Exponential growth and decay
A repeated percentage change uses a multiplier at each period. Growth uses a multiplier above 1, while decay uses a multiplier between 0 and 1
- Choose and check a model Use constant first differences for linear models, constant second differences for quadratic models and constant ratios for exponential models. State assumptions, compare predictions with observations and revise the model when necessary
- Keep the initial amount The amount at period 0 stays outside the power \(\displaystyle A_0=2{,}000\)
- Use the correct multiplier Add r divided by 100 for growth and subtract it for decay \(\displaystyle \begin{gathered}1+\frac{6}{100}=1.06\\1-\frac{6}{100}=0.94\end{gathered}\)
- Raise the multiplier to n The period count is the exponent on the multiplier because the change repeats \(\displaystyle A_3=2{,}000\left(1.06\right)^3\)
- Round once at the end Keep the exact calculation until the final amount is rounded to the stated precision \(\displaystyle \begin{gathered}2{,}000(1.06)^3=2{,}382.032\\A_3\approx2{,}382.03\end{gathered}\)
Worked example
A savings balance is $1,500 and grows by 4% each year. Find the balance after 3 years, to the nearest cent
- \(\displaystyle 1+\frac{4}{100}=1.04\)A 4% increase has multiplier 1.04
- \(\displaystyle 1.04^3=1.124864\)Raise the multiplier to the three-year period count
- \(\displaystyle A_3=1{,}500\times1.124864\)Multiply the initial balance by the repeated multiplier
- \(\displaystyle A_3=1{,}687.296\)Keep all digits before rounding
- \(\displaystyle A_3\approx\$1{,}687.30\)Round the balance once to the nearest cent
Common mistakes
Difficulty levels
Questions about this topic
Can I use fractional periods?+
Only when the context permits a continuous model between observations. A rule defined only at whole compounding periods does not by itself establish what happens between them
Is a rounded crossing the first whole period below a target?+
No. Find the continuous crossing if appropriate, then separately check the permitted observation times. For a strictly below condition, an exact whole-period equality is not yet below
What is the difference between a growth multiplier and a decay multiplier?+
A growth multiplier is greater than 1 because the amount increases. A decay multiplier is between 0 and 1 because the amount decreases
Why is the multiplier raised to the number of periods?+
The same percentage change is applied repeatedly, once for each period, so the multiplier is used as a power
How is compound interest represented by an exponential rule?+
The initial balance is multiplied by 1 plus the interest rate divided by 100, raised to the number of compounding periods
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 3: Algebra and indices