Direct & Inverse Proportion (Edexcel GCSE Maths: Foundation)

Revision Note

Test Yourself

Author

Naomi C

Expertise

Maths

Direct Proportion

What is direct proportion?

Direct proportion
- As one quantity increases/decreases by a certain rate (factor)
- The other quantity will increase/decrease by the same rate
The ratio of the two quantities is constant
- E.g., 2 boxes of cereal is 800 g of cornflakes
  Doubling the number of boxes of cereal (4 boxes) will double the amount of cornflakes (1600 g)

What does the graph look like for direct proportion?

If and are directly proportional then:
- The graph will be a straight line that passes through the origin
- The equation will be of the form
  - E.g. $y = 2 x$ or $y = \frac{1}{3} x$

Direct proportion graph

What is the unitary method?

The unitary method is when the proportion relating to just 1 unit of a quantity is found
- This can be used to find any proportional amount
This is done when two quantities in proportion are divided by the same scale factor
- This is helpful to find 1 unit of a quantity
- E.g. If 6 identical bicycles weigh 108 kg, then 1 bicycle weighs 108 ÷ 6 = 18 kg (by dividing both quantities by a scale factor of 6)
  - This means 11 bicycles weigh 11 × 18 = 198 kg etc

Exam Tip

Start a question by jotting down the key values involved.
- You'll be able to pick them out quickly and easily later on.

Worked example

The ingredients for 12 cookies are as follows:

90 g butter
130 g chocolate
180 g flour
150 g sugar
1 egg

(a)

If Tanya makes 50 cookies, how much sugar does she need?

12 cookies require 150 g of sugar
Find how much sugar is needed to make 1 cookie (by dividing 150 by 12)

150 ÷ 12 = 12.5 grams of sugar for 1 cookie

Find how much sugar is needed for 50 cookies (by multiplying 12.5 by 50)

12.5 × 50 = 625

625 grams of sugar

(b)

If Giles buys a 500 gram bag of flour to make cookies, how many whole cookies can he make?

Method 1
12 cookies require 180 g of flour
Find out how much flour is in 1 cookie (by dividing 180 by 12)

180 ÷ 12 = 15 grams of flour for 1 cookie

Find how many lots of 15 grams are in 500 grams of flour (by dividing 500 by 15)

500 ÷ 15 = 33.33… cookies

Write down how many whole cookies can be made

There is not enough for 34 whole cookies

33 cookies

Method 2
12 cookies have 180 grams of flour
Find how many lots of 12 cookies the 500-gram bag of flour contains (by dividing 500 by 180)

500 ÷ 180 = 2.77… lots of 12 cookies

Calculate how many cookies are in 2.77… lots of 12 cookies (by multiplying 2.77… by 12)

2.77… × 12 = 33.33… cookies

Write down how many whole cookies can be made

There is not enough for 34 whole cookies

33 cookies

Inverse Proportion

What is inverse proportion?

Inverse proportion
- As one quantity increases by a certain rate (factor)
- The other quantity will decrease by the same rate
This relationship applies vice versa too, if one quantity decreases the other increases
- E.g., If 2 robots take 15 hours to build a car
  Tripling the number of robots (6) would mean the time taken to build a car would be divided by 3 (5 hours)

What does the graph look like for inverse proportion?

If and are inversely proportional then:
- The graph will be a reciprocal graph that never touches the axes
- The equation will be of the form
  - E.g. $y = \frac{2}{x}$ or $y = \frac{3}{x}$

Inverse proportion graph

How do I solve inverse proportion questions?

Read through wordy inverse proportion questions carefully
- Ensure that you understand the context of the question
- Some questions may tell you the relationship between the two values as a ratio
Identify the two quantities involved
Find the factor that you will be increasing/decreasing by
- This may be given to you in the question, e.g., 'the amount is tripled'
- Alternatively, find this by dividing the 'new' quantity by the 'old' quantity
Divide the other quantity by this factor to find the required quantity
Give your final answer in context
- Round and give units where appropriate

Exam Tip

Think about the context to determine if a question is direct or inverse proportion.
- As the number of robots goes up, the time to build a car comes down (inverse proportion).
- If you buy more boxes of cereal, the amount of cereal also increases (direct proportion).

Worked example

The time taken to fill a swimming pool is inversely proportional to the number of pumps used to pump the water in.
If 3 pumps are used it will take 12 hours to fill the pool.

(i)

Work out the amount of time required to fill the pool if 9 pumps are used.

(ii)

If only 2 pumps are available find out how much extra time will be needed to fill the pool

(i)

Identify the two quantities, 'number of pumps' and 'time (hours)'

Find the factor ('new' ÷ 'old') from the number of pumps

$\frac{9}{3} = 3$

Divide the time by the factor

$12 \div 3 = 4$

Answer in context with units

If 9 pumps are used it will take 4 hours to fill the swimming pool

(ii)

We are still working with 'pumps' and 'time'

Find the factor using 'new' ÷ 'old'

$\frac{2}{3} (= 0.666 666 . . .)$

Avoid rounding, keep the exact value in your calculator (it will be stored under the ANS key)
Find the time taken by dividing by the factor

$12 \div \frac{2}{3} = 18$

Answer in context with units

If only 2 pumps are available then it will take 18 hours to fill the swimming pool

You've read 0 of your 0 free revision notes

Get unlimited access

to absolutely everything:

Downloadable PDFs
Unlimited Revision Notes
Topic Questions
Past Papers
Model Answers
Videos (Maths and Science)

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Test yourself Next topic

Did this page help you?