CMClearMathAcademy

Exponential Decay

A free Precalculus lesson from the “Exponential and Logarithmic Functions” unit, with a worked example and practice problems including step-by-step solutions.

Exponential decay uses a repeated multiplier between 0 and 1. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Exponential and logarithmic models describe growth, decay, sound, pH, finance, and scientific scales.

Worked example

Problem. A quantity starts at 120 and decreases by 20 percent each period. What multiplier is used?

  1. Worked Example: First identify exactly what the question is asking: A quantity starts at 120 and decreases by 20 percent each period. What multiplier is used?
  2. For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
  3. A 20 percent decrease leaves 80 percent.
  4. 80 percent as a decimal is 0.8.
  5. That is the repeated multiplier.
  6. Check the result by substituting or estimating: the response should match 0.8 and make sense in the original problem.

Answer: 0.8

Practice problems

1. A quantity starts at 120 and decreases by 20 percent each period. What multiplier is used?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A quantity starts at 120 and decreases by 20 percent each period. What multiplier is used?
  2. For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
  3. A 20 percent decrease leaves 80 percent.
  4. 80 percent as a decimal is 0.8.
  5. That is the repeated multiplier.
  6. Check the result by substituting or estimating: the response should match 0.8 and make sense in the original problem.

Answer: 0.8

2. A quantity starts at 160 and decreases by 20 percent each period. What multiplier is used?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A quantity starts at 160 and decreases by 20 percent each period. What multiplier is used?
  2. For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
  3. A 20 percent decrease leaves 80 percent.
  4. 80 percent as a decimal is 0.8.
  5. That is the repeated multiplier.
  6. Check the result by substituting or estimating: the response should match 0.8 and make sense in the original problem.

Answer: 0.8

3. A quantity starts at 200 and decreases by 20 percent each period. What multiplier is used?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A quantity starts at 200 and decreases by 20 percent each period. What multiplier is used?
  2. For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
  3. A 20 percent decrease leaves 80 percent.
  4. 80 percent as a decimal is 0.8.
  5. That is the repeated multiplier.
  6. Check the result by substituting or estimating: the response should match 0.8 and make sense in the original problem.

Answer: 0.8

4. For A(t) = 80(0.5^t), find A(2).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For A(t) = 80(0.5^t), find A(2).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute t = 2.
  4. 0.5^2 = 0.25.
  5. A(2) = 80 * 0.25 = 20.
  6. Check the result by substituting or estimating: the response should match 20 and make sense in the original problem.

Answer: 20

5. For A(t) = 120(0.5^t), find A(2).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For A(t) = 120(0.5^t), find A(2).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute t = 2.
  4. 0.5^2 = 0.25.
  5. A(2) = 120 * 0.25 = 30.
  6. Check the result by substituting or estimating: the response should match 30 and make sense in the original problem.

Answer: 30

Practice this interactively with instant feedback and an AI tutor.

Practice Exponential Decay Take the free placement check

More Precalculus lessons