Ratios (Cambridge O Level Maths)

What is a ratio?

• A ratio is a way of comparing one part of a whole to another
  • Fractions are used to compare one part to the whole
  • Ratios are used to compare one part to another part
  • How many parts that make the whole will depend on the question
    • Questions will involve ratios with two or three parts

What do ratios look like?

• The different numbers in a ratio are separated using a colon
  • e.g.  2 : 5,  3 : 1,  4 : 2 : 3 
• In the ratio 4 : 3
  
  • The first number 4 tells us that the first quantity involved consists of 4 parts (of the whole)
  • The second number 3 tells us the second quantity involved consists of 3 parts (of the whole)
  • In total, the whole is made up of 4 + 3 = 7 parts
• For example, in a cake recipe, flour and butter are mixed in the ratio 2 : 1
  • This would mean for every 2 'parts' of flour we would need 1 'part' of butter
  • In this case it would be appropriate that a 'part' is a gram
• In the ratio 2 : 5 : 3
  • The first quantity consists of 2 parts compared to 5 parts for the second quantity and 3 parts for the third
  • In total the whole is made up of 2 + 5 + 3 = 10 parts
• For example, in a cake recipe, flour, butter and sugar are mixed in the ratio 3 : 2 : 1
  • This would mean for every 3 'parts' of flour we would need 2 'parts' of butter and 1 'part' of sugar
  • Again, a 'part' being a gram would be appropriate
• 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 means that 'flour' is associated with '2' and 'butter' is associated with '1'

How are ratios and fractions linked?

• Fractions compare a part to the whole, ratio compares one part to another
• 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
  • As a fraction (of the whole) the second person receives  of the pizza

• • The ratio of slices of the first person to the second person is 5 : 3
    • Note how the value 8 does not appear in the ratio but is obtained by adding the parts together (5 + 3 = 8)
• Fractions could also be converted/expressed as percentages or decimals

What is an equivalent ratio?

• Like with fractions, ratios can be equivalent to each other
  • e.g.  The ratio 5 : 10 is equivalent to 20 : 40
    Can you see why?
• Equivalent ratios are frequently used when the values involved take on a real-life meaning
  • For example, a cake recipe involves flour and butter being mixed in the ratio 3 : 2
    3 grams of flour and 2 grams of butter would not lead to a very big cake!
  • More realistically, for a cake made at home, the ratio would be 300 : 200
    i.e. 300 grams of flour to 200 grams of butter

How do I find an equivalent ratio?

• Every part of the ratio would need to be multiplied (or divided, see Simplifying ratios) by the same value
  • e.g.  If we multiply (all parts) of the ratio 2 : 3 : 7 by 4, we get 8 : 12 : 28
• The size of each part, relative to the others, is still the same
  • but 8, 12 and 28 may be more meaningful in the context of the situation
• We can keep scaling ratios (up or down) to suit our needs or numbers
  • 2 : 3 : 7 is equivalent to 8 : 12 : 28 (by multiplying by 4)
  • 2 : 3 : 7 is also equivalent to 200 : 300 : 700 (by multiplying by 100)
  • So all three are equivalent and we can keep going
    • 2 : 3 : 7, 8 : 12 : 28, 10 : 15 : 35 and 200 : 300 : 700 are all equivalent ratios
• This 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
  • e.g.  If Rameen and Tori are sharing counters in the ratio 4 : 5 and Rameen receives 24 counters
    • the multiplier is 24 ÷ 4 = 6 and so Tori would receive 5 x 6 = 30 counters
    • the equivalent ratio would be 24 : 30
    • with 24 + 30 = 54 counters making the whole

What is meant by simplifying ratios?

• In a similar way to fractions, ratios can be simplified
  • This involves making the numbers used in the ratio smaller and so easier to work with
  • e.g.  The ratio 45 : 30 is equivalent to 3 : 2
    Can you see why?
• The need to simplify a ratio often arises when the initial ratio has been given or explained using (large) values that have a real life meaning
  • e.g.  A pizza is cut into 24 slices and shared between two people, the first gets 10 slices, the second gets 14 slices
    The ratio of slices that the first person gets to the second person is 10 : 14
    However this ratio can be simplified to 5 : 7

How do I simplify a ratio?

• In a very similar way to how you would simplify fractions!
• Every part of the ratio would need to be divided 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
    • But if this is not used, we can repeat the process of simplifying
  • However the HCF may take some time to work out and can be awkward to do so
    • The far majority of simplifying ratio questions will have a (highest) common factor that is easy to spot
• In the pizza example above, the ratio of slices the first person got to the second person was 10 : 14
  • It should be easy to see that 2 will divide into both 10 and 14
    • In this case, 2 is also the HCF of 10 and 14
  • So dividing both 10 and 14 by 2 gives us the simplified ratio 5 : 7
    • This is fully simplified (it cannot be simplified any further)
• In the ratio 6 : 24 : 36 we may spot that 3 is a common factor
  • Dividing all three parts by 3 we get the simplified ratio 2 : 8 : 12
    • But this is not fully simplified as we may now spot that 2 is a common factor
  • That's fine, we can just repeat the process but divide by 2
    • This gives the fully simplified ratio 1 : 4 : 6
  • If we had spotted 6 as the HCF, and divided each part by that, we would've got to 1 : 4 : 6 in one go