Multi-Step Reasoning Challenge Set 3
Multi-Step Reasoning Challenge Set 3 tackles the hardest multi-step problems in this series, including questions that must be worked backwards from a final amount to an unknown starting value, and problems that combine ratio, fractions and percentages within the same scenario. It follows on from Multi-Step Reasoning Challenge Set 2.
Before you start
Make sure you're comfortable with these topics first:
Method
- Decide first whether the problem runs forwards, with a starting amount known and a result to find, or backwards, with a final amount known and the start to find, since a backwards problem must be undone one step at a time in reverse order.
- For a backwards problem, reverse each operation described: undo a percentage decrease by dividing by the multiplier rather than multiplying, and undo a fraction removed by working out what fraction remains first.
- Where a problem combines ratio with fractions or percentages, convert the ratio into fractions of the whole amount before combining it with the other information.
- Write down what each unknown quantity in the problem represents in words, not just as a number, so you do not lose track of which value you are working with.
- Work through the reversed or combined steps one at a time, checking that each intermediate value could plausibly lead to the final amount given in the question.
- Once you have a starting value, test it by working the problem forwards from your answer to confirm it produces the final amount stated in the question.
Worked example
After a 20% discount, a jacket costs 68 pounds. What was the original price of the jacket?
- Recognise that 68 pounds represents 100% - 20% = 80% of the original price.
- Find 1% of the original price by dividing: 68 / 80 = 0.85 pounds.
- Find 100% by multiplying: 0.85 x 100 = 85 pounds.
- Check: 85 pounds reduced by 20% is 85 x 0.8 = 68 pounds, which matches.
- The original price of the jacket was 85 pounds.
Practice questions
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Q1After spending 3/5 of his money, Ben has 28 pounds left. How much money did he start with?Show answer
Answer: 70 pounds (28 pounds is 2/5 of the original amount, so 1/5 = 14 and the whole amount = 70)
Q2A price is increased by 15% to give a new price of 46 pounds. Find the original price.Show answer
Answer: 40 pounds (46 / 115 x 100)
Q3After giving away 2/7 of her sweets, Mia has 45 sweets left. How many sweets did she start with?Show answer
Answer: 63 sweets (45 is 5/7 of the original amount, so 1/7 = 9 and the whole amount = 63)
Q4A number is increased by 40%, then the result is decreased by 25%, giving a final value of 63. Find the original number.Show answer
Answer: 60 (the combined multiplier is 1.40 x 0.75 = 1.05, so the original number = 63 / 1.05)
Q5A photo is enlarged so its width increases by 20%, then reduced by 10%, giving a final width of 21.6 cm. Find the original width.Show answer
Answer: 20 cm (the combined multiplier is 1.20 x 0.90 = 1.08, so the original width = 21.6 / 1.08)
Q6In a school, the ratio of boys to girls is 4:5. There are 27 more girls than boys. How many pupils are there in total?Show answer
Answer: 243 pupils (1 ratio part = 27, since 5 - 4 = 1 part represents the difference; total = 9 parts x 27)
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
After a 15% pay rise, Priya's weekly wage is 253 pounds. Calculate her weekly wage before the rise.
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A number is decreased by 30%, then the result is increased by 50%, giving a final value of 84. (a) Find the single multiplier that is equivalent to these two combined changes. (b) Calculate the original number. (c) State whether the two changes together increase or decrease the original number, and explain how you know.
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In a fruit basket, the original number of apples is 3x and the original number of oranges is 4x. After 6 more apples are added, the ratio of apples to oranges becomes 1:1. (a) Write an equation using this information. (b) Solve your equation to find x. (c) State the original number of apples and oranges.
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Free printable worksheet
Want more practice on paper? Download the multi-step reasoning challenge set 3 worksheet pack - 11 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 20 of 11+ Maths Workbook 4, the whole course as one free printable PDF.
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