Practice Growth vs Decay in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

A town's population shrinks by

2\%

per year. After

15

years, what fraction of the original population remains?

Example 2

medium

What is the

y

-intercept of

f(x) = 9 \cdot 0.5^x

Example 3

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 4

easy

What is the initial value of

f(x) = 12 \cdot 1.4^x

Example 5

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 6

challenge

After how many half-lives does a sample drop below

10\%

of its original amount?

Example 7

medium

Caesium-137 has a half-life of

30

years. Starting with

80

g, how much remains after

90

years?

Example 8

hard

An investment grows by

4\%

per quarter. What is the effective annual growth rate?

Example 9

medium

A drug is eliminated at

20\%

per hour. If

400

mg is taken, how much remains after

5

hours?

Example 10

medium

At what

x

does

y = 100 \cdot 0.9^x

first drop below

50

Example 11

medium

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

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

Example 12

easy

y = 1.05^x

growth or decay?

Example 13

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 14

medium

A car worth $20000 loses

20\%

of its value yearly. Value after 2 years?

Example 15

medium

Rewrite

y = 5 \cdot e^{0.4x}

in the form

a \cdot b^x

and determine growth or decay.

Example 16

medium

A town of

5000

shrinks

4\%

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

Example 17

easy

What is the long-run behavior of

y = 50 \cdot 0.7^x

x \to \infty

Example 18

easy

For

y=100\cdot0.9^x

, what is the decay factor?

Example 19

easy

For

y=a\cdot b^x

with

b=1.5

, find the percent growth rate per step.

Example 20

medium

A sample of

80

mg halves every

4

hours. How much remains after

8

hours?

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Related Concepts

Exponential Function Exponential Growth Exponents