need the percent change. Sometimes a decimal multiplier hides the percent completely.

Each version comes down to two questions. Where did the quantity start? And does the question want the part added or removed, or the whole new amount?

Solution to the example

First ask: is $3.49$ the whole June amount, or only the increase?

June’s production was $3.49q$ , so it was $349\%$ of May’s production. The first $100\%$ of that is May’s production over again, so the increase was

349\%-100\%=249\%.

The answer is C. Choice D is the trap: $349\%$ describes June’s whole production compared with May’s, not the increase.

SAT example

A factory produced $q$ batteries in May and $3.49q$ batteries in June, where $q>0$ . What was the percent increase in production from May to June?

A
$2.49\%$
B
$3.49\%$
C
$249\%$
D
$349\%$

Compare the new amount with the original

Start by labeling the two amounts:

$O$ is the original amount, where the quantity starts.
$N$ is the new amount, where it ends up.

The absolute change is how many units the quantity gained or lost. It stays in the original units, like dollars, liters or people:

\begin{aligned} \text{amount of increase}&=N-O &&\text{when }N>O,\\[1.4em] \text{amount of decrease}&=O-N &&\text{when }N<O. \end{aligned}

The percent change compares that absolute change with the original:

\text{percent change} = \frac{\text{absolute change}}{\text{original}} \times100\%.

Why the original on the bottom? Percent change tells you how big the change is compared with where the quantity started, so the starting amount is what you divide by. A short way to remember it: change over original.

Say a workshop makes $160$ parts a week, and then its weekly output rises to $212$ parts. The absolute increase is

212-160=52\text{ parts}.

Now compare those $52$ extra parts with the original $160$ parts:

\frac{52}{160}\times100\%=32.5\%.

So output went up by $52$ parts, which is a $32.5\%$ increase. Both numbers describe the same change. One counts parts, and the other measures the change against where output started.

Check your understanding:

A quantity rises from $40$ to $100$ . In a separate change, a quantity falls from $100$ to $40$ . Both changes are $60$ units. What is the percent increase in the first change, and what is the percent decrease in the second? Why aren’t they the same?

Common mistake:

It’s easy to divide by the new value by accident. For a change from $80$ to $100$ , $\frac{20}{100}=20\%$ compares the increase with where the quantity ended. Percent increase compares it with where the quantity started, so it’s $\frac{20}{80}=25\%$ . Before you plug in numbers, write “original” under the fraction bar. Then check that the original times $1.25$ gives the new value: $80(1.25)=100$ .

That formula is the tool when you’re given both amounts. When you’re given the percent and asked for the new amount, a multiplier gets you there more directly.

Write the change as a multiplier

A multiplier is the one number you multiply the original by to get the new amount.

For an increase of $p\%$ , you keep all of the original and add $\frac{p}{100}$ of it:

\begin{aligned} N &=O+\frac{p}{100}O\\[1.4em] &=\left(1+\frac{p}{100}\right)O. \end{aligned}

For a decrease of $p\%$ , you keep the original minus the part taken away:

\begin{aligned} N &=O-\frac{p}{100}O\\[1.4em] &=\left(1-\frac{p}{100}\right)O. \end{aligned}

The $1$ in each formula is the original itself, $100\%$ of it. Leave it out and you get only the part added or removed, not the new amount.

For example:

\begin{aligned} 18\%\text{ increase}&\longrightarrow 1+0.18=1.18,\\[1.4em] 18\%\text{ decrease}&\longrightarrow 1-0.18=0.82. \end{aligned}

“Of” and “greater than” are different

These two phrases sound alike, but they give different multipliers. $150\%$ of the original means

1.50O.

The first $100\%$ is the original itself, and the other $50\%$ is the increase. So this new amount is $50\%$ greater than the original.

$150\%$ greater than the original means you keep the whole original and add $150\%$ of it on top:

O+1.50O=2.50O.

Discounts work the same way. A $40\%$ discount doesn’t use the multiplier $0.40$ . It takes away $40\%$ and leaves $60\%$ , so the new price is $0.60$ times the original.

Check your understanding:

Write each statement as a multiple of $x$ : (1) $y$ is $175\%$ of $x$ ; (2) $z$ is $175\%$ greater than $x$ .

Read a multiplier back as a percent

Sometimes you’re given the multiplier and need the percent. If $N=bO$ , compare the multiplier $b$ with $1$ :

If $b>1$ , the amount went up, and the percent increase is $(b-1)\times100\%$ .
If $0<b<1$ , the amount went down, and the percent decrease is $(1-b)\times100\%$ .
If $b=0$ , nothing is left, which is a $100\%$ decrease.
If $b=1$ , nothing changed.

Increases can go past $100\%$ . A multiplier of $2$ means the new amount is $200\%$ of the original, so the increase is $100\%$ . Any multiplier above $2$ means an increase above $100\%$ , like the $249\%$ in the battery example. Decreases can’t go that far: a positive amount that lost more than $100\%$ of itself would turn negative. So when an SAT question in an everyday setting has a change above $100\%$ , it’s an increase.

Example: Turn a multiplier into a percent

Worked example

After one processing step, the mass of a sample is $0.35$ times its original mass. By what percent was the mass decreased?

A
$3.5\%$
B
$35\%$
C
$65\%$
D
$135\%$

Step 1

Read what’s left

The multiplier $0.35$ means the new mass is $35\%$ of the original mass. That’s the part that’s left, not the part taken away.

Step 2

Compare the multiplier with 1

The percent decrease is the gap between the multiplier and $1$ :

p=100(1-0.35)=65.

Subtracting from $1$ leaves the part that was removed, and multiplying by $100$ turns that decimal into a percent.

Step 3

Answer what’s asked

$35\%$ is left, so $65\%$ was removed. The answer is C. Choice B gives the part that’s left, not the decrease the question asks for.

Common mistake:

Choosing $35\%$ usually comes from reading the multiplier as the part removed. A multiplier below $1$ tells you what’s left, so subtract it from $1$ to find the decrease: $1-0.35=0.65$ . Then check that taking away $65\%$ leaves $35\%$ of the original.

Use Desmos when the arithmetic is awkward

Desmos can’t tell you which value is the original, so decide that first. Then type the new amount as a percent of the original, the way you’d say it in words. If $x$ is the percent increase, enter:

(100 + x)% of original = new

The original’s $100\%$ plus the increase of $x\%$ makes the new amount. Written this way, the original sits right after of, where you can see it.

For example, say an amount rises from $286.40$ to $327.75$ . Its percent increase is

\left(\frac{327.75}{286.40}-1\right)100 =14.437849\ldots

That’s not arithmetic you want to do by hand. Enter (100+x)% of 286.40 = 327.75 exactly as written, and Desmos draws a vertical solution line. Select the line and read its x-coordinate, $x=14.437849\ldots$ . If the question asks for the nearest tenth of a percent, round only at the very end, to $14.4\%$ .

For a decrease of $x\%$ , enter (100-x)% of original = new. The original still goes right after of, and the minus sign matches the drop.

If you’d like optional calculator practice with more percent wording, see Solve percent problems in Desmos. Its reverse-percent and successive-change sections go a step further.

Calculator loads as you approach

The original stays in view right after “of,” and x is the percent increase.

Practice problems

Each problem is a little harder than the last.

Apply one direct decrease

Practice problem

A nature trail recorded $480$ hikers one week. The next week, the number of hikers decreased by $12.5\%$ . How many hikers were recorded the next week?

Calculator loads as you approach

Use Desmos if it helps you solve or check.

Calculate and round an awkward increase

Practice problem

A laboratory used $428.6$ milliliters of a solution during one trial and $513.9$ milliliters during a later trial. To the nearest tenth of a percent, by what percent did the amount used increase?

Calculator loads as you approach

Type the new amount as a percent of the original.

Find a percent increase from symbolic amounts

Practice problem

At the beginning of a data transfer, an archive contained $4m$ gigabytes of data. At the end of the transfer, it contained $15m$ gigabytes, where $m>0$ .

By what percent did the amount of data in the archive increase?

Your answer

Calculator loads as you approach

Use Desmos if it helps you solve or check.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Find the original first. Percent change is change over original.
An increase of $p\%$ multiplies by $1+\frac{p}{100}$ , and a decrease of $p\%$ multiplies by $1-\frac{p}{100}$ .
“ $p\%$ of” is the whole new amount. “ $p\%$ greater than” adds $p\%$ on top of the original.
For awkward numbers, type the new amount as a percent of the original in Desmos, solve for the change, and round only the final percent.

Related lesson

For percent change that repeats over equal intervals, see Build, identify, and interpret exponential models.

Next lesson

Reverse and combine percent changes

Work backward to an unknown original or combine changes in sequence.

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Practice

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543 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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