Practice Growth vs Decay in Math

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

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Quick Recap

Exponential growth occurs when a quantity multiplies by a factor >1> 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%2\% per year. After 1515 years, what fraction of the original population remains?

Example 2

medium
What is the yy-intercept of f(x)=9⋅0.5xf(x) = 9 \cdot 0.5^x?

Example 3

hard
Show algebraically that f(x)=e−0.3xf(x)=e^{-0.3x} is an exponential decay function by rewriting it in the form a⋅bxa\cdot b^x and confirming 0<b<10<b<1.

Example 4

easy
What is the initial value of f(x)=12⋅1.4xf(x) = 12 \cdot 1.4^x?

Example 5

challenge
An investment grows 5%5\% annually. Roughly how many years to double (use that 1.0514≈1.981.05^{14}\approx1.98 and 1.0515≈2.081.05^{15}\approx2.08)?

Example 6

challenge
After how many half-lives does a sample drop below 10%10\% of its original amount?

Example 7

medium
Caesium-137 has a half-life of 3030 years. Starting with 8080 g, how much remains after 9090 years?

Example 8

hard
An investment grows by 4%4\% per quarter. What is the effective annual growth rate?

Example 9

medium
A drug is eliminated at 20%20\% per hour. If 400400 mg is taken, how much remains after 55 hours?

Example 10

medium
At what xx does y=100⋅0.9xy = 100 \cdot 0.9^x first drop below 5050?

Example 11

medium
Is y=5⋅(3/4)xy=5\cdot(3/4)^x growth or decay, and what is its long-run value?

Example 12

easy
Is y=1.05xy = 1.05^x growth or decay?

Example 13

medium
A car's value depreciates by 15%15\% per year. If it is worth $24{,}000 today, what is its value after 66 years?

Example 14

medium
A car worth $20000 loses 20%20\% of its value yearly. Value after 2 years?

Example 15

medium
Rewrite y=5⋅e0.4xy = 5 \cdot e^{0.4x} in the form a⋅bxa \cdot b^x and determine growth or decay.

Example 16

medium
A town of 50005000 shrinks 4%4\% per year. Write the model and find population after 1 year.

Example 17

easy
What is the long-run behavior of y=50⋅0.7xy = 50 \cdot 0.7^x as x→∞x \to \infty?

Example 18

easy
For y=100⋅0.9xy=100\cdot0.9^x, what is the decay factor?

Example 19

easy
For y=a⋅bxy=a\cdot b^x with b=1.5b=1.5, find the percent growth rate per step.

Example 20

medium
A sample of 8080 mg halves every 44 hours. How much remains after 88 hours?
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