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Exponential Growth and Decay Practice Problems

Every growth or decay word problem uses the same model, y = a·bᵗ. The work is in building b correctly — 1 + r for growth, 1 − r for decay, or a power of 2 or ½ when the problem talks about doubling or half-life. These twelve problems practise each version with real-world numbers.

  1. 1.

    What is the growth factor for a quantity that grows by 6% per year?

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    Keep the whole amount and add 6%: .

  2. 2.

    What is the decay factor for a quantity that loses 15% per year?

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    Losing 15% leaves 85% each year: .

  3. 3.

    Does model growth or decay? Answer "growth" or "decay".

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    The base 1.07 is greater than 1, so the amount grows — by 7% each period.

  4. 4.

    models decay. By what percentage does it decrease each period?

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    The factor 0.82 keeps 82%, so the decrease is .

  5. 5.

    A town of 12,000 people grows by 3% per year. What is its population after 10 years, to the nearest whole person?

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    .

  6. 6.

    A $28,000 car loses 15% of its value each year. What is it worth after 5 years, in dollars?

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    . Note the total loss is about 56%, not .

  7. 7.

    A culture of 500 bacteria doubles every 4 hours. How many bacteria are there after 24 hours?

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    24 hours is doublings: .

  8. 8.

    A 200 g sample has a half-life of 10 days. How many grams remain after 30 days?

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    30 days is 3 half-lives: g.

  9. 9.

    $5,000 is invested at 4% annual interest compounded monthly. What is the balance after 3 years, in dollars?

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    .

  10. 10.

    $5,000 is invested at 4% compounded continuously. What is the balance after 3 years, in dollars?

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    .

  11. 11.

    A $900 phone loses 30% of its value each year. After how many whole years is it first worth less than $300?

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    . Year by year: , still above 300; . So it first drops below $300 after 4 years.

  12. 12.

    A patient takes 80 mg of a drug with a half-life of 6 hours. How many milligrams remain after 15 hours, to two decimal places?

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    half-lives: mg.
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Exponential Growth and Decay Practice Problems

These word problems follow the exponential growth and decay lesson. For half-life problems you can check your numbers with the half-life calculator.

The models

SituationModel
grows by rate per period
decays by rate per period
doubles every
half-life
compounded times a year
compounded continuously

Before you calculate

Match the time units: a monthly rate needs in months, and a half-life in hours needs in hours. Keep all the decimals in the power and round only the final answer.

Common mistakes to watch for

  • Wrong growth factor. Growth of per period gives ; decay gives .
  • Confusing half-life with rate. After one half-life, the amount is exactly half.
  • Mixing time units. Keep the time unit consistent with the rate.
  • Assuming linear change. Equal time steps multiply by the same factor; they do not add the same amount.

Frequently Asked Questions

How do I turn a percentage rate into a growth or decay factor?+

Write the rate as a decimal r. Growth uses 1 + r, so 6% growth is 1.06. Decay uses 1 − r, so 15% decay is 0.85. Never use r on its own as the factor.

How do doubling time and half-life fit the same model?+

Put the number of doublings or halvings in the exponent. After t hours with a doubling time of d, the amount is a·2^(t/d). With a half-life of h, it is a·(1/2)^(t/h).

Why is losing 15% a year for 5 years not a 75% loss?+

Each year's 15% is taken from a smaller amount than the year before. The remaining fraction is 0.85⁵ ≈ 0.44, so the total loss is about 56%.

What is the difference between monthly and continuous compounding?+

Monthly compounding uses A = P(1 + r/12)^(12t); continuous compounding uses A = Pe^(rt). Continuous always gives slightly more, but the difference is small — about $1.12 on $5,000 over 3 years at 4%.

How should I enter money answers?+

Type the number without commas, for example 12423.75. A dollar sign and the comma-formatted version are also accepted, and answers within a cent are marked correct.

Calculators for this topic

Exponential Growth Calculator

More practice problems

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