Exponential growth and decay.
A repeated percentage change creates a curve, because the amount changes each period.
Model growth and depreciation · use repeated multipliers · reverse exponential change and find a rate
Repeat a multiplier, not a fixed addition
Imagine a population that grows by 10% each year. The second year’s 10% is calculated from the new population, not the original one. Repeating a percentage change therefore means repeating a multiplication.
A growth factor is larger than 1; a decay factor lies between 0 and 1. Write out the first few stages to see the pattern, then replace repeated equal multipliers with a power. Equal percentage changes do not mean equal numerical changes.
Exponential change uses the same multiplier in each equal time interval. Growth has a multiplier greater than 1; decay has a multiplier between 0 and 1.
n is the number of periods. A rate stated per year needs n in years; a rate per month needs n in months. You do not need the number e for this topic.
1000 × 1.24 = 2073.60 (to 2 d.p.). The increase per period changes because the starting amount for each period changes.
Pause and explainDoes 20% annual decay mean subtracting the same amount every year?
No. Each year retains 80% of the current amount, so the numerical decrease becomes smaller as the amount shrinks.
Population growth and depreciation
Depreciation is a decrease in value over time. Population change can increase or decrease. In each case, begin with the original amount and decide what fraction remains after one period.
A 15% decrease leaves 85%, giving multiplier 0.85. Three periods mean multiplying by 0.85 three times. Count the periods carefully: the starting value is before any change, so it does not use up the first multiplier.
A population of 12 000 grows by 3% per year for 4 years.
Estimated population after 4 years = 13 506 people, to the nearest whole person. This is a mathematical model; round at the end.
A $9000 machine depreciates by 15% each year. Find its value after 3 years.
Value to the nearest cent = $5527.13. Total loss in value = 9000 − 5527.125 = $3472.88 to the nearest cent.
Each year’s decrease is 15% of that year’s starting value, rather than 15% of the original $9000 every year.
Pause and explainAfter two years of 5% growth, why is the multiplier 1.05²?
There are two separate annual changes. Each multiplies the current value by 1.05, so the combined factor is 1.05 × 1.05.
Reverse the process and find the rate
To work backward through repeated growth, undo the whole repeated multiplier. Subtracting the percentage of the final amount would use the wrong starting value and would not reverse the process.
When the rate is unknown, first compare final amount with original amount to find the overall multiplier. An nth root then finds the equal multiplier used in each of n periods. Only after that do you convert the multiplier into a percentage increase or decrease.
After two years of 10% annual growth, a quantity is 2420. Find its starting value.
Divide by the complete repeated multiplier. Check: 2000 × 1.10² = 2420.
A value grows from 500 to 605 in two years at a constant annual rate.
If the annual multiplier is k, then 500k² = 605.
k = √1.21 = 1.10
The annual rate is 10%. The overall 21% increase is not split into two equal 10.5% annual increases.
Pause and explainAn amount doubles over two equal growth periods. Is the growth rate 50% each period?
No. The period multiplier is √2, about 1.414, so the rate is about 41.4%. Multiplying by 1.5 twice would give 2.25 times the original.
Your turn. Choose the multiplier.
Identify the initial value, the change per period and the number of periods. Write the one-period multiplier before using a power.
If working backward, divide by the full growth or decay factor. If finding a rate, distinguish the overall multiplier from the per-period multiplier. Check that a decay calculation gives a smaller positive amount.
A population of 5000 grows by 4% annually. Estimate the population after 3 years to the nearest whole number.
Reveal worked answer
A $640 item loses 25% of its value each year. Find its value after 2 years.
Reveal worked answer
After three annual decreases of 20%, a value is 512. Find the initial value.
Reveal worked answer
A value falls from 800 to 648 in two years at a constant annual percentage rate. Find the rate.