Library / Foundations: Maths / FMA-02
Foundations: Maths
Repeated percentage change
Value after the same percentage rate is applied for n periods.
- Ref
- FMA-02
- Rev
- v1
- Field
- Foundations: Maths
- Result
- B
A is the start, r the percentage rate per period (negative for a decrease) and n the number of periods. For a reverse percentage divide the final value by the multiplier.
Each period multiplies the value by the same factor, so after n periods the starting value has been multiplied by that factor n times.
Calculator
- Total change62.89 %
- Multiplier per period1.05
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Compounding means growth accelerates; at 7% a quantity roughly doubles in 10 periods.
- Using the same multiplier for a decrease (a rate below zero) gives exponential decay, as in depreciation and decay of signals.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Starting value | 10 to 10,000 | 16.29 | 16,290 | 16,270 | +10% | +10 % |
| r Rate per period | 0.5 to 20 % | 1,051 | 6,192 | 5,141 | +10% | +4.87 % |
| n Number of periods | 1 to 30 | 1,050 | 4,322 | 3,272 | +10% | +5 % |
Sweep of Number of periods
| Number of periods | Final value |
|---|---|
| 1 | 1,050 |
| 4.625 | 1,253 |
| 8.25 | 1,496 |
| 11.88 | 1,785 |
| 15.5 | 2,130 |
| 19.13 | 2,542 |
| 22.75 | 3,034 |
| 26.38 | 3,621 |
| 30 | 4,322 |
Sweep one variable
Variables and how each one contributes
Final value
What the equation gives you. Everything else feeds it.
Starting value
Amount at the start. Directly proportional.
Rate per period
Percentage change each period. Strong: the effect compounds, so a small rate gives a large change over many periods.
Number of periods
Number of times the rate is applied. Exponential: each extra period multiplies the value again.
Worked example with the values above
- Multiplier per period: 1 + 5/100 = 1.05.
- Raise to the power 10: 1.6289.
- B = 1,000 × 1.6289 = 1,629.
Assumptions and limits
- The rate is constant each period.
- Percentages are applied to the running value, not the original.
Common mistakes
- Multiplying the rate by n instead of compounding it.
- Entering a decrease as a positive rate.
Statistics and uncertainty
Errors in the rate are amplified by the power n: a 1% error in the multiplier becomes about n% in the result, so small rate errors matter over many periods.
Where it is used
- Cost, mass and performance changes applied repeatedly.
- Decay of battery capacity or signal strength per stage.
Maths you need
- Powers: repeated multiplication
- Percentages: multiplier 1 + r/100