2.2: Growth and Decay Factors | AP Precalculus

How it works

A percent increase uses a factor one plus the rate; a percent decrease uses one minus the rate. The rate must match the time unit in the exponent. Repeated percent change multiplies rather than adds.

P(t)=P_0(1+r)^t\quad\text{or}\quad P_0(1-r)^t

Worked examples

Example 1

A population starts at $\displaystyle 200$ and grows $\displaystyle 5\%$ each year. Write a model.

$\displaystyle P(t)=200(1.05)^t$ with t in years.

Example 2

A value of $\displaystyle 500$ loses $\displaystyle 20\%$ each year. Find the value after two years.

$\displaystyle V(2)=500(0.8)^2=320$ .

Problems

Problem 2.2.1

An initial amount of $\displaystyle 100$ grows by $\displaystyle 10\%$ per year. Find the amount after three years.

Show worked solution

$\displaystyle 100(1.1)^3=133.1$ units.

Problem 2.2.2

A model is $\displaystyle P(t)=80(0.93)^t$ . State the percent change each time unit.

Show worked solution

The factor is $\displaystyle 1-0.07$ , so this is a $\displaystyle 7\%$ decrease per time unit.

Problem 2.2.3

Why does increasing by $\displaystyle 20\%$ and then decreasing by $\displaystyle 20\%$ not restore the original value?

Show worked solution

The combined factor is $\displaystyle 1.2(0.8)=0.96$ , a net $\displaystyle 4\%$ decrease. The percentages use different starting amounts.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.2.1

Problem 2.2.2

Problem 2.2.3

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