College Algebra

Exponential Growth and Decay

Lesson

Anything that gets multiplied by the same factor every fixed time period grows or decays exponentially. Populations, money in an account, radioactive material, drugs in the bloodstream — all follow the same shape.

The formula:

A(t) = A_0 \cdot b^{\,t/p}

$A_0$ is the starting amount, $b$ is the multiplier per period, $p$ is the period length, and $t$ is elapsed time. The exponent $t/p$ counts how many periods have passed.

Doubling means $b = 2$ .
Tripling means $b = 3$ .
Half-life means $b = \tfrac{1}{2}$ .
A growth rate of $r$ per period means $b = 1 + r$ ; decay rate $r$ means $b = 1 - r$ .

Worked example 1 — growth

A bacteria culture doubles every 5 hours. If you start with 6 cells, how many are there after 15 hours?

Number of doubling periods: $15 / 5 = 3$ .

A = 6 \cdot 2^3 = 6 \cdot 8 = 48

Worked example 2 — decay

A radioactive substance has a half-life of 6 years. Starting with 240 grams, how much remains after 18 years?

Number of half-lives: $18 / 6 = 3$ .

\begin{aligned} A &= 240 \cdot \left(\tfrac{1}{2}\right)^3 \\ &= 240 \cdot \tfrac{1}{8} \\ &= 30 \end{aligned}

How to type your answer

Type a single number — the amount, or the number of years/periods, depending on the question. Examples: 800, 30, 4.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Population doubles each year. Start: 5. After 3 years?}

Problem 2

\text{Bacteria triple each hour. Start: 4. After 2 hours?}

Problem 3

\text{Substance halves each year. Start: 80\,g. After 3 years (g)?}

Problem 4

\text{Investment doubles each decade. Start: \$50. After 30 years (\$)?}

Practice

Standard problems matching the lesson.

Problem 5

\text{Population doubles every 5 years. Start: 100. After 15 years?}

Problem 6

\text{A car's value halves every 8 years. Start: \$32{,}000. After 24 years (\$)?}

Problem 7

\text{Bacteria triple every 4 hours. Start: 6. After 12 hours?}

Problem 8

\text{Half-life 6 years. Start: 240\,g. After 18 years (g)?}

Problem 9

\text{Doubles every 2 hours. Start: 7. After 6 hours?}

Problem 10

\text{Triples every 20 years. \$1{,}000 start. After 40 years (\$)?}

Problem 11

\text{Half-life 5 days. Start: 64\,g. After 25 days (g)?}

Problem 12

\text{Doubles every 30 minutes. Start: 5. After 2 hours?}

Problem 13

\text{Population doubles each decade. 200 in 1990. In 2030?}

Problem 14

\text{A drug halves in the body every 4 hours. Start: 48\,mg. After 12 hours (mg)?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\text{Investment grows by factor }\tfrac{3}{2}\text{ each year. \$128 start. After 4 years (\$)?}

Problem 16

\text{Triples every 2 hours. Start: 4. After 8 hours?}

Problem 17

\text{Loses 1/3 of mass each year (so 2/3 remains). Start: 81\,g. After 4 years (g)?}

Problem 18

\text{Doubles each year. Start: 5. After how many years does it reach 80?}

Problem 19

\text{Half-life 10 years. Start: 32\,g. How many years until 1\,g remains?}

Problem 20

\text{Triples each decade. Start: 8. How many decades to reach 648?}

Problem 21

\text{Bacteria double each hour. Start: 3. After 5 hours?}

Problem 22

\text{100\% annual rate, compounded yearly. \$1{,}000 start. After 3 years (\$)?}