Ratio Problems with Totals and Differences
Ratio problems become harder once a question no longer gives the total amount directly, since sometimes only one share is known and sometimes only the difference between two shares is known. The key idea stays the same: work out what a single ratio part is worth from whatever information is given, a share or a difference, then use that value to build every other amount the question asks for, including the total.
Method
- Read the problem carefully to identify what is directly known: is it the total amount, a single share, or the difference between two shares?
- If a single share and its number of parts are known, divide that share by its number of parts to find the value of one part.
- If the difference between two shares is known, subtract the smaller ratio number from the larger ratio number to find how many parts that difference represents, then divide the difference by this to find the value of one part.
- Once the value of one part is known, multiply it by any number in the ratio to find that share, or add all the parts and multiply to find the total.
- Check the answer by reconstructing the whole problem: do the shares add up to the total, and does the difference between them match what was given?
- Watch for problems that mix ratio with an extra step, such as one group then spending or receiving more - do the ratio part first, then apply the extra step.
- Label which number in the ratio belongs to which person or group before doing any arithmetic - it is the most common place marks are lost.
Worked example
Two allotment holders, Priya and Callum, share a delivery of compost in the ratio 4 : 9. Callum receives 105 kg more compost than Priya. Work out the total mass of compost delivered.
- Find the difference in ratio parts: 9 - 4 = 5 parts.
- This difference of 5 parts is equal to 105 kg, so divide to find the value of one part: 105 / 5 = 21 kg.
- Priya's share = 4 x 21 = 84 kg.
- Callum's share = 9 x 21 = 189 kg.
- Total mass = 84 + 189 = 273 kg (or 13 parts x 21 kg = 273 kg).
Practice questions
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Q1A grandmother leaves money to her two grandchildren, Femi and Isla, in the ratio 3 : 5. Femi receives 96 pounds. Work out the total amount left to both grandchildren.Show answer
Answer: 256 pounds (one part = 96 / 3 = 32 pounds, total = 8 x 32)
Q2A vineyard's grape harvest is split between red and white grapes in the ratio 5 : 2. There are 420 kg more red grapes than white grapes. Work out the mass of white grapes harvested.Show answer
Answer: 280 kg (difference = 3 parts = 420 kg, so one part = 140 kg, white = 2 x 140)
Q3A quiz prize fund of 168 pounds is shared between the winning team and the runner-up team in the ratio 5 : 3. How much more does the winning team receive than the runner-up team?Show answer
Answer: 42 pounds (one part = 168 / 8 = 21 pounds, winners = 105, runners-up = 63, difference = 42)
Q4A relay baton company divides an order of 250 batons between three schools in the ratio 2 : 3 : 5. Work out how many batons the second school receives.Show answer
Answer: 75 batons (one part = 250 / 10 = 25 batons, second school = 3 x 25)
Q5Two hikers, Tomasz and Bethan, split the mass of their shared tent and cooking gear in the ratio 7 : 5. Bethan carries 6.5 kg. Work out the total mass of the tent and cooking gear.Show answer
Answer: 15.6 kg (one part = 6.5 / 5 = 1.3 kg, total = 12 x 1.3)
Q6A market gardener plants carrots and parsnips in a field in the ratio 11 : 4. There are 245 more carrot plants than parsnip plants. Work out the total number of plants in the field.Show answer
Answer: 525 plants (difference = 7 parts = 245, so one part = 35, carrots = 385, parsnips = 140, total = 525)
Q7A choir's sheet music budget is split between the sopranos and the altos in the ratio 9 : 7. The sopranos receive 24 pounds more than the altos. Work out how much the altos receive.Show answer
Answer: 84 pounds (difference = 2 parts = 24 pounds, so one part = 12 pounds, altos = 7 x 12)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A cafe orders coffee beans and tea leaves in the ratio 8 : 3 by mass for a new seasonal menu. The coffee beans order is 150 kg heavier than the tea leaves order. Work out the total mass of the order.
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Two market stalls, run by Grace and Oluwaseun, share the cost of a gazebo hire in the ratio 7 : 4, based on how many days each stall uses it. The total hire cost is 176 pounds. Oluwaseun then pays an extra 15 pounds for a repair to the section he used. Work out the total amount Oluwaseun pays altogether.
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Free printable worksheet
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This topic is chapter 37 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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