Growth/decay factors | Year 10

01 / LEARN

Understand the idea.

The factor includes the original amount

A percentage change acts on the current amount. An increase of $r %$ keeps the original $100 %$ and adds $r %$ , giving factor $1 + \frac{r}{100}$ . A decrease uses factor $1 - \frac{r}{100}$ .

When the same factor $b$ applies each period, $A = A_{0} b^{n}$ . The exponent counts repeated multiplications. The amount added or removed can change each time even though the percentage stays constant.

Identify the period

Label the starting amount $A_{0}$ at time $0$ , then match the factor to the exponent. An hourly factor must be raised to the number of hours, and an annual factor to the number of years.

Interpret the calculated value in context. A continuous mass can be fractional, while a population model may predict a non-integer count. The formula remains a model whose fixed percentage assumption may cease to be reasonable.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Growth

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

Worked solution

1
Convert the annual percentage into a factor.
$b = 1 + \frac{8}{100} = 1.08$
The new population includes the starting population plus an $8 %$ increase.
2
Count the annual growth periods.
$P = 500 {(1.08)}^{2}$
Two years apply the yearly factor twice.
3
Evaluate and interpret.
$P = 583.2$
This is the supplied model value; it is not an exact whole-person count.

Answer

583.2 in the model

EXAMPLE 02

Decay

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

Worked solution

1
Find the retained hourly fraction.
$b = 1 - \frac{15}{100} = 0.85$
Losing $15 %$ leaves $85 %$ of the current mass.
2
Apply two hourly factors.
$M = 200 {(0.85)}^{2}$
The second loss is based on the reduced mass after the first hour.
3
Evaluate with units.
$M = 200 (0.7225) = 144.5 g$
The initial mass and resulting mass use the same unit.

Answer

144.5 g

03 / YOUR TURN

Try it yourself.

Check your understanding

What factor represents a $12 %$ decrease?

Show answer & reasoning

0.88

A $12 %$ decrease retains $100 % - 12 % = 88 %$ , so its multiplier is $1 - 0.12 = 0.88$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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
$1 + \frac{6}{100} = 1.06$ .
1.06
Write the multiplier for a $23 %$ decrease.
2 marks
$1 - \frac{23}{100} = 0.77$ .
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 $\frac{t}{2}$ growth periods. At 6 hours there are 3 periods: $20 \times 27 = 540$ .
$N = 20 \times 3^{\frac{t}{2}}$ ; 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 ↗

In this lesson

Understand the idea.

The factor includes the original amount

Identify the period

See the method in action.

Growth

Convert the annual percentage into a factor.

Count the annual growth periods.

Evaluate and interpret.

Decay

Find the retained hourly fraction.

Apply two hourly factors.

Evaluate with units.

Try it yourself.

Check your understanding

Keep practising