Growth/decay models | Year 11

01 / LEARN

The idea, step by step

Choose the correct factor

Convert the percentage to a decimal before forming the multiplier. Growth uses $1 + r$ ; ordinary decay uses $1 - r$ , where $0 < r < 1$ . The exponent counts the stated intervals, so convert time before using the rule.

A fixed-percentage model assumes the same mechanism continues and that no extra deposits, withdrawals or changed conditions intervene. State these assumptions when giving a practical prediction.

P = A {(1 + r)}^{n}

P = A {(1 - r)}^{n}

Recover a starting value or time

To recover a starting amount, divide the later amount by the full compound factor. To find time, a table, a graph or logarithms can be used. Decide whether time is continuous or whether the question asks for the first whole reporting interval.

Keep full calculator precision until the final rounding step. Money is normally rounded to cents. Modelled populations may be decimal estimates, but an answer about a number of actual objects needs a suitable whole-number interpretation.

A common trap

Removing the same fraction of the original value each year in a compound model.

Instead: The percentage acts on the current amount, producing repeated multiplication.

02 / WORKED EXAMPLES

See the thinking

Example 1

Repeated growth

A culture starts at 500 units and increases 15% per hour. Predict its size after 3 h.

The growth factor is 1.15.
Use $P = 500 {(1.15)}^{t}$ .
At 3, $P = 760.4375$ , or about 760 units.

About 760 units.

Example 2

Asset depreciation

An asset worth 18000 dollars loses 12% per year. Find its value after two years.

The factor is $1 - 0.12 = 0.88$ .
$V = 18000 {(0.88)}^{2} = 13939.2$ .
Round the monetary answer to cents.

13939.20 dollars.

Check your understanding

An amount becomes 1250 after one year of 25% growth. Find its original value.

Show answer

1000.

Reverse the growth factor: $1250 / 1.25 = 1000$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write a model for 600 increasing by 8% per year.
2 marks
A machine worth 20000 dollars depreciates 10% annually. Find its value after 3 years.
2 marks
A quantity reaches 864 after two steps of 20% growth. Find its start.
2 marks
A quantity starts at 40 and doubles every 3 h. Find it after 9 h.
2 marks
A quantity changes from 500 to 320 over two equal steps. Find its positive step factor and percentage decrease.
3 marks
Use a scientific calculator to find when $100 {(1.1)}^{t}$ reaches 150. Give continuous time and first whole interval.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write a model for 600 increasing by 8% per year.
2 marks
Convert 8% to 0.08; use factor 1.08.
$P = 600 {(1.08)}^{n}$ .
A machine worth 20000 dollars depreciates 10% annually. Find its value after 3 years.
2 marks
$20000 {(0.9)}^{3} = 14580$ .
14580 dollars.
A quantity reaches 864 after two steps of 20% growth. Find its start.
2 marks
$A = 864 / {(1.2)}^{2} = 864 / 1.44 = 600$ .
600.
A quantity starts at 40 and doubles every 3 h. Find it after 9 h.
2 marks
9 h contains three doubling intervals.
$40 {(2)}^{3} = 320$ .
320.
A quantity changes from 500 to 320 over two equal steps. Find its positive step factor and percentage decrease.
3 marks
$b^{2} = 320 / 500 = 0.64$ .
Thus $b = 0.8$ .
Factor 0.8; decrease 20%.
Use a scientific calculator to find when $100 {(1.1)}^{t}$ reaches 150. Give continuous time and first whole interval.
3 marks
$t = log (1.5) / log (1.1) \approx 4.2542$ .
The first integer at or above this time is 5.
About 4.25 intervals; first whole interval 5.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: algebraic and graphical modelling of linear/exponential change, with interpretation and evaluation.

Official curriculum source ↗

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