IGCSE · Practice questions · Exponential growth and decay

Multiply, then repeat.

Extended. Six original Cambridge-style questions on exponential growth and decay: forward, backward and finding the number of years.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Start value times multiplier to the power n.

Multiplier: growth $1 + \dfrac{r}{100}$; decay $1 - \dfrac{r}{100}$.
After n steps: start $\times$ multiplier$^{n}$.
Number of steps: try values of $n$ and show the values either side of the target.
Backwards: divide by multiplier$^{n}$.
Rate: multiplier $= \sqrt[n]{\text{end} \div \text{start}}$.

Recall

[2 marks]

Write down the multiplier for (a) growth of 7% per year (b) a decrease of 12% per year.

(a) 1.07 ✓
(b) 0.88 ✓

Calculation

[2 marks]

A town has a population of 24 000. It grows by 3.5% each year. Calculate the population after 10 years, to the nearest hundred.

$24\,000 \times 1.035^{10} = 33\,854.37...$ ✓
33 900 to the nearest hundred ✓

Calculation

[2 marks]

A machine costs $\$45\,000$. Its value decreases by 22% each year. Calculate its value after 5 years.

Multiplier 0.78. ✓
$45\,000 \times 0.78^{5} = \$12\,992.28$ ✓

Calculation

[2 marks]

A colony of 300 bacteria doubles every 20 minutes. How many bacteria are there after 2 hours?

2 hours $= 6$ lots of 20 minutes. ✓
$300 \times 2^{6} = 19\,200$ ✓

Analysis

[3 marks]

A house is worth $\$250\,000$. Its value increases by $r\%$ each year. After 3 years it is worth $\$281\,216$. Find $r$.

$250\,000 \times m^{3} = 281\,216$, so $m^{3} = 1.124\,864$. ✓
$m = \sqrt[3]{1.124\,864} = 1.04$. ✓
$r = 4$ ✓

Analysis

[4 marks]

There are 12 000 fish in a lake. The number decreases by 6% each year.

(a) How many fish are there after 5 years?
(b) After how many complete years is the number of fish first less than half of 12 000?

(a) $12\,000 \times 0.94^{5} = 8806.8...$, about 8810 fish (3 s.f.). ✓
(b) Half is 6000. $12\,000 \times 0.94^{11} = 6075.6$, still above 6000. ✓
$12\,000 \times 0.94^{12} = 5711.0$, below 6000. ✓
12 years. ✓

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