Exponential growth and decay

Model repeated percentage change with multipliers, compound interest and exponential functions.

Also known as: compound interest, depreciation, exponential functions.

SAT · Advanced Math GCSE Maths · Ratio, proportion and rates of change IGCSE Maths · Number

Key ideas

Growth by $r\%$ uses the multiplier $1 + \frac{r}{100}$ ; decay uses $1 - \frac{r}{100}$ .
After $n$ periods: $\text{value} = \text{start} \times \text{multiplier}^n$ .
$f(t) = a \cdot b^t$ is growth when $b > 1$ and decay when $0 < b < 1$ .
Exponential models change by a constant factor; linear models change by a constant amount.

Multipliers

Increasing by $15\%$ is the same as multiplying by $1.15$ . Decreasing by $15\%$ means multiplying by $0.85$ .

Repeated change uses powers. A value of $1200$ that falls by $15\%$ each year is worth $V = 1200 \times 0.85^t$ after $t$ years.

Compound interest

Compound interest adds interest to the balance, so each year's interest is larger. At $3\%$ per year on $2000$ : $2000 \times 1.03^2 = 2121.80$ after two years, compared with $2120$ using simple interest.

Reading exponential functions

In $f(t) = a \cdot b^{t/k}$ , $a$ is the starting value and the quantity is multiplied by $b$ every $k$ units of time. For $f(t) = 80(1.5)^{t/2}$ , the value is multiplied by $1.5$ every $2$ units, so by $1.5^2 = 2.25$ every $4$ units.

Worked examples

Example 1 · Depreciation

A car bought for 15 000 loses 20% of its value each year. What is it worth after 3 years?

Step 1.Multiplier for a $20\%$ decrease: $0.8$ .
Step 2. $15000 \times 0.8^3 = 15000 \times 0.512$ .

Answer: 7680

Example 2 · Doubling

A population of 500 doubles every year. What is it after 3 years?

Step 1. $500 \times 2^3 = 500 \times 8$ .

Answer: 4000

Common mistakes

Using $0.15$ instead of $0.85$ for a $15\%$ decrease.
Applying simple interest (adding the same amount each year) to a compound problem.
Treating $1200(0.85)^t$ as a linear decrease of $0.15$ per year.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Exponential growth and decayEasier

A population of 500 bacteria doubles every hour. How many bacteria are there after 3 hours?

Question 2Exponential growth and decayEasier

Which of the following functions shows exponential growth?