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Solve Exponential Growth and Decay Problems

In this worksheet, students will be able to find an amount after repeated percentage changes as well as solve growth and decay problems.

'Solve Exponential Growth and Decay Problems' worksheet

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

QUESTION 1 of 10

car

You bought your first car for £2300. Your insurance will probably cost you more!  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 it lost 25% of its value in the first year and 18% of its value in the second year.

What is the value of your car at the end of the 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.

 All you are going to do is apply a percentage increase or decrease or (maybe even both) more than once.

You could decrease by 25% and then again by 18% but that could be time consuming.

Mathematicians are always looking for something a little quicker.. a bit like your car.

boy

 

A quick recap

To find a percentage increase or decrease  you use a multiplier.  

Increase 65 by 23% 65 x 1.23 = 79.95

Decrease 80 by 42% 80 x 0.58 = 46.40 (100% - 42% = 58% which gives our multiplier)

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

 

Top tip - use your multipliers at the same time.

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

 

100 - 25 = 75  (0.75)  For the first decrease

100 - 18 = 82 (0.82) For the second decrease

0.75 x 0.82 = 0.615

£2300 x 0.615 = £1414.50

crying baby

It is enough to make you cry like a baby isn't it.. all that money lost.

 

 

Like wise to increase anything  e.g  320 by  2% and the 3.5%

2 x 3.5 = 7 (Multiplier is 1.07)

320 x 1.07 =342.40

Top tip - don't be tempted to round your multipliers it could throw out your final answers

Match the following to their multipliers

Column A

Column B

Increase of 5% followed by an increase of 2%
1.10
Decrease of 6% followed by a decrease of 4%
0.6336
Increase of 8% followed by an increase of 3.5%
0.97375
Decrease of 28% followed by an increase of 12%
0.9024
Increase of 6% followed by an decrease of 9%
1.28
Decrease of 5% followed by an increase of 2.5%
0.9646

When this tomato plant was planted it was 6 cm tall.  At the end of the first week it had increased its height by 4% and in the second week it had grown a further 6%.  What is the height of the tomato plant at the end of the second week?

The answer has been rounded to 2 decimal places (2 d.p)

 

7.24cm

6.61cm

7.25cm

6.62cm

Andy has an annual (yearly) salary of £35,000.  At the end of his first year he is given an increase of 3%.  The following year he is given an increase of  4.5%.  What is his salary at the end of the second year?

£39,825

£40,250

£39,725

£40, 500

 

house

Rashid buys 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 is the value of the house after 2 years?

Start your answer £...

 

An ambulance crew drove 452 miles to and from  Accident and Emergency on Monday.

 On Tuesday there is a decrease of 8% but on Wednesday there is an increase of 32%.

What was the mileage covered by the end of Wednesday? 

Round your answer to the nearest mile

 

 

 

 

551

548

550

549

The rabbit population in my village are eating all of the carrots. In my garden there were 348 carrots. Each week these have declined  at a rate of 8% followed by 6% followed by 2%.  How many carrots do I have left?

Round your answer to the nearest whole number

 

 

296

295

294

293

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

 YesNo
Increase of 22% followed by an increase 15% is the same as an increase of 37%

 

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

Round your answer to 2 decimal places (2 d.p)

I paid £350 for an aircraft ticket to the sunshine.  Since I have bought it the price has fluctuated. 

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

If I bought the ticket now how much would I pay.

Round your answer to 2 decimal places( 2.dp)

Start your answer £....

 

 

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 2020?

Round your answer to the nearest whole number.

 

  • Question 1

Match the following to their multipliers

CORRECT ANSWER

Column A

Column B

Increase of 5% followed by an inc...
1.10
Decrease of 6% followed by a decr...
0.9024
Increase of 8% followed by an inc...
1.28
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 thing. You just have to remember how to calculate your multipliers. 5 x 2 = 10 multiplier 1.10 0.94 x 0.96 = 0.9024 (multiplier) 8 x 3.5 = 28 multiplier 1.28 0.72 x 0.88 =0.6336 (multiplier) 1.06 x 0.91 =00.9646 0.95 x 1.025 = 0.97375 You can now solve growth and decay problems until your heart is content.
  • Question 2

When this tomato plant was planted it was 6 cm tall.  At the end of the first week it had increased its height by 4% and in the second week it had grown a further 6%.  What is the height of the tomato plant at the end of the second week?

The answer has been rounded to 2 decimal places (2 d.p)

 

CORRECT ANSWER
6.61cm
EDDIE SAYS
1.04 x 1.6= 1.1024 Multiplier 6 x 1.1024 =6.6144 cm That is not much after two weeks. I hope I get some tomatoes from it. Being self sufficient isn't easy.
  • Question 3

Andy has an annual (yearly) salary of £35,000.  At the end of his first year he is given an increase of 3%.  The following year he is given an increase of  4.5%.  What is his salary at the end of the second year?

CORRECT ANSWER
£39,725
EDDIE SAYS
First find the multiplier. 4.5 x 3 =13.5 (multiplier 1.135) 35,000 x 1.135 = 39,725 Handy to know when negotiating a pay rise.
  • Question 4

 

house

Rashid buys 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 is the value of the house after 2 years?

Start your answer £...

 

CORRECT ANSWER
£255,304
EDDIE SAYS
Find your multiplier. Be careful as one of these is a decrease so we need that extra step at the beginning. 1.12 x 0.97 = 1.0864 1.0864 x 235000 = £255,304 Not bad - still in profit even after the decrease.
  • Question 5

An ambulance crew drove 452 miles to and from  Accident and Emergency on Monday.

 On Tuesday there is a decrease of 8% but on Wednesday there is an increase of 32%.

What was the mileage covered by the end of Wednesday? 

Round your answer to the nearest mile

 

 

 

 

CORRECT ANSWER
549
EDDIE SAYS
Find the multiplier 0.9x x 1.32 = 1.2144 1.2144 x 452 = 549 I think it must have been sports day at school on Wednesday with an increase like that. Shouldn't have the javelin in school!!
  • Question 6

The rabbit population in my village are eating all of the carrots. In my garden there were 348 carrots. Each week these have declined  at a rate of 8% followed by 6% followed by 2%.  How many carrots do I have left?

Round your answer to the nearest whole number

 

 

CORRECT ANSWER
295
EDDIE SAYS
Greedy little things 0.92 x 0.94 x0.98 = 0.847504 0.847504 x 348 = 295 (nearest whole number) I thought rabbits were supposed to be fluffy and cute
  • Question 7

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

CORRECT ANSWER
 YesNo
Increase of 22% followed by an increase 15% is the same as an increase of 37%
EDDIE SAYS
As always start with a the multiplier. Find the multiplier for 22% increase followed by 15% increase. 1.22 x 1.15 =1.403 this is not the same as using 37 as your multiplier 1.37
  • Question 8

 

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

Round your answer to 2 decimal places (2 d.p)

CORRECT ANSWER
135.12
EDDIE SAYS
1.14 x 1.23 = 1.4022 1.4022 x 400 =560.88 1.16 x 1.20 = 1.392 1.392 x 696 The word difference should remind you that a subtraction is required. 696 - 560.88 = 135.12
  • Question 9

I paid £350 for an aircraft ticket to the sunshine.  Since I have bought it the price has fluctuated. 

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

If I bought the ticket now how much would I pay.

Round your answer to 2 decimal places( 2.dp)

Start your answer £....

 

 

CORRECT ANSWER
£368.87
EDDIE SAYS
Its always the same with these airlines. You never know when the right time to buy is. Find the multiplier first 1.06 x 0.97 x 1.025 = 1.053905 1.053905 x 350 = £368.87 I bought it at the right time..this means more ice cream on holiday..yey.
  • 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 2020?

Round your answer to the nearest whole number.

 

CORRECT ANSWER
EDDIE SAYS
Now this is easy to get incorrect. Did you try the multiplier as 1.08 x 6? This is where mistakes can be made. Multiplier is 1.08 (power of 6) = 1.586874323 1.586874323 x 180 = 286 to the nearest whole number That steak is making me feel hungry, even though I am a veggie.
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