Ratios (CIE IGCSE Maths: Core)

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 of the pizza
    • The second person receives  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

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 
• 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

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