IGCSEMaths: CoreCIERevision NotesNumberRatio & ProportionRatios

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First teaching 2023

First exams 2025

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Ratios (CIE IGCSE Maths: Core)

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Naomi C

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Naomi C

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Maths

Introduction to Ratio

What is a ratio?

  • A ratio is a way of comparing one part of a whole to another
    • Ratios are used to compare one part to another part
      • How many parts that make the whole will depend on the question

What do ratios look like?

  • Ratios involve two or three different numbers separated using a colon
    • E.g.,  2 : 5,  3 : 1,  4 : 2 : 3 
  • In all ratio questions, who or what is mentioned first in the question, will be associated with the first part of the ratio
    • E.g.,  The cake recipe with flour and butter in the ratio 2 : 1
      • 'Flour' is associated with '2' and 'butter' is associated with '1'
  • The numbers in a ratio tell us, for each quantity involved, its proportion of the whole 
    • In the ratio 4 : 3
      • The first quantity comprises 4 parts (of the whole)
      • The second quantity comprises 3 parts (of the whole)
      • In total, the whole is made up of 4 + 3 = 7 parts
    • In the ratio 2 : 5 : 3
      • The first quantity comprises 2 parts (of the whole)
      • The second quantity comprises 5 parts (of the whole)
      • The third quantity comprises 3 parts (of the whole)
      • In total, the whole is made up of 2 + 5 + 3 = 10 parts

How are ratios and fractions linked?

  • Fractions and ratios are closely linked and are used to compare a part of a whole
    • A fraction compares a part to the whole
    • A ratio compares one part to another part
  • For example, a pizza is sliced into 8 pieces, and shared between two people such that the first person receives 5 slices, and the second person receives 3 slices
    • As a fraction (of the whole)
      • The first person receives 5 over 8 of the pizza
      • The second person receives 3 over 8 of the pizza
    • The ratio of slices of the first person to the second person is 5 : 3
      • The 8 does not appear in the ratio but is obtained by adding the parts together (5 + 3 = 8)
  • Unlike ratios, fractions can be converted/expressed as percentages or decimals

Exam Tip

  • Start by writing down any information given in words as values, and use abbreviations to refer to them.
    • This will make it easier to look back at the information later 
    • E.g.  For a flour and butter cake mix, where twice as much flour is needed as butter, you could write

 straight F space colon space straight B 2 space colon space 1

Worked example

A pot of money is shared between three friends, Dave, John and Mary.
Dave receives $450, John receives $200 and Mary receives $350.

(a)

Find the total amount of money in the pot.

Add up the three separate amounts

450 plus 200 plus 350 equals 1000

$1000

 

(b)

Write down the ratio of money received by Dave, John and Mary.
(There is no need to simplify the ratio.)

Be careful with the order
Dave gets mentioned first, so 450 will be the first part of the ratio, then John and finally Mary

450 : 200 : 350

 

(c)
Write down the fraction of the pot of money that Mary receives.

(There is no need to simplify the fraction.)

Fractions are compared to the whole, so this will be 'Mary's money' "out of" 'total money'

bold 350 over bold 1000

Equivalent Ratios

What is an equivalent ratio?

  • Equivalent ratios are two ratios that represent the same proportion of quantities within a whole
    • E.g.,  The ratio 5 : 10 is equivalent to 20 : 40
  • Equivalent ratios are frequently used when the values involved take on a real-life meaning
    • E.g., a cake recipe involves flour and butter being mixed in the ratio 3 : 2
      • 3 g of flour and 2 g of butter would not lead to a very big cake
      • An equivalent ratio of 300 : 200 gives a more realistic 300 g of flour and 200 g of butter

How do I find an equivalent ratio?

  • You can find an equivalent ratio by multiplying (or dividing) each part of the ratio by the same value
    • E.g.,  Multiply each part of the ratio 2 : 3 : 7 by 4 to find an equivalent ratio of 8 : 12 : 28
    • Ratios can be scaled up or down to suit the context of a question
  • The size of each part in the ratio, relative to the others, is still the same
    • The actual values in the equivalent ratio may be more meaningful in the context of the situation
  • Finding an equivalent ratio is similar to finding equivalent fractions
    • However it is crucial to remember that 1 : 4 is not equivalent to 1 fourth
  • In questions you will be given information that will allow you to find the multiplier
    • The information may be about the one part of the ratio, or it may be about the whole 
    • You can then use this to find the other parts of the equivalent ratio and answer the question

Exam Tip

  • Writing down what you are doing to each part of the ratio helps show your working and makes it easier to keep track of what you are doing.
    • E.g.,

 A:B  3:4 ×5↓   ↓×5 15:20 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true,"color":"#FFFFFF"}

Worked example

The ratio of cabbage leaves eaten by two rabbits, Alfred and Bob, is 7 : 5.


 

(a)

Write down an equivalent ratio that would involve a total of 48 cabbage leaves being eaten.

We have information about the whole so first add up the parts of the ratio to find how many parts make the whole

7 + 5 = 12

We require a total of 48 so divide this by 12 to find the multiplier

48 ÷ 12 = 4

Answer the question by multiplying each part of the ratio by this multiplier

table row space A colon B space row space 7 colon 5 space row cell table row cell cross times 4 end cell downwards arrow end table end cell space space space cell table row downwards arrow cell cross times 4 end cell end table end cell row space 28 colon 20 space end table

28 : 20

  

(b)

On another occasion Bob eats 35 leaves.  Find out how many leaves Alfred eats.

Use the information about Bob to find the multiplier using division

35 ÷ 5 = 7

Now multiply Alfred's part of the ratio by the multiplier to answer the question

table row space A colon B space row space 7 colon 5 space row cell table row cell cross times 7 end cell downwards arrow end table end cell space space space cell table row downwards arrow cell cross times 7 end cell end table end cell row space 49 colon 35 space end table

Alfred eats 49 cabbage leaves

Simplifying Ratios

What is a simplified ratio?

  • Simplifying a ratio involves finding an equivalent ratio where the numbers involved are smaller
    • E.g.,  The ratio 45 : 30 is equivalent to 9 : 6
  • The need to simplify a ratio often arises when the initial ratio has been given using large values from a real-life context
  • A ratio is in its simplest form when
    • All of the values in the ratio are integers
    • There are no common factors between each of the values in the ratio
    • E.g., The simplest form of the ratio 45 : 30 is 3 : 2

How do I simplify a ratio?

  • Divide each part of the ratio by the same value
    • This value should be a common factor of all parts of the ratio
      • Ideally, the highest common factor (HCF) should be used to get the ratio into its simplest form in one go
      • If the HCF is not used, we can repeat the process of simplifying
    • E.g., Divide all parts of the ratio 30 : 66 : 12 by 6 to find the ratio in its simplest form 5 : 11 : 2

Exam Tip

  • Make the most of your calculator in the exam.
    • If you type the ratio in as though it was a fraction (although remember it is not) your calculator will give you the answer in its simplified form.

Worked example

Amber and Naomi are sharing a large cake that has been cut into 48 pieces.
Amber receives 30 of these pieces, Naomi receives the rest.

Write down the ratio, in its simplest form, of the number of pieces of cake that Amber receives to the number of pieces of cake that Naomi receives.

First use subtraction to find the number of pieces that Naomi receives

48 - 30 = 18

Next write down the ratio without simplifying
Make sure you have the order correct ('Amber to Naomi')

30 : 18

Look for a common factor of both 30 and 18 to simplify the fraction
6 is the highest common factor
Divide both parts of the fraction by 6

table row space A colon N space row space 30 colon 18 space row cell table row cell divided by 6 end cell downwards arrow end table end cell space space space cell table row downwards arrow cell divided by 6 end cell end table end cell row space 5 colon 3 space end table

5 : 3

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    Naomi C

    Author: Naomi C

    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.