Repeated Percentage Change and Multiplicative Growth in Secondary Mathematics

This is the structure behind compound growth, repeated discounts, depreciation, population-style models and many financial calculations. The key mathematical object is not the percentage by itself but the multiplier applied repeatedly.

One percentage move becomes one multiplier

An increase of 8% means multiply by 1.08. A decrease of 8% means multiply by 0.92. Repeating the same change for n periods gives multiplier 1.08ⁿ or 0.92ⁿ.

This is why repeated percentage change is not handled by simply multiplying the percentage by the number of periods.

Why adding percentages fails

Start with 100 and increase by 10% twice. After the first increase, the value is 110. The second 10% is 11, giving 121. The total increase is 21%, not 20%.

The second percentage uses 110 as its base rather than the original 100.

The general growth model

If an initial value P grows by r per period, written as a decimal, then after n equal periods the model is A=P(1+r)ⁿ, provided the same percentage rate applies each period and the stated model assumptions hold.

For repeated decrease, A=P(1−r)ⁿ.

Worked example: repeated growth

An initial quantity is 800 and grows by 5% per period for three periods. A=800(1.05)³=926.10.

The increases are 40, then 42, then 44.10. The percentage is constant but the absolute increase grows because the base grows.

Worked example: repeated decrease

A value of 2000 falls by 12% per period for two periods. A=2000(0.88)²=1548.80.

The total decrease is 451.20, or 22.56%, not 24%.

Different percentages in sequence

If a value rises by 20% and then falls by 10%, the combined multiplier is 1.2×0.9=1.08. The net result is an 8% increase.

Percentage changes combine by multiplying multipliers, not by adding signed percentage numbers.

Order of percentage multipliers

For pure multiplication, 1.2×0.9=0.9×1.2, so a 20% increase followed by a 10% decrease gives the same final multiplier as the reverse order. The intermediate values differ, but the final product is the same.

This should not be confused with contexts where additional fixed amounts, thresholds or fees are inserted between percentage changes; then order can matter.

Compound interest as a percentage model

Under a simplified compound-interest model with rate r per compounding period and no deposits, withdrawals or fees, balance after n periods is P(1+r)ⁿ.

For example, $1000 at 4% per period for three periods gives 1000(1.04)³≈$1124.86. This is a mathematical illustration, not a current financial product claim.

Depreciation uses the same mathematics

A model that depreciates by 15% per period uses multiplier 0.85. An initial value 10,000 becomes 10,000(0.85)³≈6141.25 after three periods under the simplified model.

Growth and decay are structurally the same: repeated multiplication by a factor above or below 1.

Tables reveal multiplicative growth

Start 500 and multiply by 1.1 repeatedly: 500, 550, 605, 665.5. The differences are 50,55,60.5 and are not constant. The ratios between successive terms are constant at 1.1.

This distinguishes exponential-style multiplicative growth from linear additive growth.

Graphs curve because the absolute change changes

With constant percentage growth, larger current values produce larger absolute increases. The graph therefore bends upward rather than following a constant-gradient straight line. With repeated percentage decay, the amount lost each period becomes smaller as the base shrinks.

Reverse repeated percentage change

If a value after three 5% increases is 926.10, solve P(1.05)³=926.10. Dividing by 1.05³ gives P=800.

This connects directly to Reverse Percentage Through Equations.

Equivalent single percentage change

Two increases of 10% have combined multiplier 1.21, equivalent to one 21% increase. A 20% increase followed by a 10% decrease has multiplier 1.08, equivalent to one 8% increase.

To find the equivalent percentage, multiply all factors first and compare the final factor with 1.

Common misconceptions

“10% for three periods means 30%.” False under compounding. It means multiplier 1.1³=1.331, a 33.1% increase.

“The same percentage means the same amount each period.” False because the base changes.

“Growth and decay need different mathematics.” Both use repeated multipliers.

“Every real-world growth process follows a constant percentage forever.” False. The model is valid only while its assumptions are reasonable.

A first-principles teaching sequence

Start with a table and recalculate the base each period. Ask students to record the multiplier beside every step. Then compress repeated multiplication into index notation. Only after that introduce the general form P(1+r)ⁿ.

Next, mix additive and multiplicative sequences. Students should decide whether constant differences or constant ratios control the pattern.

Diagnostic checkpoints

• Can the student convert percentage changes to multipliers?
• Can they explain why repeated percentages multiply?
• Can they distinguish constant difference from constant ratio?
• Can they find an equivalent single percentage change?
• Can they reverse repeated change to recover the initial value?

Practice set

1. 6% growth for 4 periods → multiplier 1.06⁴≈1.2625, about 26.25% total increase.

2. 10% decrease for 3 periods → multiplier 0.9³=0.729, total decrease 27.1%.

3. +25% then −20% → multiplier 1.25×0.8=1, so the original value is restored.

4. Start 400, grow 5% twice → 400×1.05²=441.

5. Final 729 after three 10% decreases → original=729/0.9³=1000.

The deeper mathematical habit

Repeated percentage change teaches students to recognise multiplicative systems. That idea later supports exponential functions, finance, population models, decay, probability and many algorithmic processes.

Build from The Percentage Base, Percentage Change Is Not Symmetric and Reverse Percentage Through Equations. Return to the Secondary Mathematics Master Index.