How can I calculate repeated percent change with an exponential rule?
Quick answer
Turn the percent into a decimal, then add it to 1 for an increase or subtract it from 1 for a decrease. Use that multiplier once per step: final amount = starting amount × multiplier^number of steps. For example, a 15% decrease uses 0.85 each step, not 0.15. The exponent counts how many times the change happens.
The question, the details, the next step.
I know something changes by the same percentage each time, but I’m not sure how to turn that into an exponential rule. What’s the basic setup?
Use this rule:
amount after n steps = starting amount × multiplier^n
First convert the percent to a decimal. For an increase of p%, the multiplier is 1 + p/100; for a decrease, it is 1 − p/100. A repeated percentage change is multiplicative, so the multiplier is applied again at each equal step. [S1]
For a 10% increase, the multiplier is 1.10. That means the whole amount is multiplied by 1.10 each step; 10% is the rate, while 1.10 is the multiplier. If the starting amount is 100, after two steps the calculation is 100 × 1.10² = 121. [S1]
How do I handle a decrease? I keep wanting to use the percent itself as the multiplier.
For a decrease, subtract the decimal rate from 1. A 15% decrease is 0.15, so its multiplier is 1 − 0.15 = 0.85. Multiplying by 0.85 keeps 85% of the amount for that step. Using 0.15 instead would calculate 15% of the amount, which is not the same as reducing it by 15%.
Worked example: an amount starts at 240 and decreases by 15% for three steps.
- Multiplier:
0.85 - Rule:
A(n) = 240 × 0.85^n - After three steps:
A(3) = 240 × 0.85³ = 147.39
Each step acts on the amount left after the previous step, rather than subtracting 15% of the original 240 every time.
What if the increase is only a few percent and there are several steps? How should I show the calculation and rounding?
Keep the multiplier and exponent visible, and round only the final result if you can. For example, suppose a starting amount of 500 increases by 4% for five steps. The multiplier is 1 + 0.04 = 1.04, so the rule is A(n) = 500 × 1.04^n. After five steps:
A(5) = 500 × 1.04⁵ ≈ 608.33
The starting amount, rate, and five steps here are an illustrative example, not a forecast. The rule assumes the same percentage is applied for each step. It does not establish what rate or number of steps will apply in a real situation.
Can you give me a quick way to check that I set it up correctly, especially when the number of steps is zero?
Check the direction and the endpoints. For an increase, the multiplier should be greater than 1; for a decrease between 0% and 100%, it should be between 0 and 1. At zero steps, no change has been applied, so the amount is the starting amount. For a positive multiplier, the formula shows this as starting amount × multiplier⁰ = starting amount. Then calculate one step separately to confirm the multiplier changes the starting amount in the intended direction. If the rate changes from step to step, one fixed multiplier raised to a power does not describe that pattern; write out the changing multipliers instead.
A practical checklist
Write down the starting amount and how many equal steps occur.
Convert the percentage to a decimal by dividing by 100.
For an increase, add the decimal rate to 1; for a decrease, subtract it from 1.
Write
starting amount × multiplier^number of stepsbefore calculating.Check that zero steps returns the starting amount and that the result moves in the expected direction.
Keep extra digits during calculation and round the final result to a sensible precision.
A few more useful answers
Why is a 10% increase a multiplier of 1.10 rather than 0.10?
Because the new amount includes the original whole amount plus an extra 10%: 100% + 10% = 110%, or 1.10 as a multiplier. The 0.10 represents only the increase, not the entire new amount. [S1]
Can I use the same exponential rule if the percentage changes each step?
Not with one fixed multiplier. If the rate varies, each step has its own multiplier, so show the sequence of multiplications rather than using one multiplier raised to the number of steps.
Does the exponent count the starting amount as a change?
No. The starting amount is the value at step zero; the exponent counts how many times the multiplier is applied. For example, step three means three applications.
Sources
References used for this guide.
- [S1] OpenStax Precalculus 2e: Exponential Growth, Decay, and Multipliers openstax.org · Checked