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Here we will learn about direct and indirect proportion, including what direct proportion is and what indirect proportion is. We will look at solving some real life word problems involving these different proportional relationships. We will also look at some GCSE maths revision and exam style questions (which are also in the IGCSE).

There are also direct and indirect proportion* *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Direct and indirect proportion **are two different proportional relationships. They are two ways in which quantities are related to each other.

**Direct proportion**is a relationship between two quantities where as one quantity increases, so does the other quantity.

For example,

The cost of a banana is 70p. As the number of bananas increases, so does the cost; 3 bananas would cost 3 times the cost of one banana (£2.10).

If y is directly proportional to x \ (y\propto{x}), then y=kx where k is the constant of proportionality.

**Step-by-step guide:** Direct proportion

**Indirect proportion (inverse proportion)**is a relationship between two quantities where as one quantity increases, the other quantity decreases and vice-versa.

For example, it takes 1 worker 9 hours to dig a hole. As the number of workers increases, the number of hours it takes to dig the same hole decreases. 3 workers would take a third of the time ( 3 hours).

To calculate indirect proportion problems we need to appreciate that multiplication and division are inverse operations of each other.

Indirect proportion is sometimes known as inverse variation.

If y is indirectly proportional to x \ (y\propto\frac{1}{x}), then y=\frac{k}{x} where k is the constant of proportionality.

**Step-by-step guide:** Inverse proportion

In order to answer word problems involving direct and inverse proportion:

**Determine the type of proportionality relationship between the two quantities.****Calculate the constant of proportionality,**k.**Calculate the unknown value.****Write the solution.**

Get your free direct and inverse proportion worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free direct and inverse proportion worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEA t-shirt costs £4. How much do 5 t-shirts cost?

**Determine the type of proportionality relationship between the two quantities.**

As the number of t-shirts increases, so does the cost. This is a direct proportion problem.

2**Calculate the constant of proportionality,** k.

For direct proportion, the constant of proportionality k is the cost of one t-shirt. As this is already given (a t-shirt costs £4 ) we can say k=4 and so y=4x where y would be the total cost of x number of t-shirts.

3**Calculate the unknown value.**

Substituting x=5 into y=4x, we have

y=4\times{5}=20.4**Write the solution.**

The cost of 5 t-shirts is £20.

7 bags of sweets weigh 350 grams. How much do 10 bags of sweets weigh?

**Determine the type of proportionality relationship between the two quantities.**

As the number of bags of sweets increases, so does the weight. This is a direct proportion problem.

**Calculate the constant of proportionality,** k.

For direct proportion, the constant of proportionality k is the weight of one bag of sweets.

Using k=\frac{y}{x} where y is the weight of a bag of sweets and x is the number of bags of sweets, we can calculate the value of k.

k=\frac{350}{7}=50

k=50 and so a bag of sweets weighs 50g and we can say y=50x.

**Calculate the unknown value.**

Substituting x=10 into y=50x, we have

y=50\times{10}=500.

**Write the solution.**

The weight of 10 bags of sweets is 500g.

8 laps of a race track has a total of 12 \ km. What would the distance be for 20 laps of the race track?

**Determine the type of proportionality relationship between the two quantities.**

As the number of laps of the track increases, so does the total distance. This is a direct proportion problem.

**Calculate the constant of proportionality,** k.

For direct proportion, the constant of proportionality k is the distance of one lap of the track.

Using k=\frac{y}{x} where y is the distance travelled and x is the number of laps, we can calculate the value of k.

k=\frac{12}{8}=1.5

k=1.5 and so one lap of the track is 1.5 \ km and we can say y=1.5x.

**Calculate the unknown value.**

Substituting x=20 into y=1.5x, we have

y=1.5\times{20}=30.

**Write the solution.**

The distance covered in 20 laps is 30 \ km.

A worker takes 10 days to fit a bathroom. How long would it take 2 workers to fit a bathroom?

**Determine the type of proportionality relationship between the two quantities.**

As the number of workers increases, the time taken to fit a bathroom decreases. This is an indirect proportion problem.

**Calculate the constant of proportionality,** k.

For indirect proportion, the constant of proportionality k is the time it takes one person to fit a bathroom.

As one worker takes 10 days to fit a bathroom, we can say that k=10 and so we have the equation y=\frac{10}{x} where y is the time taken for x number of workers to complete a bathroom.

**Calculate the unknown value.**

Substituting x=2 into y=\frac{10}{x}, we have

y=10\div{2}=5.

**Write the solution.**

It takes 2 workers 5 days to complete a bathroom.

An oil tank takes 25 hours to be filled by 3 hose pipes. How long does it take 5 hose pipes to fill the same oil tank?

**Determine the type of proportionality relationship between the two quantities.**

As the number of hose pipes increases, the time taken to fill the oil tank decreases. This is an indirect proportion problem.

**Calculate the constant of proportionality,** k.

For indirect proportion, the constant of proportionality k is the time it takes one hose to fill the oil tank.

Using k=xy where x is the number of hoses and y is the time taken to fill the oil tank, we can calculate the value of k.

k=3\times{25}=75

k=75 and so one hose would take 75 hours to fill the oil tank and we can say y=\frac{75}{x}.

**Calculate the unknown value.**

Substituting x=5 into y=\frac{75}{x}, we have

y=75\div{5}=15.

**Write the solution.**

It takes 5 hoses 15 hours to fill an oil tank.

10 computers can do a task in 15 minutes. How long does it take 3 computers to do the same task?

**Determine the type of proportionality relationship between the two quantities.**

As the number of computers increases, the time taken to do a task decreases. This is an indirect proportion problem.

**Calculate the constant of proportionality,** k.

For indirect proportion, the constant of proportionality k is the time it takes one computer to complete a task.

Using k=xy where x is the number of computers and y is the time taken to complete the task, we can calculate the value of k.

k=10\times{15}=150

k=150 and so one computer would take 150 hours to complete a task and we can say y=\frac{150}{x}.

**Calculate the unknown value.**

Substituting x=3 into y=\frac{150}{x}, we have

y=150\div{3}=50.

**Write the solution.**

It takes 3 computers 50 hours to complete a task.

**Modelling assumption**

Whenever you solve word problems for proportion you assume everything has the same value. If the question involves the costs of pencils, we assume each pencil costs the same. If the question involves the number of people working, we assume all the workers work at the same rate.

**Indirect proportion has a negative rate of change**

Direct proportion is referred to as “as one value increases, so does the other”. Indirect proportion is therefore considered to be the opposite where “as one value decreases, so does the other”. This is not true.

An indirectly proportional relationship shows that when one value increases, the other decreases. As a graph, this would look like a reciprocal graph.

**Indirect proportion is treated as direct proportion**

For example, if 3 people take 12 hours to build a wall, 6 people take 24 hours to build the same size wall. This is not true as we assume everyone works at the same rate and so the wall should be built in less time if more people are building it.

As the number of people increases, the time taken to build the wall decreases and so if we have 6 builders (double the original amount), the time it takes to build the wall should be 6 hours (half of the original amount). The type of proportion must be determined for every proportionality question.

**There may be several ways to solve problems involving proportion**

There may be several ways to get the correct answer for proportion questions. Some ways are more efficient than others depending on the numbers involved.

**Take care with writing money**

Money is used in some proportional word problems. If an answer is 4.1 you may be tempted to write it as £4.1, but the correct way of writing it would be £4.10.

**Take care with writing time**

Time is used in some proportional word problems. If an answer is 7.25 you may be tempted to write it as 7 hours 25 minutes, but it would be 7 hours 15 minutes. (Remember there are 60 minutes in an hour).

**Direct and indirect proportion** is part of our series of lessons to support revision on **proportion**. You may find it helpful to start with the main proportion lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

1. One tennis ball weighs 57 grams. Find the weight of 4 tennis balls.

7 grams

228 grams

14.25 grams

61 grams

k=57 and y=kx where y is the weight of x number of tennis balls. This means that y=57x. When x=4 ,

y=57\times 4=228 \ grams.

2. One worker takes 30 hours to build a wall. Find the time it would take 5 workers to build a similar wall.

150 hours

35 hours

6 hours

120 hours

k=30 and y=\frac{k}{x} where y is the time taken to build a wall with x number of people. This means that y=\frac{30}{x}. When x=5,

y=30\div{5}=6 hours.

3. 4 computer games cost £18. Find the cost of 5 computer games.

£23.50

£22

£24.50

£22.50

If 4 computer games cost £18, 1 computer game will cost £4.50.

k=4.5 and y=kx where y is the cost of x number of computer games. This means that y=4.5x. When x=5 ,

y=4.5\times 5=£22.50.

4. 7 workers take 20 weeks to build a house. How long would it take 10 workers to build the same house?

17 weeks

28.6 weeks

14 weeks

140 weeks

k=xy. When x=7, \ y=20 and so k=7 \times 20=140. This means that it would take 1 person 140 weeks to build the house and so y=\frac{140}{x}.

When x=10,

y=140 \div 10=14 weeks.

5. 5 pens cost 65p. Find the cost of 8 pens.

£1.04

£0.13

£0.82

£104

k=\frac{y}{x}. When x=5, \ y=65 and so k=65 \div 5=13. This means that 1 pen costs £0.13 and so y=0.13x.

When x=8,

y=0.13\times{8}=\pounds{1.04}.

6. 4 machines take 15 hours to complete a job. Find how long it would take 3 machines to complete the same job.

45 hours

5 hours

60 hours

20 hours

k=xy. When x=4, \ y=15 and so k=4 \times 15=60. This means that it would take 1 machine 60 hours to complete the job and so y=\frac{60}{x}.

When x=3,

y=60\div{3}=20 hours.

1. 5 sacks of potatoes cost £40.

Find the cost of 7 sacks of potatoes.

**(2 marks)**

Show answer

40\div 5=8

**(1)**

**(1)**

2. A small town has four rubbish trucks to collect its rubbish.

It takes four trucks 18 hours to collect the rubbish.

One of the trucks breaks down.

Find how long it would take 3 trucks to collect the rubbish in the town.

**(2 marks)**

Show answer

18\times{4}=72

**(1)**

**(1)**

3. A recipe for lemon cheesecake needs 250 grams of soft cheese.

The lemon cheesecake will have 6 proportions.

Soft cheese is sold in 300 gram packets.

The packets cost £1.25 each.

Samir wants to make enough lemon cheesecake for 15 portions.

Calculate the cost of soft cheese for Samir to make 15 portions of cheese cake.

**(4 marks)**

Show answer

250\div 6 \times 15

**(1)**

625 grams

**(1)**

625\div{300}=2.08\dot{3} and 3 packets.

**(1)**

**(1)**

You have now learned how to:

- Solve problems involving direct and inverse proportion, including algebraic representations

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