Printable · GCSE Foundation · ages 14-16
Ratios, fractions and linear functions worksheet — GCSE Foundation
Fifteen questions on "ratios, fractions and linear functions" — DfE statement R8. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Ratios, fractions and linear functions worksheet — GCSE Foundation
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- 1.The ratio of two lengths is 0.6 : 1.5. Write this ratio in its simplest whole-number form.
- 2.A shop sells identical pens. The total cost, C pence, of n pens is given by C = 12n. Write the ratio n : C in its simplest form.
- 3.A taxi journey of m miles costs C pounds, where C = 2.5m + 3. A driver says the ratio C : m is the same for every journey. Is the driver correct? Give a reason for your answer.
- 4.A scale drawing of a garden uses a scale of 1 : 20. Write down a formula for the real length, L metres, in terms of the length on the drawing, d centimetres, given that 1 metre = 100 centimetres.
- 5.In a fruit squash mix, the ratio of squash to water is 2 : 9. A shop wants to make up the mix using 3.4 litres of squash. Work out how many litres of water are needed.
- 6.In a bag the ratio of red counters to blue counters is 7 : 4. Write the number of red counters as a fraction of the number of blue counters.
- 7.Given that a is 40% of b, write the ratio a : b in its simplest form.
- 8.A recipe requires flour and butter in the ratio 3 : 2. Write down which of these amounts could be used in the recipe.
- 9.A ratio is 3 : 5. Write the first part of the ratio as a fraction of the whole.
- 10.Write the ratio 6a : 9a in its simplest form, where a is a positive number.
- 11.In a class, 50% of the students study French, 30% study Spanish and the rest study German. Write down the ratio of French : Spanish : German students, in its simplest form.
- 12.A stall sells bottles of juice. Two bottles cost £3.00, five bottles cost £7.50 and eight bottles cost £12.00. The cost, C pounds, is in a fixed ratio to the number of bottles, n. Write down a formula for C in terms of n.
- 13.Write the ratio 2/3 : 1/2 as a ratio of whole numbers, in its simplest form.
- 14.The ratio of y to x is always 3 : 4. Write down which equation could represent this relationship.
- 15.In a fruit bowl, the ratio of apples to oranges is 5 : 3. Work out what percentage of the fruit in the bowl is apples.
Answer key
- (d) 2:5 — Multiply both parts by 10 to clear the decimals: 6 : 15. Divide both parts by their common factor 3: 2 : 5.
- (b) 1:12 — C = 12n means that for every 1 pen there are 12 pence of cost, so n : C = 1 : 12, and the highest common factor of 1 and 12 is 1, so this is already in its simplest form. Writing C : n instead of n : C gives 12 : 1, the ratio the wrong way round. Reading C = 12n as '12 more than n' instead of '12 times n', so n = 1 gives C = 13, gives 1 : 13, from adding instead of multiplying. Choosing n = 12, so C = 12 × 12 = 144, gives the correct pair of values n : C = 12 : 144, but the ratio the right way round left unsimplified — 12 and 144 share a common factor of 12, which has not been cancelled.
- (b) The ratio C : m is not constant because the formula includes a fixed charge of £3 as well as the charge per mile. — For the ratio C : m to stay constant, C must be directly proportional to m, i.e. C = km with no constant term. Because of the +3 fixed charge, C is not directly proportional to m: for example m = 1 gives C = 5.5 (ratio 5.5 : 1), while m = 10 gives C = 28 (ratio 2.8 : 1) — the ratio has changed.
- (c) L = d/5 — The scale 1 : 20 means each cm on the drawing represents 20 cm in real life, so the real length in cm is 20d. Converting to metres by dividing by 100: L = 20d/100 = d/5.
- (a) 15.3 litres — Squash : water = 2 : 9, so water is 9 ÷ 2 = 4.5 times the amount of squash. Multiply: 3.4 × 4.5 = 15.3 litres. Using the multiplier upside down — treating squash as 9 ÷ 2 times water, when it is water that is 9 ÷ 2 times squash — and calculating 3.4 × (2 ÷ 9) gives about 0.8 litres (to 1 d.p.); that would be the squash needed for 3.4 litres of water, not the water needed for 3.4 litres of squash. Adding the difference between the ratio parts, 9 − 2 = 7, to the squash amount, 3.4 + 7 = 10.4, mistakes a ratio for a fixed extra amount. Using the total number of parts, 2 + 9 = 11, so the multiplier 11 ÷ 2 = 5.5, gives 3.4 × 5.5 = 18.7 litres — that finds the total mix from the squash amount, not the water alone.
- (d) 7/4 — A part-to-part ratio a : b gives the fraction a/b when the first quantity is written as a fraction of the second, so 7 : 4 gives 7/4. Writing 4/7 puts the parts the wrong way round — blue as a fraction of red, not red as a fraction of blue. Writing 7/11 uses the total number of counters, 7 + 4 = 11, as the denominator instead of the number of blue counters — that is red as a fraction of the whole bag, not red as a fraction of blue. Writing 11/7 has both the wrong denominator and the parts inverted.
- (b) 2 : 5 — 40% as a fraction is 40/100 = 2/5, so a = (2/5)b, giving a : b = 2 : 5. Swapping the two numbers gives 5 : 2, which is the ratio b : a instead. Treating 40% as the fraction of the total (a + b) rather than of b alone gives a : (a + b) = 2 : 5, which rearranges to a : b = 2 : 3 — a different statement from the one in the question. Inverting that same mistaken ratio gives 3 : 2.
- (a) 180 g flour and 120 g butter — The ratio 3 : 2 means flour and butter must scale by the same factor. Scaling both parts by 60 gives 180 g flour and 120 g butter, since 3 × 60 = 180 and 2 × 60 = 120, which is the correct pair. Swapping the amounts gives 120 g flour and 180 g butter, which is in the ratio 2 : 3 — butter to flour, not flour to butter. Scaling the flour by 60 but the butter by only 45 gives 180 g flour and 90 g butter, using two different scale factors on the two parts of the ratio. Scaling the flour by 60 but the butter by 70 gives 180 g flour and 140 g butter, the opposite inconsistency.
- (d) 3/8 — The ratio 3 : 5 has 3 + 5 = 8 parts in total. The first part as a fraction of the whole is 3 out of 8, or 3/8. Writing 3/5 gives the first part compared to the second part, not to the whole. Writing 5/8 gives the second part as a fraction of the whole, not the first. Writing 8/3 has the fraction upside down — the whole must be on the bottom.
- (c) 2:3 — Divide both parts by the common factor 3a: 6a ÷ 3a = 2 and 9a ÷ 3a = 3, giving the ratio 2 : 3.
- (d) 5:3:2 — German = 100% − 50% − 30% = 20%. The ratio 50 : 30 : 20 simplifies by dividing every part by 10 to give 5 : 3 : 2.
- (c) C = 1.5n — Method: a fixed ratio between C and n means C is always the same multiple of n, and that multiple is the cost of one bottle. Working: 3.00 ÷ 2 = 1.5, 7.50 ÷ 5 = 1.5 and 12.00 ÷ 8 = 1.5, so every bottle costs £1.50 and C = 1.5n. Answer: C = 1.5n. The distractors: C = n + 1 comes from subtracting on the first row, 3 − 2 = 1, and adding that difference instead of multiplying; it fits the first row and fails the other two, which is why three rows are given; C = 3n reads the £3.00 as the price of one bottle when it is the price of two; C = n/1.5 divides the number of bottles by the price of one bottle, which works out how many bottles a pound buys instead of what n bottles cost.
- (a) 4:3 — Multiply both parts by the lowest common denominator, 6: (2/3) × 6 = 4 and (1/2) × 6 = 3, giving the ratio 4 : 3, which is already in simplest form.
- (d) y = 3x/4 — y : x = 3 : 4 means y/x = 3/4. Rearranging to make y the subject gives y = (3/4)x = 3x/4. A student who mixes up which quantity goes on top gets y = 4x/3. A student who treats the ratio numbers as the coefficient and constant of a linear equation instead of a proportional relationship gets y = 3x + 4. A student who mistakes the relationship for inverse proportion gets y = 3/(4x).
- (b) 62.5% — Total parts = 5 + 3 = 8. Apples make up 5 parts, so the percentage is 5/8 × 100 = 62.5%. A student who finds the oranges' share instead gets 3/8 × 100 = 37.5%. A student who assumes an even split gets 50%. A student who inverts the fraction gets 8/5 × 100 = 160%.
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