Problem Solving with Ratios (Cambridge International IGCSE Mathematics)
Working with Ratios
How do I share an amount in a given ratio?
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Add together all parts in the ratio to find the total number of parts in the whole
- $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
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Divide the amount being shared by the total number of parts
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$200 must be split into 8 parts, so this means that 1 part must be worth $25
- 200 ÷ 8 = 25
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Multiply the amount each part is worth by the number of parts for each quantity in the ratio
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Person A receives 5 parts, each worth $25
- 5 × 25 = $125 for person A
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Person B receives 3 parts, each worth $25
- 3 × 25 = $75 for person B
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Check the values in the new ratio add up to the total amount being shared
- $125 + $75 = $200
Exam Hint
- Adding labels to your ratios will help make your working clearer and help you remember which number represents which quantity, e.g.
Worked Example A particular shade of pink paint is made using three parts red paint, to two parts white paint. Mark needs 60 litres of pink paint in order to decorate rooms in a school.
Calculate the volume of red and white paint that Mark needs to purchase in order to have enough paint to decorate the room.
Answer:
The ratio of red to white is
3 : 2
Adding these together gives the total number of parts
3 + 2 = 5
The total amount of paint is 60 litres
∴ 5 parts = 60 litres
Divide both sides by 5 to find out the number of litres in one part
The ratio was 3:2, so multiply both number of parts by 12
Answer in context, making sure you make it clear which value is associated with which colour paint
Final answer: Mark will need to buy 36 litres of red paint and 24 litres of white paint
What type of ratio problems could I be asked to solve?
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Simple ratio problems are discussed earlier, including
- Writing ratios
- The link between ratios and fractions
- Equivalent ratios
- Simplifying ratios
- Sharing an amount in a given ratio
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Further problems involving ratio include
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Ratios where you are given the difference between the two parts
- E.g. Kerry is given $30 more than Kacey who is given $50
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Ratios where one quantity is given and you have to find the other quantity
- E.g. Kerry and Kacey are sharing money in the ratio 8 : 5, Kacey gets $50
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Situations where you are given two separate (two-part) ratios but can combine them in to one (three-part) ratio
- E.g. Kerry and Kacey are sharing money in the ratio 8 : 5 whilst Kacey is also sharing money with Kylie in the ratio 1 : 2
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How do I solve a ratio problem when given the difference between two parts?
- Find the difference in the number of parts between the two quantities in the ratio
- Compare the difference in the number of parts with the difference between the actual numbers
- Simplify to find out the value of one part
- Multiply the value of one part by the number of parts for each quantity in the ratio
- Multiply the value of one part by the total number of parts to find the total amount
Given one quantity of a ratio, how can I find the other quantity?
- Compare the given quantity with the relevant number of parts in the ratio
- Simplify to find the value of one part
- Multiply the value of one part by the number of parts in the remaining quantity in the ratio
- Multiply the value of one part by the total number of parts to find the total amount
How do I combine two ratios to make a three-part ratio?
- Identify the link between the two different ratios
- **Find equivalent ratios **for both original ratios, where the value of the link is the same
- Join the two, two-part ratios into a** three-part ratio**
Worked Example (a) The ratio of cabbage leaves eaten by two rabbits, Alfred and Bob, is 7 : 3 respectively. It is known that Alfred eats 12 more cabbage leaves than Bob for a particular period of time.
Find the total number of cabbage leaves eaten by the rabbits and the number that each rabbit eats individually.
Answer:
The difference in the number of parts is
7 - 3 = 4 parts
This means that
4 parts = 12 cabbage leaves
Dividing both by 4
1 part = 3 cabbage leaves
Find the total number of parts
7 + 3 = 10 parts
Find the total number of cabbage leaves
10 × 3 = 30
Final answer: 30 cabbage leaves in total
Find the number eaten by Alfred
7 × 3 = 21
Final answer: 21 cabbage leaves
Find the number eaten by Bob
3 × 3 = 9
Final answer: 9 cabbage leaves
(b) A particular shade of pink paint is made using 3 parts red paint, to two parts white paint.
Mark already has 36 litres of red paint, but no white paint.
Calculate the volume of white paint that Mark needs to purchase in order to use all of his red paint, and calculate the total amount of pink paint this will produce.
Answer:
The ratio of red to white is
3 : 2
Mark already has 36 litres of red, so
36 litres = 3 parts
Divide both sides by 3.
12 litres = 1 part
The ratio was 3 : 2 Find the volume of white paint, 2 parts
2 × 12 = 24
Final answer: 24 litres of white paint
In total there are 5 parts, so the total volume of paint will be
5 × 12 = 60
Final answer: 60 litres in total
(c) In Jamie’s sock drawer the ratio of black socks to striped socks is 5 : 2 respectively. The ratio of striped socks to white socks in the drawer is 6 : 7 respectively.
Calculate the percentage of socks in the drawer that are black.
Answer:
Write down the ratios
B : S = 5 : 2 S : W = 6 : 7
S features in both ratios, so we can use it as a link Multiply the B : S ratio by 3 to find an equivalent ratio Both ratios are now comparing to 6 striped socks
B : S = 15 : 6 S : W = 6 : 7
Link them together
B : S : W = 15 : 6 : 7
Find the total number of parts
15 + 6 + 7 = 28
This means 15 out of 28 socks are black Find 15 out of 28 as a decimal by completing the division
Convert to a percentage Multiply by 100 and round to 3 significant figures
Final answer: 53.6 % of the socks are black
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