 # 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:   ### QUESTION 1 of 10 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.

Match each repeated proportional change below to its correct multiplier.

## Column B

Increase of 5% followed by an increase of 2%
1.1178
Decrease of 6% followed by a decrease of 4%
0.6336
Increase of 8% followed by an increase of 3.5%
0.9646
Decrease of 28% followed by an increase of 12%
1.071
Increase of 6% followed by an decrease of 9%
0.9024
Decrease of 5% followed by an increase of 2.5%
0.97375 When this tomato plant was planted it was 6 cm tall.

By the end of the first week, it had increased its height by 4%.

In the second week, it had grown a further 6%.

What was the height of the tomato plant by the end of the second week?

The answer has been rounded to 2 decimal places.

7.24 cm

6.61 cm

7.25 cm

6.62 cm

Andy has an annual salary of £35,000.

At the end of his first year he is given an increase of 3% to his salary.

The following year he is given an increase of 4.5%.

What is his salary at the end of this second year to the nearest £?

£39,825

£37,672

£39,725

£38,454 Rashid bought a house for £235,000.

In the first year, the house increased in value by 12%.

In the second year, the house decreased by 3%.

What was the value of the house after 2 years?

Type your answer using only numbers with no spaces or other characters, or you may be marked incorrectly.

£39,825

£37,672

£39,725

£38,454 Yusuf placed some cheese in his cupboard rather than his fridge and forgot about it.

After one week, there was 452 microorganisms on it.

After the second week, there was a decrease of 8% in this total, but in the next week there was an increase of 32%.

What was the total number of microorganisms after three weeks?

551

548

550

549 The rabbit population in a village has been eating all of the carrots!

In the Smiths' garden, there were 348 carrots.

Each week these have declined at a rate of 8% followed by 6% then 2%.

How many carrots did they have left at the end of the three weeks?

296

295

294

293

Is an increase of 15% followed by an increase of 22%, the same as an increase of 37%?

Yes

No

What is the difference between 400 being increased by 14% then 23% and 500 being increased by 16% and then 20%? Emma pays £350 for an aircraft ticket to the sunshine!

Since she bought it, the price has fluctuated.

There has been a 6% increase, followed by a 3% decrease, and then another 2.5% increase.

If Emma bought the ticket now, how much would she pay? A cafe opened in 2014 with 180 outlets in the UK.

The number of outlets is increasing by a rate of 8% each year.

How many cafe's will there be by the end of 2020?

• Question 1

Match each repeated proportional change below to its correct multiplier.

## Column B

Increase of 5% followed by an inc...
1.071
Decrease of 6% followed by a decr...
0.9024
Increase of 8% followed by an inc...
1.1178
Decrease of 28% followed by an in...
0.6336
Increase of 6% followed by an dec...
0.9646
Decrease of 5% followed by an inc...
0.97375
EDDIE SAYS
This should get you into the swing of things! We need to remember how to calculate our combined multipliers when we have to apply more than one percentage change. We need to find the two separate multipliers, then multiply them together to get the overall change. Remember that with a decrease we need to subtract our target % from 1.00 (which represents 100%), whereas with an increase we need to add our target % to 1.00. 1.05 × 1.02 = 1.071 0.94 × 0.96 = 0.9024 1.08 × 1.035 = 1.1178 0.72 × 0.88 = 0.6336 1.06 × 0.91 = 0.9646 0.95 × 1.025 = 0.97375 You can now use this skill to solve growth and decay problems to your heart's content.
• Question 2 When this tomato plant was planted it was 6 cm tall.

By the end of the first week, it had increased its height by 4%.

In the second week, it had grown a further 6%.

What was the height of the tomato plant by the end of the second week?

The answer has been rounded to 2 decimal places.

6.61 cm
EDDIE SAYS
As we are dealing with an increase here, we need to use our percentages in addition to a whole or 100%. 1.04 × 1.06 = 1.1024 (multiplier) 6 × 1.1024 = 6.6144 cm Which option is closest once this figure has been rounded to two decimal places? That is not much growth after two weeks, let's hope the plant grows more rapidly after this!
• Question 3

Andy has an annual salary of £35,000.

At the end of his first year he is given an increase of 3% to his salary.

The following year he is given an increase of 4.5%.

What is his salary at the end of this second year to the nearest £?

£37,672
EDDIE SAYS
First let's find our multiplier: 1.03 × 1.045 = 1.07635 Then multiply his starting salary by this: 35000 × 1.07635 = 37672.25 What is this amount rounded to the nearest £?
• Question 4 Rashid bought a house for £235,000.

In the first year, the house increased in value by 12%.

In the second year, the house decreased by 3%.

What was the value of the house after 2 years?

Type your answer using only numbers with no spaces or other characters, or you may be marked incorrectly.

EDDIE SAYS
Be careful as we have an increase, followed by a decrease here. Let's find our multiplier to start: 1.12 × 0.97 = 1.0864 Then let's multiply our value by this: 235000 × 1.0864 = £255,304 That's not bad though - he is still in profit even after the decrease.
• Question 5 Yusuf placed some cheese in his cupboard rather than his fridge and forgot about it.

After one week, there was 452 microorganisms on it.

After the second week, there was a decrease of 8% in this total, but in the next week there was an increase of 32%.

What was the total number of microorganisms after three weeks?

549
EDDIE SAYS
Let's start by finding our multiplier: 0.92 × 1.32 = 1.2144 Now let's multiply this with our starting value: 1.2144 × 452 = 548.9088 What is this amount rounded to the nearest whole number? That was pretty silly of him, wasn't it? Hopefully he realised before it was too disgusting!
• Question 6 The rabbit population in a village has been eating all of the carrots!

In the Smiths' garden, there were 348 carrots.

Each week these have declined at a rate of 8% followed by 6% then 2%.

How many carrots did they have left at the end of the three weeks?

295
EDDIE SAYS
Those greedy little rabbits! As this is a decrease, we need to subtract the percentages from 1.00. Your working should look a little like this: 0.92 × 0.94 × 0.98 = 0.847504 0.847504 × 348 = 294.931392 What is this number rounded to the nearest whole? Don't forget that it is important not to round our multiplier before we have used it, or this could seriously affect the answer we reach.
• Question 7

Is an increase of 15% followed by an increase of 22%, the same as an increase of 37%?

No
EDDIE SAYS
As always, let's start with our the multiplier. If we find the multiplier for a 22% increase followed by a 15% increase, we reach: 1.22 × 1.15 =1.403 This is not the same as an increase of 37%, as this multiplier would be 1.37. This logic does not work because the second change is not applied to the same original value, as this value has been changed by the first increase. Does that make sense?
• Question 8

What is the difference between 400 being increased by 14% then 23% and 500 being increased by 16% and then 20%?

135.12
EDDIE SAYS
If we are asked to find the 'difference', we need to subtract one of these values from the others. Let's work them each out separately then subtract them. 400 being increased by 14% then 23%: 1.14 × 1.23 = 1.4022 (multiplier) 400 × 1.4022 = 560.88 500 being increased by 16% and then 20%: 1.16 × 1.2 = 1.392 (multiplier) 500 × 1.392 = 696 Difference: 696 - 560.88 = 135.12 Did you get that answer?
• Question 9 Emma pays £350 for an aircraft ticket to the sunshine!

Since she bought it, the price has fluctuated.

There has been a 6% increase, followed by a 3% decrease, and then another 2.5% increase.

If Emma bought the ticket now, how much would she pay?

EDDIE SAYS
It's always the same with these airlines - you never know when the best time to buy a ticket is! Here is our working to calculate this: 1.06 × 0.97 × 1.025 = 1.053905 350 × 1.053905 = 368.86675 What is this value rounded to two decimal places? She bought it at the right time...great news! This means more ice cream for Emma on holiday.
• Question 10 A cafe opened in 2014 with 180 outlets in the UK.

The number of outlets is increasing by a rate of 8% each year.

How many cafe's will there be by the end of 2020?

EDDIE SAYS
Now this is easy to get incorrect. Did you try calculating the multiplier using 1.08 × 6? This is where mistakes can be made. Our multiplier is calculated like this: 1.086 = 1.58687432294 This is because there are 6 years between 2014 and 2020, but the increase is 8% for each year. 1.58687432294 × 180 = 285.63737813 What is this number rounded to the nearest whole? (As we cannot have part of a cafe open!) Congratulations on completing this activity - it was not an easy one!
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