Percent Growth Modeling

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Convert percent growth or decay descriptions into exponential functions.
  • Use \(1 + r\) and \(1 - r\) correctly for growth and decay factors.
  • Interpret different percent growth rates and compare their effects.

Key Ideas

When a quantity grows or shrinks by a fixed percentage each time period, the situation is modeled with an exponential function.

For growth by \(r\%\) per time period:

\[ \text{factor} = 1 + \frac{r}{100} \]

For decay by \(r\%\) per time period:

\[ \text{factor} = 1 - \frac{r}{100} \]

General model:

\[ Q(t) = Q_0 \cdot (1 \pm r)^t \]

where:

  • \(Q_0\) = initial amount
  • \(r\) = rate in decimal form (e.g., \(0.08\) for \(8\%\))
  • \(t\) = number of periods

This is simply exponential growth/decay expressed in terms of percent change.

Common Problem Types

1. Writing exponential models

Identify \(Q_0\), convert the percent to a decimal, and use \(1 \pm r\).

2. Converting percent to a factor

  • Growth: multiply by \(1+r\)
  • Decay: multiply by \(1-r\)

3. Evaluating the model

Compute \(Q(t) = Q_0(1 \pm r)^t\) for specific \(t\) values.

4. Identifying growth vs decay

Growth factors are greater than 1; decay factors are between 0 and 1.

Strategies

  • Always convert percent to a decimal before building the factor.
  • Remember that growth/decay happens each time period, not just once.
  • Leave expressions exact unless specifically asked to approximate.
  • Identify whether the context represents increasing or decreasing behavior.
  • Check the factor:
    • If greater than 1 → growth
    • If between 0 and 1 → decay

Worked Examples

Example 1 — Population Growth

A town’s population is 25,000 and increases by 3% each year. Write a function for the population after \(t\) years and estimate the population after 5 years.

Solution:

  1. Initial amount: \(Q_0 = 25{,}000\)
  2. Growth rate: \(r = 0.03\)
  3. Growth factor: \(1 + r = 1.03\)

Model:

\[ P(t) = 25{,}000 \cdot 1.03^t \]

After 5 years:

\[ P(5) = 25{,}000 \cdot 1.03^5 \]

Leave exact or approximate numerically if needed.

Example 2 — Depreciation (Decay)

A machine is worth $50,000 and depreciates by 12% each year. Write a model for its value after \(t\) years.

Solution:

  1. Initial value: \(Q_0 = 50{,}000\)
  2. Decay rate: \(r = 0.12\)
  3. Decay factor: \(1 - r = 0.88\)

Model:

\[ V(t) = 50{,}000 \cdot 0.88^t \]

This is exponential decay.

WarningCommon Mistakes
  • Using \(r\) instead of \(1+r\) or \(1-r\) as the growth/decay factor.
  • Confusing a percent (like \(5\%\)) with its decimal form (0.05).
  • Applying the percent change only once instead of applying the factor every period (treating exponential as linear).
  • Forgetting to check whether the factor represents growth (>1) or decay (<1).

Practice Problems

  1. A city’s population of 80,000 grows by 2.5% per year. Write a function \(P(t)\) for the population after \(t\) years.

  2. Using your function from (1), write an expression for the population after 10 years.

  3. A car worth $30,000 loses 15% of its value each year. Write a model for \(V(t)\).

  4. A company’s sales are $200,000 and grow 7% each year.

    1. Write a function for sales after \(t\) years.
    2. Write an expression for the sales after 3 years.

  5. A population decreases by 4% each year, starting at 12,000.

    1. Write the exponential model.
    2. Is this growth or decay?
    3. What is the decay factor?

1.
2.5% growth → \(r = 0.025\)
Factor: \(1.025\)

\[ P(t) = 80{,}000 \cdot 1.025^t \]

2.
Population after 10 years:

\[ P(10) = 80{,}000 \cdot 1.025^{10} \]

3.
15% decay → \(r = 0.15\)
Factor: \(0.85\)

\[ V(t) = 30{,}000 \cdot 0.85^t \]

4.
(a) \(r = 0.07\) → factor \(1.07\)

\[ S(t) = 200{,}000 \cdot 1.07^t \]

  1. After 3 years:

\[ S(3) = 200{,}000 \cdot 1.07^3 \]

5.
(a) \(r = 0.04\) → factor \(0.96\)

\[ P(t) = 12{,}000 \cdot 0.96^t \]

  1. Since the factor is less than 1, this is decay.

  2. Decay factor: 0.96

Summary

  • Percent growth and decay always use the structure \(Q(t) = Q_0(1 \pm r)^t\).
  • Convert percent \(r\%\) into a decimal before using the formulas.
  • Growth factors are greater than 1; decay factors are between 0 and 1.
  • Apply the factor every period to model exponential change correctly.
  • Expressions can be left exact or evaluated numerically if needed.
  • Growth: multiply by \(1+r\).
  • Decay: multiply by \(1-r\).
  • Always convert percent → decimal.
  • Check whether the factor indicates growth (>1) or decay (<1).