Ratios (CIE IGCSE Maths: Extended)

What is a ratio?

• A ratio is a way of comparing one part of a whole to another
• A ratio can also be expressed as a fraction (of the whole)
• We often use a ratio (instead of a fraction) when we are trying to show how things are shared out or in any situation where we might use scale factors
• If a pizza were sliced into 8 pieces, and shared in the ratio 6:2, this means that person A receives 6 slices, and person B receives 2 slices

How do I simplify a ratio or find an equivalent ratio?

• Using the same example of 8 slices of pizza shared in the ratio 6:2
• We could also express this ratio as 3:1, even though we still have 8 slices
  • Both sides of the ratio have been divided by 2
    
    • The amount that each person gets relative to the other is still the same
    • In this case, person A receives 3 times as much as person B
    • 3:1 is "simpler" than 6:2, so we can say that 6:2 simplifies to 3:1
• We could also write this ratio as 12:4, 18:6, 24:8, 30:10 and so on
  • When finding an equivalent ratio, we must multiply or divide both sides of the ratio by the same number 
  • For a giant pizza with 800 slices, person A would receive 600 slices and person B would receive 200 slices
    • Both sides of the ratio have been multiplied by 100
  • We can keep doing this as long as person A receives 3 times as much as person B
• You can think of this process as similar to finding equivalent fractions, or simplifying fractions
  • However it is important to note that 1:4 is NOT equivalent to 

How do I use a ratio to find a fraction?

• We can use a ratio to find a fraction of the whole amount
• Using the same example of 8 slices of pizza shared in the ratio 6:2
• Person A receives 6 slices out of 8, or  of the pizza
  • This could be simplified to  of the pizza
• Person B received 2 slices out of 8, or  of the pizza
  • This could be simplified to  of the pizza
• These fractions could also then be converted to percentages if needed

How do I share an amount into a ratio?

• Suppose that $200 is to be shared between two people; A and B, in the ratio 5:3
• There are 8 “parts” in total, as A receives 5 parts and B receives 3 parts
• $200 must be split into 8 parts, so this means that 1 part must be worth $ 25, as 200 ÷ 8 = 25
• Some students find it helpful to show this in a simple diagram
  • 
• Person A receives 5 parts, each worth $25
  • 5 × 25 = $125 for person A
• Person B receives 3 parts, each worth $25
  • 3 × 25 = $75 for person B
• It is worth checking that the amount for each person sums to the correct total
  • $125 + $75 = $200

What do I do when given the difference in a ratio problem?

• Rather than being told the total amount to be shared, you could be told the difference between two shares
• Suppose that in a car park the ratio of blue cars compared to silver cars is 3:5, and we are told that there are 12 more silver cars than blue cars
• Some students find it helpful to show this in a simple diagram
  • 
• The difference in the number of parts of the ratio is 2 (5 – 3 = 2)
  • 
• The difference in the number of cars is 12
  • = 12 cars
• This means that 2 parts = 12 cars
• We can simplify this to 1 part = 6 cars (by dividing both sides by 2)
  • 
• Now that we know how much 1 part is worth, we can find how many cars of each colour there are, and the total number of cars
  • 
  • 3 parts are blue
    
    • 3×6=18 blue cars
  • 5 parts are silver
    
    • 5×6=30 silver cars
  • 8 parts in total
    
    • 8×6=48 cars in total

Given one part of a ratio, how can I find the other part?

• Rather than being told the total amount to be shared, you could be told the value of one side of the ratio
• Suppose that a fruit drink is made by mixing concentrate with water in the ratio 2:3, and we want to find how much water needs to be added to 5 litres of concentrate
• Some students find it helpful to show this in a simple diagram
• 
• We are told that there are 5 litres of concentrate, and it must be mixed in the ratio 2:3
• This means that the two parts on the left, are equivalent to 5 litres
• = 5 litres
• This means that 1 part must be equal to 2.5 litres (5 ÷ 2 = 2.5)
• = 2.5 litres
• Now that we know how much 1 part is worth, we can find how many litres of water are required, and the total amount of fruit drink produced
• 
• 3 parts are water
• 3 × 2.5 = 7.5 litres of water
• 5 parts in total
• 5 × 2.5 = 12.5 litres of fruit drink produced in total

How do I combine two ratios to make a 3-part ratio?

• Sometimes you may be given two separate ratios, that link together in some way, so that you can form a 3-part ratio
• Suppose that on a farm with 85 animals
• The ratio of cows to sheep is 2:3
• The ratio of sheep to pigs is 6:7
• We want to find the number of each animal on the farm
• We can’t just share 85 in the ratio 2:3 or 6:7, because these ratios don’t account for all the animals on the farm on their own
• We need to find a combined, 3-part ratio that shows the relative portions of all the animals together
• Notice that sheep appear in both ratios, so we can use sheep as the link between the two
• C:S = 2:3 and S:P = 6:7
• We can multiply both sides of the C:S ratio by 2, so that both ratios are comparing relative to 6 sheep
• C:S = 4:6 and S:P = 6:7
• These can now be joined together
• C:S:P = 4:6:7
• We can now use this to share the 85 animals in the ratio 4:6:7
• There are 17 parts in total (4 + 6 + 7 = 17)
• Each part is worth 5 animals (85 ÷ 17 = 5)
• There are 20 cows (4 × 5), 30 sheep (6 × 5), and 35 pigs (7 × 5)