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Exponential Growth & Decay

LK20.103 min read

Exponential growth and decay problems challenge students to think beyond linear patterns, where quantities multiply rather than add by fixed amounts. In LK20 Trinn 10, students work with population models that double every 3 hours or car values that depreciate 15% annually, building critical thinking skills for real-world scenarios.

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Why it matters

Exponential models appear everywhere in science and finance. Bacteria colonies double every 20 minutes during optimal conditions, requiring healthcare workers to predict infection spread. Investment portfolios growing at 7% annually turn $10,000 into $19,672 after 10 years through compound interest. Radioactive materials with 8-year half-lives help doctors calculate safe radiation doses, while car values depreciating 12% yearly affect insurance calculations. Population growth at 2.1% annually means cities must plan infrastructure for 50% more residents every 33 years. Students who master these concepts understand pandemic modeling, retirement planning, and environmental decay rates that directly impact their future careers and personal decisions.

How to solve exponential growth & decay

Exponential Growth

  • General form: y = a Β· bΛ£, where a is the starting value and b is the growth factor.
  • If b > 1, the quantity grows; if 0 < b < 1, it decays.
  • Percent growth of r% means b = 1 + r/100.
  • To find y after x periods, substitute and evaluate.

Example: A population of 500 grows 10% per year. After 3 years: y = 500 Β· 1.10Β³ β‰ˆ 665.5.

Worked examples

Beginner

A bacteria colony starts with 500 bacteria and doubles every hour. How many bacteria are there after 5 hours?

Answer: 16000

  1. Identify the doubling pattern β†’ 500 Γ— 2⁡ β€” The colony doubles 5 times, so multiply by 2⁡.
  2. Calculate the power β†’ 2⁡ = 32 β€” 2 multiplied by itself 5 times is 32.
  3. Multiply by the starting amount β†’ 500 Γ— 32 = 16000 β€” There are 16000 bacteria after 5 hours.
Easy

A town has 8,000 people and grows by 10% per year. How many people live there after 2 years?

Answer: 9680

  1. Find the growth factor β†’ 1 + 10/100 = 1.1 β€” A 10% increase means multiplying by 1.1 each year.
  2. Year 1 β†’ 8000 Γ— 1.1 = 8800 β€” After year 1 the population is 8800.
  3. Year 2 β†’ 8800 Γ— 1.1 = 9680 β€” After year 2 the population is 9680.
  4. Verify with formula β†’ A = 8000 Γ— 1.1Β² = 9680 β€” Using A = P Γ— (1 + r)α΅— confirms the answer.
Medium

A radioactive sample of 120 g has a half-life of 5 years. How much remains after 10 years?

Answer: 30 g

  1. Find number of half-lives β†’ 10 Γ· 5 = 2 β€” In 10 years, the sample halves 2 times.
  2. Halving 1 β†’ 120 Γ· 2 = 60 β€” After 5 years: 60 g remaining.
  3. Halving 2 β†’ 60 Γ· 2 = 30 β€” After 10 years: 30 g remaining.
  4. Verify with formula β†’ 120 Γ— (1/2)Β² = 30 β€” Using the half-life formula confirms the answer.

Common mistakes

  • βœ—Students often add the growth rate instead of multiplying: for 500 bacteria doubling 3 times, they calculate 500 + 500 + 500 + 500 = 2000 instead of 500 Γ— 2Β³ = 4000.
  • βœ—Confusing growth factor with growth rate: for 15% growth, students use b = 0.15 instead of b = 1.15, calculating 1000 Γ— 0.15Β² = 22.5 instead of 1000 Γ— 1.15Β² = 1322.5.
  • βœ—Mixing up half-life periods: with 80g decaying over 12 years (4-year half-life), students calculate 80 Γ· 12 = 6.67g instead of 80 Γ— (1/2)Β³ = 10g.
  • βœ—Using wrong base for time conversion: when population doubles every 3 hours, students calculate 6 hours as 2Β² instead of 2⁢/Β³ = 2Β² = 4 times growth.

Practice on your own

Generate unlimited exponential growth and decay practice problems with varied difficulty levels using MathAnvil's free worksheet creator.

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Frequently asked questions

How do I know when to use exponential versus linear models?β–Ύ
Look for multiplicative language like "doubles," "grows by 15%," or "half-life." Linear models use additive terms like "increases by 50 people yearly." If the rate depends on the current amount, use exponential: 200 bacteria becoming 400 suggests doubling, not adding 200 more.
What's the difference between growth factor and growth rate?β–Ύ
Growth rate is the percentage (15%), while growth factor is what you multiply by (1.15). For 8% growth, the rate is 0.08 but the factor is 1.08. For 20% decay, the rate is -0.20 but the factor is 0.80. Always add 1 to positive growth rates.
How do I handle fractional time periods in exponential problems?β–Ύ
Use the formula y = a Γ— b^(t/p) where p is the period length. If bacteria double every 3 hours and you want the amount after 5 hours, calculate 500 Γ— 2^(5/3) β‰ˆ 500 Γ— 3.17 = 1587 bacteria.
Why do some exponential decay problems use (1/2)ⁿ instead of 0.5ⁿ?β–Ύ
Both are equivalent: (1/2)Β³ = 0.5Β³ = 0.125. Using fractions like (1/2) often makes half-life calculations clearer visually, showing exactly how many times the quantity halves. Choose whichever format your students find easier to track during multi-step problems.
How do I teach students to identify the starting value and growth factor?β–Ύ
Look for "starts with" or "initially" for the starting value a. For the growth factor b, find percentage language: "increases 12%" means b = 1.12, "decreases 8%" means b = 0.92, "doubles" means b = 2, "halves" means b = 0.5.

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