A phone costs $640. It loses 25% of its value each year. Find its value after 3 years.
- A 25% decrease leaves 75%, so the multiplier is 0.75.
- value = 640 × 0.753
- 0.753 = 0.421875
- 640 × 0.421875 = 270
Answer: $270
Exponential growth and decay
Maths · Number. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Some quantities change by the same percentage in every equal period of time. A population may rise by 5% each year, or a car may lose 12% of its value each year. This is exponential growth or exponential decay.
Each period you multiply by the same number, the multiplier. For 5% growth it is 1.05. For a 12% loss it is 0.88. After n periods you have multiplied n times, so the multiplier is raised to the power n.
The change is not the same amount each time, because each percentage is taken from the new value. Doubling means a multiplier of 2 and halving means a multiplier of 0.5.
Rules to remember
- growth multiplier = 1 + r/100, so 5% growth gives 1.05
- decay multiplier = 1 − r/100, so a 12% decrease gives 0.88
- value after n periods = starting value × multipliern
- number of periods n = total time ÷ length of one period
- In y = A × bn, A is the starting value; b > 1 means growth and 0 < b < 1 means decay
- To go back n periods in time, divide by multipliern
Common mistake
Students multiply the percentage by the number of years, for example treating 12% a year for 2 years as a 24% decrease. Use the multiplier to the power n instead: 0.882, not 0.76.
Worked example
Ask about this
Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.