Growth, Decay and Iteration Problems

What you will learn

• How growth and decay use percentage multipliers.
• How to write repeated percentage change using powers such as 1.05³.
• How depreciation, appreciation, reverse repeated change and mixed changes work in value contexts.
• How to model simple population growth or decrease.
• How iteration means repeating a rule step by step.
• How to generate values from a recurrence-style rule and keep them in order.

Key facts before you start

Why this matters

Many GCSE contexts change by a percentage each year, month or step. A savings account may grow, a car may lose value, and a population may increase or decrease.

Iteration uses the same repeated-step idea. Instead of a percentage multiplier, you are given a rule such as “multiply by 0.8, then add 6” and asked to keep applying it.

Prior knowledge

You should already be comfortable with:

• changing percentages into decimals,
• multiplying by decimal multipliers,
• using powers on a calculator,
• rounding money and whole-number contexts sensibly,
• substituting a value into a simple rule.

Growth, decay and iteration problems infographic

Clear explanation

Growth and decay multipliers

Growth means the amount increases. Decay means the amount decreases. For percentage change, use a multiplier.

Repeated percentage change

If the same change happens again and again, apply the same multiplier again and again. Powers are a compact way to show this.

The percentage is taken from the new amount each time, not from the original amount each time.

If a question has different percentage changes, use the multipliers in the order given. A decrease of 10% followed by a decrease of 8% is not the same as one decrease of 18%.

Reverse repeated-change questions work backwards from the final amount. Divide by the multiplier power instead of multiplying by it.

Appreciation and depreciation

Appreciation means something gains value. Depreciation means something loses value.

Population and context examples

In population questions, round in a way that makes sense. You cannot have part of a person, so a final population answer is usually rounded to a whole number.

Iteration as repeated steps

Iteration means repeating a rule. The output from one step becomes the input for the next step.

GCSE iteration questions usually reward careful substitution and clear organisation. A small table can stop you losing track.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For growth, decay and iteration, marks usually come from choosing the correct repeated multiplier, using powers for several equal time periods, following iteration steps in order, and rounding only when the question asks or at the final stage. For mixed changes, show each multiplier separately so the order of the changes is clear.

Common mistakes

• Using the percentage itself as the multiplier: 6% growth uses 1.06, not 0.06.
• Adding the same amount each time: repeated percentage change applies the multiplier to the latest amount.
• Using a growth multiplier for decay: a 12% decrease leaves 88%, so use 0.88.
• Rounding too early: keep full calculator values until the final answer unless the question says otherwise.
• Skipping the first iteration: find the next value, then use that new value in the rule again.

Extension challenge

A camera depreciates by 15% in the first year and then appreciates by 15% in the second year. Is its final value the same as its original value?

Exam-board guidance

Growth, decay and simple iteration questions are common GCSE problem-solving skills. Expect practical contexts, repeated multipliers, calculator powers, recurrence-style steps, sensible rounding and clear step-by-step reasoning.

Next lesson

Next, move into Algebra Notation and Substitution.