Growth vs Decay Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Growth vs Decay.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Exponential growth occurs when a quantity multiplies by a factor $> 1$ repeatedly; exponential decay when it multiplies by a factor between 0 and 1.

Growth compounds: each period's increase is larger than the last. Decay shrinks: each period's decrease is smaller than the last, never quite reaching zero.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Repeatedly multiplying by a factor above 1 grows; multiplying by a factor between 0 and 1 decays.

Common stuck point: The procedure for growth vs decay is the easy part; the trap is reading $b=0.8$ as growth because 0.8 is positive. Asking "Is the quantity multiplied by the same factor each period, and is that factor above or below 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the quantity multiplied by the same factor each period, and is that factor above or below 1?

Worked Examples

Example 1

easy

Classify each function as growth or decay, and find its value at

x=3

: (a)

f(x)=4\cdot2^x

, (b)

g(x)=100\cdot(0.5)^x

Answer

(a) Growth,

f(3)=32

; (b) Decay,

g(3)=12.5

First step

(a) Base

b=2>1

: exponential growth.

f(3)=4\cdot8=32

Full solution

2
(b) Base $b=0.5$ , $0<0.5<1$ : exponential decay. $g(3)=100\cdot(0.5)^3=100\cdot0.125=12.5$ .
3
Interpretation: (a) doubles with each unit increase; (b) halves with each unit increase.

For

y=a\cdot b^x

with

a>0

: if

b>1

the function grows exponentially; if

0<b<1

it decays exponentially. The base

b

determines direction; the coefficient

a

sets the initial value.

Example 2

medium

A radioactive substance has a half-life of

10

years. Starting with

200

g, write the decay function and find the amount remaining after

35

years.

Example 3

medium

A car's value depreciates by

15\%

per year. If it is worth $24{,}000 today, what is its value after

6

years?

Example 4

medium

Compare a

\$1000

investment growing at

6\%

per year for

10

years to the same amount growing by a flat

\$60

per year for

10

years. Find both final values.

Example 5

medium

Caesium-137 has a half-life of

30

years. Starting with

80

g, how much remains after

90

years?

Example 6

medium

At what

x

does

y = 100 \cdot 0.9^x

first drop below

50

Example 7

hard

A population doubles every

7

years. How long, to the nearest tenth of a year, does it take to grow by a factor of

5

Example 8

hard

An investment grows by

4\%

per quarter. What is the effective annual growth rate?

Example 9

hard

A radioactive sample loses

30\%

of its mass in

5

years. Find its half-life.

Example 10

challenge

A culture follows

P(t) = 100 \cdot 2^{t/3}

but a competing species removes a constant

20

organisms per hour. Write a recursion for the population per hour and find

P_1, P_2, P_3

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A population grows by

8\%

per year. Starting at

5000

, write the growth function and find the population after

10

years.

Example 2

hard

Show algebraically that

f(x)=e^{-0.3x}

is an exponential decay function by rewriting it in the form

a\cdot b^x

and confirming

0<b<1

Example 3

easy

y=0.8^x

growth or decay?

Example 4

easy

y=3^x

growth or decay?

Example 5

easy

y=2^x

, what is the growth factor per step?

Example 6

easy

Does exponential decay ever produce negative values for

y=5\cdot0.5^x

Example 7

easy

For

y=100\cdot0.9^x

, what is the decay factor?

Example 8

easy

Linear decrease by

5

each step vs. exponential decay: which loses a fixed amount?

Example 9

easy

For

y=a\cdot b^x

with

b=1.5

, find the percent growth rate per step.

Example 10

easy

A bacteria count doubles each hour. Is this growth or decay?

Example 11

medium

A $1000 investment grows

10\%

per year. Write the model and find the value after 2 years.

Example 12

medium

A car worth $20000 loses

20\%

of its value yearly. Value after 2 years?

Example 13

medium

Which is bigger after

5

steps: starting at

100

with

+10

linear, or

\times1.1

exponential?

Example 14

medium

A sample of

80

mg halves every

4

hours. How much remains after

8

hours?

Example 15

medium

For

y=2\cdot3^x

, find

y

x=3

Example 16

medium

y=5\cdot(3/4)^x

growth or decay, and what is its long-run value?

Example 17

medium

A population grows from

200

242

in one step. Find the growth factor.

Example 18

medium

Compare

0.5^x

and

0.9^x

: which decays faster?

Example 19

challenge

A culture triples every

2

hours, starting at

50

. Write the model and find the count at

6

hours.

Example 20

challenge

After how many half-lives does a sample drop below

10\%

of its original amount?

Example 21

challenge

An investment grows

5\%

annually. Roughly how many years to double (use that

1.05^{14}\approx1.98

and

1.05^{15}\approx2.08

Example 22

medium

A town of

5000

shrinks

4\%

per year. Write the model and find population after 1 year.

Example 23

easy

y = 1.05^x

growth or decay?

Example 24

easy

y = 7 \cdot 0.25^x

growth or decay?

Example 25

easy

A quantity loses

10\%

each year. What is its decay factor?

Example 26

easy

A savings account grows by

3\%

per year. Write the growth factor.

Example 27

easy

Evaluate

f(2)

where

f(x) = 6 \cdot 3^x

Example 28

medium

A bacteria culture triples every

4

hours. Starting with

200

bacteria, write the population function and find the count after

12

hours.

Example 29

medium

A drug is eliminated at

20\%

per hour. If

400

mg is taken, how much remains after

5

hours?

Example 30

medium

A town's population shrinks by

2\%

per year. After

15

years, what fraction of the original population remains?

Example 31

medium

A coffee at

90^\circ

C cools toward room temperature (

20^\circ

C) so that the gap halves every

10

minutes. What is the temperature after

30

minutes?

Example 32

hard

Convert

A(t) = 800 \cdot e^{-0.08t}

to an equivalent annual decay percent.

Example 33

hard

Two populations:

A(t) = 200 \cdot 1.05^t

and

B(t) = 500 \cdot 1.02^t

. When does

A

first exceed

B

Example 34

hard

A bank advertises

6\%

APR compounded monthly. What is the effective annual yield?

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Related Concepts

Exponential Function Exponential Growth Exponents

Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential function