IGCSEMaths: CoreCIERevision NotesNumberRatio & ProportionDirect & Inverse Proportion

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Direct & Inverse Proportion (CIE IGCSE Maths: Core)

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Naomi C

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CIE Maths: CoreCambridge (CIE) International Maths: CoreCambridge (CIE) International Maths: Extended

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)

How do I solve direct proportion questions?

  • Read through wordy direct 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
    • E.g. Hours worked and pay
  • Find the factor that you will be increasing/decreasing by
    • This may be given to you in the question, e.g., 'the amount is doubled'
      • E.g. Three times as many hours are worked
    • Alternatively, find this by dividing the 'new' quantity by the 'old' quantity
  • Multiply the other quantity by this factor to find the required quantity
    • E.g. If three times as many hours are worked, the pay will be three times more in total
  • Give your final answer in context
    • Round and give units where appropriate

What is the unitary method?

  • The unitary method means finding one of something (1 unit of something)
    • This can be a useful strategy
  • For example, find the weight of 7 boxes if 8 boxes weigh 60 kg
    • Find the weight of 1 box (1 unit) using division
      • 60 kg ÷ 8 boxes = 7.5 kg per box
    • Scale this unit up using multiplication
      • 7.5 kg per box × 7 boxes = 52.5 kg

How do I find which item is the best value?

  • A common proportion question is to find which product is the best value for money
    • Which is best value for money?
      • 1.5 kg of flour for $0.84
      • or 5 kg of flour for $2.65?
  • To do this, compare the prices of 1 kg (1 unit):
    • Divide the price in dollars by the weight this buys
      • $0.84 ÷ 1.5 kg = $0.56 per kg
      • $2.65 ÷ 5 kg = $0.53 per kg
    • Make sure you compare prices for the same size unit of both items 
      • The lowest unit price is the best value (0.53 < 0.56)
      • 5 kg of flour for $2.65 is better value for money than 1.5 kg of flour for $0.84

Examiner 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 bonus received by an employee is directly proportional to the profit made by the company they work for.
Bonuses are paid at a rate of $250 per $3000 profit the company makes.

(i)

Work out the bonus an employee receives if the company makes a profit of $18 000.

(ii)
If the company makes less than $600 profit, no bonus is paid.  Find the lowest bonus an employee could receive.

(i)

Identify the two quantities 'profit' and 'bonus'

Find the factor ('new' ÷ 'old') from the profit

fraction numerator 18 space 000 over denominator 3000 end fraction equals 6

Multiply the bonus by the factor

250 cross times 6 equals 1500

Answer in context with units

An employee should receive a bonus of $1500

 

(ii)

We are still working with profit and bonus

Find the factor using 'new' ÷ 'old'

600 over 3000 equals 1 fifth

Find the amount of bonus by multiplying by the factor

250 cross times 1 fifth equals 50

Answer in context with units

The lowest amount of bonus an employee could receive is $50

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)

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

Examiner 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

9 over 3 equals 3

Divide the time by the factor

12 divided by 3 equals 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'

2 over 3 space space open parentheses equals 0.666 space 666 space... close parentheses

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 divided by 2 over 3 equals 18

Answer in context with units

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

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    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.