Growth/decay factors | Year 10

01 / LEARN

The idea, step by step

The factor includes the original amount

An increase of $r %$ multiplies an amount by $1 + r / 100$ . A decrease multiplies by $1 - r / 100$ . The original $100 %$ must remain in the calculation.

Repeating the same percentage change uses repeated multiplication, not repeated addition. After $n$ equal periods, $A = A_{0} b^{n}$ .

Identify the period

The factor and exponent must refer to the same time interval. An hourly factor needs the number of hours, not a number of days.

A model is appropriate only while the assumed factor remains reasonable. State the initial time and the meaning of the quantity before substituting.

A common trap

An $8 %$ increase uses factor $0.08$ .

Instead: $0.08$ gives only the increase. The new total uses $1.08$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Growth

A population model starts at $500$ and grows by $8 %$ per year. Find the model value after 2 years.

The yearly factor is $1.08$ .
$P = 500 {(1.08)}^{2} = 583.2$ .
This is a model value; a population count may require a whole-number interpretation.

583.2 in the model

Example 2

Decay

A sample starts at $200$ g and loses $15 %$ of its current mass each hour. Find its mass after 2 hours.

The hourly factor is $0.85$ .
$M = 200 {(0.85)}^{2} = 144.5$ .

144.5 g

Check your understanding

What factor represents a $12 %$ decrease?

Show answer

0.88

$1 - 0.12 = 0.88$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write the multiplier for a $6 %$ increase.
2 marks
Write the multiplier for a $23 %$ decrease.
2 marks
Increase $800$ by $5 %$ for each of 2 periods.
2 marks
Reduce $600$ by $10 %$ for each of 3 periods.
2 marks
A model is $V = 900 {(0.92)}^{t}$ with $t$ in years. State the annual percentage change.
2 marks
A bacteria model triples every 2 hours from $20$ units. Write a model using hours $t$ and find the value at $t = 6$ .
3 marks

Read the worked solutions

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

Write the multiplier for a $6 %$ increase.
2 marks
$1 + 6 / 100 = 1.06$ .
1.06
Write the multiplier for a $23 %$ decrease.
2 marks
$1 - 23 / 100 = 0.77$ .
0.77
Increase $800$ by $5 %$ for each of 2 periods.
2 marks
$800 {(1.05)}^{2} = 882$ .
882
Reduce $600$ by $10 %$ for each of 3 periods.
2 marks
$600 {(0.9)}^{3} = 437.4$ .
437.4
A model is $V = 900 {(0.92)}^{t}$ with $t$ in years. State the annual percentage change.
2 marks
$1 - 0.92 = 0.08$ .
8% decrease
A bacteria model triples every 2 hours from $20$ units. Write a model using hours $t$ and find the value at $t = 6$ .
3 marks
There are $t / 2$ growth periods. At 6 hours there are 3 periods: $20 \times 27 = 540$ .
$N = 20 \times 3^{t / 2}$ ; 540 units.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

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