  # Solve Exponential Growth and Decay Problems

In this worksheet, students will use combined multipliers to quickly calculate repeated proportional change over time in both numerical sums and real-life problems. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR,

Curriculum topic:   Ratio, Proportion and Rates of Change

Curriculum subtopic:   Ratio, Proportion and Rates of Change Discrete Growth and Decay

Difficulty level:   #### Worksheet Overview Perhaps you are looking forward to having your first car.

Your insurance will probably cost you more than this! Honestly it will.

You will pay more for your insurance than you will the car, but you will love your first car.

To make matters worse, your car will lose 25% of its value in its first year and 18% of its value in its second year.

What is the value of your car at the end of this second year?

Can you even bear to think about it?

This is a great example of repeated proportional change, which can be used to predict changes over a period of time.

We are going to apply a percentage increase or decrease or (maybe even both) more than once.

We could decrease by 25% and then again by 18%, but this method could be time consuming.

Mathematicians are always looking for something a little quicker... a bit like your car! A quick recap...

To find a percentage increase or decrease we use a multiplier:

Increase 65 by 23% = 65 × 1.23 = 79.95

Decrease 80 by 42% = 100% - 42% = 58% which gives our multiplier = 80 × 0.58 = 46.40

With growth and decay problems, we need to apply these formulas more than once.

Top tip:

Use your multipliers at the same time by combining them.

Now back to trying to calculate the value of your car:

100 - 25 = 75  (0.75) > This is our multiplier for the first decrease

100 - 18 = 82 (0.82) > This is our multiplier for the second decrease

0.75 × 0.82 = 0.615 > This is our combined multiplier for the two years

£2300 × 0.615 = £1414.50 It is enough to make you cry like a baby isn't it... all that money lost!

Likewise to increase anything by more than one percentage, we can apply the same method.

e.g  Increase 320 by 2% and then 3.5%.

1.02 × 1.035 = 1.0557  -->  So our combined multiplier is 1.0557

320 × 1.0557 = 337.824

Top tip:

In this activity, we will use combined multipliers to quickly calculate repeated proportional change over time in both numerical sums and real-life problems.

You may want to have a calculator handy so that you can focus on applying the correct methods, rather than testing your mental arithmetic.

You may also need a pen and paper to record your working, so that you can compare this to our examples written by a maths teacher.

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