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Lesson 21 of 46 · Arithmetic

Percents, percent change and successive changes

Turn every percent into a multiplier, find the right base for a percent change, and chain or undo changes without the classic GRE slips.

About 20 minutes

In this lesson

  • Write any percent increase or decrease as a single multiplier.
  • Find a percent change from the correct base, and tell it apart from a change in percentage points.
  • Combine successive changes and reverse a change without adding or subtracting percents.

The idea

A percent is a fraction with 100 underneath

"Percent" means "per hundred", so 35%35\% is 35100=0.35\frac{35}{100} = 0.35. Almost every percent question reduces to one sentence:

part=percent×whole\text{part} = \text{percent} \times \text{whole}

Read the question for the whole, the thing the percent is of. "30% of the students" has the students as its whole; "30% more than last year" has last year as its whole. Once you know the whole, the arithmetic is one multiplication or one division.

Changes are multipliers

An increase of p%p\% multiplies a quantity by 1+p1001 + \frac{p}{100}; a decrease of p%p\% multiplies it by 1−p1001 - \frac{p}{100}.

Change Multiplier
up 8% 1.081.08
up 150% 2.52.5
down 12% 0.880.88
down 100% 00

Thinking in multipliers does three jobs at once. It finds a new value in one step ($60\$60 up 8% is 60×1.08=$64.8060 \times 1.08 = \$64.80). It chains changes: successive changes multiply, they never add. And it runs backwards: to recover an original value, divide by the multiplier.

Percent change needs the right base

percent change=new−oldold×100%\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%

The base is the old value, the one the change starts from. A rise from 40 to 50 is a 25% increase (1040\frac{10}{40}), but the fall from 50 back to 40 is a 20% decrease (1050\frac{10}{50}). Same gap, different base, different percent.

Percentage points are not percents

When the quantity being compared is itself a percent, two different changes can be asked for. If an approval rate moves from 40% to 50%, it rose by 10 percentage points (a subtraction) and by 25 percent (a percent change of the rate, 1040\frac{10}{40}). The two phrasings ask for different numbers; read which one the question asks for.

Percent of a percent, and the order of changes

Because multiplication is commutative, the order of successive changes never matters: up 10% then down 20% gives 1.1×0.8=0.881.1 \times 0.8 = 0.88, exactly the same as down 20% then up 10%. Taking "40% of 25% of xx" is 0.4×0.25×x=0.1x0.4 \times 0.25 \times x = 0.1x, which is 10% of xx.

A check that saves marks

Before you choose, estimate: is the answer bigger or smaller than the starting value, and by roughly how much? A discount question whose answer is larger than the original price, or a two-step increase smaller than one of its steps, tells you a multiplier went the wrong way.

Worked examples

Example 1: Up 25%, then down 20%

A jacket's price is raised by 25%. Later the new price is cut by 20%. Compared with the original price, is the final price higher, lower or the same, and by what percent?

  1. Write each change as a multiplier: up 25% is ×1.25\times 1.25, down 20% is ×0.8\times 0.8.

  2. Successive changes multiply: 1.25×0.8=1.001.25 \times 0.8 = 1.00.

  3. A combined multiplier of exactly 1 means no net change. A quick check with a price of $100\$100: up 25% gives $125\$125, and 20% of $125\$125 is $25\$25, so the final price is $100\$100 again.

Answer

The final price equals the original price: a net change of 0%. Adding the percents (+25−20=+5+25 - 20 = +5) would have given the wrong answer of a 5% increase.

Example 2: Recovering the price before a discount

After a 15% discount, a laptop costs $680\$680. What was its price before the discount?

  1. A 15% discount is the multiplier 0.850.85, so original×0.85=680\text{original} \times 0.85 = 680.

  2. Undo a multiplier by dividing: original=6800.85\text{original} = \frac{680}{0.85}.

  3. Clear the decimal: 6800.85=68 00085=800\frac{680}{0.85} = \frac{68\,000}{85} = 800.

  4. Check forwards: 15% of $800\$800 is $120\$120, and 800−120=680800 - 120 = 680.

Answer

The laptop cost $800\$800. Adding 15% to $680\$680 instead gives $782\$782, which is wrong because the 15% was taken of the original price, not of $680\$680.

Example 3: Percentage points or percent?

In 2024, 40% of a firm's 250 employees worked remotely. In 2025 the firm had 300 employees, and 45% of them worked remotely. By what percent did the number of remote workers increase?

  1. The question asks about the number of remote workers, not the remote rate, so find both numbers first.

  2. 2024: 0.40×250=1000.40 \times 250 = 100 remote workers. 2025: 0.45×300=1350.45 \times 300 = 135 remote workers.

  3. Percent change from the old value: 135−100100=0.35\frac{135 - 100}{100} = 0.35.

Answer

The number of remote workers rose by 35%. The rate rose by 5 percentage points, and by 12.5%12.5\% as a percent of the rate (540\frac{5}{40}); neither is what was asked.

Common traps

  • Adding successive percents

    "Up 10%, then up 10%" is 1.1×1.1=1.211.1 \times 1.1 = 1.21, an increase of 21%, not 20%. The second change is taken of a bigger number. Whenever changes come one after another, multiply the multipliers.

  • Undoing a change with the same percent

    A 20% rise is not undone by a 20% fall: 1.2×0.8=0.961.2 \times 0.8 = 0.96. To return to the start you need to divide by 1.2, which is a fall of 16\frac{1}{6}, about 16.7%. "20%" is the tempting wrong choice in such a question.

  • Taking the percent of the wrong whole

    "AA is 25% more than BB" uses BB as the base: A=1.25BA = 1.25B. It does not follow that BB is 25% less than AA; in fact B=0.8AB = 0.8A, 20% less. Name the base before you compute.

  • Confusing percentage points with percent

    A rate moving from 4% to 6% has risen by 2 percentage points and by 50%. If the answer choices contain both 2 and 50, the question is testing whether you read which one it asked for.

Check yourself

Answer each question before you open its answer.

Question 1: Up 15%, down 15%

Percent

A price of $240\$240 is increased by 15%, and the new price is then decreased by 15%. What is the final price?

  1. A.$204.00
  2. B.$234.60
  3. C.$240.00
  4. D.$276.00
Show the answer

Answer: B. $234.60

Working

The combined multiplier is 1.15×0.85=0.97751.15 \times 0.85 = 0.9775, so the final price is 240×0.9775=$234.60240 \times 0.9775 = \$234.60. Step by step: 240×1.15=276240 \times 1.15 = 276, then 276×0.85=234.6276 \times 0.85 = 234.6.

Why the other options are wrong

  • A. This is only the 15% decrease applied to $240; the increase was skipped.
  • C. Assumes a 15% rise and a 15% fall cancel. They do not: the fall is taken of the larger, increased price.
  • D. This is the price after the increase only.

What carries over

Equal percent rises and falls never cancel; together they always leave the quantity lower.

Question 2: Which change was asked for?

Percent

A survey found that 30% of 400 households owned a bicycle. A year later, 25% of 560 households owned one. By what percent did the number of households owning a bicycle change?

  1. A.A decrease of 5%
  2. B.A decrease of about 16.7%
  3. C.An increase of about 16.7%
  4. D.An increase of 20%
  5. E.An increase of 40%
Show the answer

Answer: C. An increase of about 16.7%

Working

Owners before: 0.30×400=1200.30 \times 400 = 120. Owners after: 0.25×560=1400.25 \times 560 = 140. Percent change from the old value: 140−120120=20120=16≈16.7%\frac{140 - 120}{120} = \frac{20}{120} = \frac{1}{6} \approx 16.7\%, an increase.

Why the other options are wrong

  • A. The ownership rate fell by 5 percentage points, but the question asks about the number of households, which rose.
  • B. This is the percent change of the rate, 25−3030\frac{25 - 30}{30}. It ignores that the number of households grew from 400 to 560.
  • D. The number of owners rose by 20 households; 20 is a count, not a percent of the starting 120.
  • E. This is the growth in the number of households surveyed (160400\frac{160}{400}), not in the number that owned a bicycle.

What carries over

When the rate and the total both change, compute the two counts before you compute any percent change.

Question 3: Undoing a 20% increase

Percent

After a 20% increase, a quantity equals 360. By what percent must it now decrease to return to its original value?

  1. A.1623%16\frac{2}{3}\%
  2. B.20%20\%
  3. C.80%80\%
  4. D.8313%83\frac{1}{3}\%
Show the answer

Answer: A. 1623%16\frac{2}{3}\%

Working

The original value is 3601.2=300\frac{360}{1.2} = 300. The decrease needed is 360−300=60360 - 300 = 60, and as a percent of the current value 360 that is 60360=16=1623%\frac{60}{360} = \frac{1}{6} = 16\frac{2}{3}\%.

Why the other options are wrong

  • B. A 20% fall is taken of 360, the larger number, so it removes 72 and overshoots to 288.
  • C. 0.8 is the multiplier for a 20% decrease; it is not a percent change, and it would not return 360 to the original value anyway.
  • D. 300 is 8313%83\frac{1}{3}\% of 360: that is the fraction that remains, not the size of the decrease.

What carries over

To reverse a change, divide by its multiplier, then measure the new change from the value you are starting at now.

Key points

  • Every percent change is a multiplier: up p%p\% is ×(1+p100)\times\left(1+\frac{p}{100}\right), down p%p\% is ×(1−p100)\times\left(1-\frac{p}{100}\right).

  • Successive changes multiply; they never add.

  • To undo a change, divide by its multiplier.

  • Percent change is measured from the old value; a change in percentage points is a subtraction.

All questions here are original and written for UniExamer; none comes from the test maker.

Written by the UniExamer team with AI assistance. Every worked example and check question was re-solved by an independent checker before publication. No subject expert has reviewed this lesson yet. If you find a mistake, write to support@uniexamer.com.

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