# Identify Inverse Proportion

In this worksheets, students will identify inverse proportion and solve simple 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, Direct and Inverse Proportion

Difficulty level:

### QUESTION 1 of 10

If one quantity decreases as the other increases, then they are said to be in inverse proportion.

Inverse is just another word for doing the opposite.

For example , if one quantity is halved the other is doubled.

Inverse proportion questions don't always appear that straightforward though.

Don't bury your head in the sand with this type of question.  They are more straightforward than you think.

Four dogs take five hours to did a big hole.

How long would it take ten dogs to dig the same sized hole at the same rate?

 4 dogs Take 5 hours 1 dog 5 x 4 20 hours 10 dogs 20 ÷10 2 hours

It takes 5 bakers, 4 hours to bake 125 cakes.

How long would it take 8 bakers to make 200 cakes if they all work at the same rate?

 5 Bakers Take 4 hours 1 baker 4 x 5 20 hours 8 bakers 20 ÷ 8 2.5  hours (for 125 cakes) 1 cake 2.5 ÷ 125 0.02 hours 200 cakes 0.02 x 100 4 hours

Don't you think it is unfair that you always have to do that bit extra before you can get some cake?

It takes three people two hours to paint a shed.

How long will it take six people to paint the shed at the same rate.

1.5 hours

30 minutes

45 minutes

1 hour

It takes four people knitting three and a half hours to make a blanket

How long will it take seven people knitting to make the same blanket if they all work at the same rate?

2 hours 30 minutes

1 Hour 30 minutes

2 hours

1 hour

Four chefs can prepare a meal in 20 minutes.  They hire another chef.

How long will it take five chefs to prepare the same meal, if they all work at the same rate as the original four?

A journey  takes two hours when travelling at a speed of 30 mph.

How long will it take when travelling at a speed of 45 mph?

It takes fifty six builders nine days to complete a particular project.

It takes two groundsmen three hours to mow six  identical lawns.

How long will it take five groundsmen to mow 20 lawns of the same size?

Write you answer in figures without the units.

In the  Super Sandwich shop it takes 2 assistants  to make up 72 identical sandwiches.

The total time it takes is 144 minutes.

 68 minutes 72 minutes 78 minutes 80 minutes How long would it take 5 assistants to make up 90 of these sandwiches if they all worked at the same rate?

It takes twenty minutes for three people to shovel gravel onto two identical driveways.

 75 minutes 76 minutes 78 minutes How long will it take for two people to shovel gravel onto five identical driveways?

It takes fourteen art students to paint 35 identical canvases in two hours.

How long would it take for twenty art students to paint 60 identical canvases, if they all work at the same rate?

It takes twelve students three weeks to complete four projects.

How long will it take fifteen students to complete ten projects?

The ten projects need to be completed in under four weeks.  What is the minimum number of students that would be needed to complete the projects on time, assuming they all worked at the same rate?

• Question 1

It takes three people two hours to paint a shed.

How long will it take six people to paint the shed at the same rate.

1 hour
EDDIE SAYS
Using the calculation you would do the following 1 person 3 x 2 = 6 6 people 6 ÷6 = 1 hour You may have spotted that the numbers 3 and 6 are factors of each other, so you could have just halved the time with double the people.
• Question 2

It takes four people knitting three and a half hours to make a blanket

How long will it take seven people knitting to make the same blanket if they all work at the same rate?

2 hours
EDDIE SAYS
Find the time it would take one person 1 person 4 x 3.5 = 14 hours 7 people 14 ÷ 7 = 2 hours There is nothing nicer than snuggling up in a warm blanket on a cold day. Mind you it would probably take me more than 14 hours to make one on my own.
• Question 3

Four chefs can prepare a meal in 20 minutes.  They hire another chef.

How long will it take five chefs to prepare the same meal, if they all work at the same rate as the original four?

16
EDDIE SAYS
It is obviously going to be quicker with an extra chef. Which is always good as if you are like me, when you are hungry you don't like waiting for food. 1 chef would take 4 x 20 minutes = 80 minutes 80 ÷ 5 = 16 minutes
• Question 4

A journey  takes two hours when travelling at a speed of 30 mph.

How long will it take when travelling at a speed of 45 mph?

1 hour 20 minutes
EDDIE SAYS
You need to be careful on questions such as this as you are dealing with time and will need to convert this back to get the minutes. 2 hours x 30mph = 60mph 60mph ÷ 45 = 1 .33 minutes Now this is the bit that can catch us out. 1.33 minutes does not mean 1 hour and 33 minutes. The point 33 means 1/3 of an hour. 60 ÷3 = 20 This is where we get 1 hour 20 minutes.
• Question 5

It takes fifty six builders nine days to complete a particular project.

EDDIE SAYS
Lets find out how long one builder would take. 5 x 62 = 310 days Wow nearly a year to build a house.(They are a good builder doing that on their own) so for 2 builders 310 ÷ 2 = 155 Thats better it is always good to have a mate help you. For the minimum number of builders take the time for one builder and divide by the number of days you want the project completed. 310 ÷ 50 = 6.2 Therefore 7 builders
• Question 6

It takes two groundsmen three hours to mow six  identical lawns.

How long will it take five groundsmen to mow 20 lawns of the same size?

Write you answer in figures without the units.

4
EDDIE SAYS
Hopefully you are getting used to the calculations. This is one of the two step questions. 1 person 3 x 2 = 6 hours 5 people 6 ÷ 5 = 1.2 hours (that is for 6 lawns) Find out how long it would take to cut 1 lawn 1.2 ÷ 6 = 0.2 hours Now we can find out how long it would take 5 groundsmen to cut 20 lawns 0.2 x 20 = 4 hours. No minutes to have to convert which is even better.
• Question 7

In the  Super Sandwich shop it takes 2 assistants  to make up 72 identical sandwiches.

The total time it takes is 144 minutes.

 68 minutes 72 minutes 78 minutes 80 minutes How long would it take 5 assistants to make up 90 of these sandwiches if they all worked at the same rate?
EDDIE SAYS
It must be getting nearer lunchtime as I am thinking about food again. 2 x144 = 288 minutes for one assistant (the queue would be massive) 288 ÷ 5 = 57.6 minutes (for 72 sandwiches) 57.6 ÷ 72 = 0.8 90 x 0.8 = 72 minutes
• Question 8

It takes twenty minutes for three people to shovel gravel onto two identical driveways.

 75 minutes 76 minutes 78 minutes How long will it take for two people to shovel gravel onto five identical driveways?
EDDIE SAYS
This is just really a case of getting used to the calculations and getting the multiplication and division the correct way around. 20 minutes for 3 people 1 person will take 20minutes x 3 = 60 minutes 2 people 60 ÷ 2 = 30 minutes For 1 driveway 30 ÷ 2 = 15 15 x 5 = 75minutes That is a lot of digging just to get to the answer. But I expect you are an expert by now.
• Question 9

It takes fourteen art students to paint 35 identical canvases in two hours.

How long would it take for twenty art students to paint 60 identical canvases, if they all work at the same rate?

2 hours 24 minutes
EDDIE SAYS
Using time can often confuse on these questions. So lets not get into a muddle with these. It is just a sneaky way of seeing if you are paying attention. 1 student would take 14 x 2 = 28 minutes 20 students would take 28 ÷20 = 1.4 hours 1 canvas would take 1.4 ÷35 = 0.04 hours 60 x 0.04 = 2.4. Here's the thing, you need to convert .4 of an hour into minutes 0.4 x 60 = 24 Hence 2 hours 24 minutes
• Question 10

It takes twelve students three weeks to complete four projects.

How long will it take fifteen students to complete ten projects?

The ten projects need to be completed in under four weeks.  What is the minimum number of students that would be needed to complete the projects on time, assuming they all worked at the same rate?

EDDIE SAYS
12 x 3 = 36 weeks for 1 student 36 ÷ 15 = 2.4 weeks 1 project will take 2.4 ÷ 4 = 0.6 weeks 10 projects would take 0.6 x 10 = 6 weeks So 15 students x 6 weeks = 90 weeks 90 weeks ÷ 4 weeks = 22.5 (23 students)
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