Repeated Percent Change on the SAT: Why Percentages Multiply Instead of Add

When a quantity changes by a percentage more than once, the percentages usually do not simply add.

You multiply the change factors.

If a value rises 10% and then rises another 10%:

$$1.10\times1.10=1.21.$$

So the total increase is:

$$21\%,$$

not 20%.

Why? The second 10% is calculated from the new, larger amount, not from the original.

The core SAT rule is:

$$\boxed{\text{final}=\text{initial}\times(\text{factor}_1)(\text{factor}_2)\cdots}$$

This is the same logic behind exponential growth, depreciation, repeated discounts, population change, and compound percentage models.

Percent change factors

Translate every percentage change into a multiplier.

Increase by r%

$$1+r$$

where $r$ is a decimal.

Decrease by r%

$$1-r.$$

Examples:

For a deeper explanation of the difference between a rate and a factor, see Growth Rate vs Growth Factor.

Worked example 1: two equal increases

A price starts at $200 and increases 10% twice.

First increase:

$$200(1.10)=220.$$

Second:

$$220(1.10)=242.$$

So final:

$$\$242.$$

Total increase:

$$242-200=42.$$

Relative increase:

$$\frac{42}{200}=0.21.$$ $$\boxed{21\%}$$

Factor shortcut:

$$200(1.10)^2=242.$$

Why 10% + 10% becomes 21%

The first 10% adds:

$$20.$$

The second 10% is calculated from:

$$220,$$

so it adds:

$$22.$$

Total added:

$$42.$$

The extra 2 comes from earning 10% on the first increase.

That is compounding.

Worked example 2: repeated decrease

A machine loses 20% of its value each year for 2 years. It starts at $5,000.

Decay factor:

$$0.80.$$

After 2 years:

$$5000(0.80)^2.$$ $$5000(0.64)=3200.$$

Final value:

$$\boxed{\$3,200}.$$

Overall decrease:

$$5000-3200=1800.$$ $$\frac{1800}{5000}=36\%.$$

Two 20% decreases create a 36% total decrease, not 40%.

Increase then decrease by the same percentage

This is one of the most common percentage traps.

Start at 100.

Increase 20%:

$$100(1.20)=120.$$

Then decrease 20%:

$$120(0.80)=96.$$

Final:

$$96.$$

Overall change:

$$-4\%.$$

Factor method:

$$1.20(0.80)=0.96.$$ $$\boxed{4\%\text{ decrease overall}}$$

Equal percentage up and down do not cancel because the second percentage uses a different base.

Worked example 3: 25% down then 25% up

Start at 400.

Decrease:

$$400(0.75)=300.$$

Increase:

$$300(1.25)=375.$$

Final:

$$375.$$

Overall change:

$$-25.$$ $$\frac{25}{400}=6.25\%.$$ $$\boxed{6.25\%\text{ decrease}}$$

Factor:

$$0.75(1.25)=0.9375.$$

Which increase exactly reverses a decrease?

Suppose a quantity falls 20%.

Factor:

$$0.80.$$

To return to the original, we need a second factor $f$ such that:

$$0.80f=1.$$

So:

$$f=\frac1{0.80}=1.25.$$

That means a:

$$25\%$$

increase.

So:

A 20% decrease requires a 25% increase to recover.

The reverse percentage is larger because it is applied to a smaller base.

Worked example 4: recover after a 30% decrease

After a 30% decrease, the value is 70% of original:

$$0.70.$$

Recovery factor:

$$\frac1{0.70}\approx1.42857.$$

Required increase:

$$42.857\%.$$

So a 30% loss needs about a:

$$\boxed{42.9\%\text{ gain}}$$

to return to the original.

Different successive percentages

Suppose a value increases 15%, then 8%.

Combined factor:

$$1.15(1.08).$$ $$=1.242.$$

Overall increase:

$$24.2\%.$$

Not:

$$23\%.$$

The factors multiply.

Worked example 5: discount then tax

An item costs $250.

It receives a 20% discount, then 8% sales tax is applied.

Discount factor:

$$0.80.$$

Tax factor:

$$1.08.$$

Final:

$$250(0.80)(1.08)$$ $$=250(0.864)$$ $$\boxed{\$216}.$$

Overall change from original:

$$216-250=-34.$$ $$\frac{34}{250}=13.6\%.$$

So the final price is 13.6% below the original.

Does order matter?

For pure percentage multipliers applied to the same evolving amount, multiplication is commutative:

$$1.20(0.80)=0.80(1.20).$$

So a 20% increase then 20% decrease gives the same final factor as the reverse order.

But be careful: in real contexts, a fee, tax, discount, or fixed amount may apply to a different base or at a different stage. Then the model may not be simple multiplier multiplication.

Repeated percentage change over many periods

If the same rate repeats:

$$\text{final}=\text{initial}(1+r)^n$$

for growth.

For decay:

$$\text{final}=\text{initial}(1-r)^n.$$

This is exponential modeling.

See SAT Exponential Growth and Decay.

Worked example 6: annual population growth

A population starts at 80,000 and grows 3% annually for 4 years.

$$P=80000(1.03)^4.$$

Compute:

$$1.03^4\approx1.12550881.$$

So:

$$P\approx90040.7.$$

Depending on context:

$$\boxed{\approx90,041}$$

people.

The total increase is about 12.55%, not simply 12%.

Worked example 7: depreciation

A car is worth $32,000 and loses 12% each year for 3 years.

Retained factor:

$$0.88.$$ $$V=32000(0.88)^3.$$ $$0.88^3=0.681472.$$ $$V=21807.104.$$ $$\boxed{\$21,807.10\text{ approximately}}$$

Overall loss is:

$$1-0.681472=0.318528.$$

About:

$$31.85\%.$$

Not 36%.

Repeated percent vs fixed amount change

Compare:

Add 10 each year

$$A(t)=A_0+10t.$$

Linear.

Increase 10% each year

$$A(t)=A_0(1.10)^t.$$

Exponential.

This distinction is central to Linear vs Exponential Models.

Repeated percentages from a table

Suppose:

Ratios:

$$\frac{230}{200}=1.15.$$ $$\frac{264.5}{230}=1.15.$$

So the model grows by:

$$15\%$$

each year.

Constant ratio, not constant difference.

Reverse repeated percentage problems

Suppose a value grows 5% per year for 3 years and ends at 926.1.

Let original be $P$.

$$926.1=P(1.05)^3.$$ $$926.1=P(1.157625).$$ $$P=800.$$ $$\boxed{800}$$

For the general reverse-percent strategy, see Reverse Percent Problems.

Worked example 8: two-step reverse problem

After a 10% increase and then a 20% decrease, a quantity is 176.

Combined factor:

$$1.10(0.80)=0.88.$$

Let original be $x$:

$$176=0.88x.$$ $$x=200.$$ $$\boxed{200}$$

Percentage points are different

Suppose a rate changes from 40% to 50%.

That is:

• +10 percentage points;
• +25% relative increase.

Repeated-percent multiplication applies to relative percent changes, not to simply adding percentage points in a proportion.

See Percent Increase vs Percentage Points.

Growth by more than 100%

If a value increases by 150%, the growth factor is:

$$1+1.50=2.50.$$

A repeated 150% increase for two periods gives:

$$2.5^2=6.25.$$

The final amount is 625% of the original.

Large percentages still use the same multiplier logic.

Worked example 9: repeated 50% growth

Initial:

$$64.$$

Increase 50% three times:

$$64(1.5)^3.$$ $$1.5^3=3.375.$$ $$64(3.375)=216.$$ $$\boxed{216}$$

Overall increase:

$$\frac{216-64}{64} = \frac{152}{64} = 2.375.$$ $$237.5\%.$$

Common trap 1: adding repeated percent changes

Two 8% increases are not 16% overall.

$$1.08^2=1.1664.$$

Overall:

$$16.64\%.$$

Common trap 2: equal increase/decrease cancel

They do not unless their factors multiply to 1.

Common trap 3: using the rate as the factor

10% growth factor is:

$$1.10,$$

not 0.10.

Common trap 4: using the lost percentage as the remaining factor

20% decrease leaves:

$$80\%.$$

Factor:

$$0.80.$$

Common trap 5: multiplying percentages instead of multipliers

Do not compute:

$$10\%\times20\%.$$

Convert to:

$$1.10\times1.20.$$

Common trap 6: adding a fixed amount and treating it as a percentage

A $20 increase and a 20% increase are not the same unless the original happens to be $100.

Common trap 7: rounding too early

In multi-period problems, keep the multiplier calculation precise until the end.

Common trap 8: forgetting the number of periods

If 4% monthly growth lasts 18 months, exponent is:

$$18,$$

not 1.5 unless the model is explicitly rewritten in years.

A fast repeated-percent routine

1. Convert every change to a multiplier.
2. Multiply the factors.
3. Apply to the original.
4. If asked for total percent change: $(\text{combined factor}-1)\times100\%.$
5. If combined factor is below 1, interpret as a decrease.
6. Keep units and time periods consistent.

SAT-style practice

Question 1

A value increases 20% twice.

Combined factor:

$$1.2^2=1.44.$$ $$\boxed{44\%\text{ increase}}$$

Question 2

A value decreases 10% twice.

$$0.9^2=0.81.$$ $$\boxed{19\%\text{ decrease}}$$

Question 3

A price rises 30% then falls 30%.

$$1.3(0.7)=0.91.$$ $$\boxed{9\%\text{ decrease overall}}$$

Question 4

A quantity falls 25%. What increase restores the original?

$$0.75f=1.$$ $$f=\frac43=1.3333\dots$$ $$\boxed{33\frac13\%\text{ increase}}$$

Question 5

An investment of $5,000 grows 4% annually for 2 years.

$$5000(1.04)^2 = 5000(1.0816) = \boxed{\$5,408}.$$

FAQ: repeated percent change

Why don't repeated percent changes add?

Because each later percentage is applied to the new amount, not the original amount.

How do I combine two percentage increases?

Multiply the factors:

$$(1+r_1)(1+r_2).$$

Does a 20% increase followed by a 20% decrease return to the start?

No. The combined factor is:

$$1.2(0.8)=0.96.$$

How much increase is needed after a 20% decrease?

25%.

What is the formula for the same percent growth over n periods?

$$A=A_0(1+r)^n.$$

Is repeated percent change linear or exponential?

A constant repeated percentage rate creates an exponential model.

How do I reverse repeated percent changes?

Divide the final value by the combined multiplier.

How to practice this on MastaPrep

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Related MastaPrep guides

• SAT Percentages
• Growth Rate vs Growth Factor
• Linear vs Exponential Models
• Percent Increase vs Percentage Points
• Reverse Percent Problems
• SAT Exponential Growth and Decay

Final takeaway

For repeated percentage changes, think in multipliers, not added percentages.

$$\boxed{\text{final}=\text{initial}\times\prod(\text{change factors})}$$

That one habit handles:

• compounding growth;
• repeated decay;
• discount + tax;
• increase-then-decrease traps;
• reverse multi-step percentage problems.

The percentages may look additive in the wording, but the underlying process is multiplicative.

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https://satsuite.collegeboard.org/practice/student-question-bank/math

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