Number, Ratio and Algebra Depth
Number, Ratio and Algebra Depth is a Common Entrance 13+ revision set that draws the harder end of Number and Calculation, Fractions, Ratio and Proportion, and Algebra together into the kind of multi-step question found in the later parts of the Core and Additional Mathematics papers.
Before you start
Make sure you're comfortable with these topics first:
Method
- Read the whole question before starting and identify every skill it draws on, for example underlining where a ratio is used and separately where a percentage is applied.
- Break a combined question into a clear sequence of smaller steps, writing down what each step finds before moving on to the next, rather than trying to do it in one jump.
- When a question mixes numbers and letters, decide early whether to find a numerical value first and then substitute it into an expression, or to work with the algebra first.
- Keep exact fractions rather than converting to decimals too early, since rounding partway through a multi-step calculation is a common source of a lost accuracy mark.
- Label each intermediate answer clearly, such as writing 'one share =' before a value, so that even if the final answer is wrong, the correct steps can still earn method marks.
- After finishing, reread the original question and check the final answer actually answers what was asked, since a common depth-question trap is finding a correct intermediate value and stopping there.
Worked example
A school raises money for two charities, A and B, in the ratio 5 : 3. In total the school raises 640 pounds. Charity A then donates 20% of what it receives to a third charity, C. Work out how much charity C receives.
- Find the total number of parts in the ratio: 5 + 3 = 8.
- Find the value of one part: 640 / 8 = 80.
- Find charity A's share: 5 x 80 = 400 pounds.
- Find 20% of charity A's share: 400 x 0.2 = 80.
- So charity C receives 80 pounds.
Practice questions
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Q1Two numbers are in the ratio 4 : 7. Their sum is 66. Work out the difference between the two numbers.Show answer
Answer: 18 (one part = 66 / 11 = 6, the numbers are 24 and 42, difference = 18)
Q2After a 12% pay rise, Priya's monthly salary is 2,240 pounds. Work out her monthly salary before the pay rise.Show answer
Answer: 2,000 pounds (2240 / 1.12 = 2000)
Q33/4 of a number is 54. Work out 5/6 of the same number.Show answer
Answer: 60 (the number is 54 divided by 3/4 = 72, then 5/6 of 72 = 60)
Q4Solve 3(2x - 1) = x + 17.Show answer
Answer: x = 4 (expand to 6x - 3 = x + 17, so 5x = 20, x = 4)
Q5A recipe for 8 people uses flour, sugar and butter in the ratio 4 : 3 : 1 by weight, totalling 800 g. Work out how many grams of sugar are needed to make the recipe for 12 people.Show answer
Answer: 450 g (one part = 800 / 8 = 100 g, sugar for 8 people = 300 g, scaled to 12 people = 300 x 12/8 = 450 g)
Q6Two joggers start together at a park gate. One returns to the gate every 18 minutes, the other every 24 minutes. After how many minutes will they next be at the gate together?Show answer
Answer: 72 minutes (the lowest common multiple of 18 and 24 is 72)
Q7A shop increases a price by 25%, then later reduces the new price by 20%. Work out the overall percentage change from the original price.Show answer
Answer: 0% overall change (the combined multiplier is 1.25 x 0.8 = 1.0, so the price returns exactly to its original value)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A youth theatre group has 260 members, split between Junior and Senior sections in the ratio 7 : 6. (a) Work out the number of members in the Senior section. (b) Membership of the Senior section then grows by 15%. Work out the new number of Senior members. (c) The Junior section stays the same size. Work out the new ratio of Junior to Senior members, giving your answer in its simplest form.
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A gardener has a rectangular plot of length (2x + 3) m and width x m. The perimeter of the plot is 36 m. (a) Form an equation in x and solve it to find the value of x. (b) Hence find the area of the plot in square metres.
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Free printable worksheet
Want more practice on paper? Download the number, ratio and algebra depth worksheet pack - 15 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 13 of 13+ Maths Workbook, the whole course as one free printable PDF.
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