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.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.
- 2.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.
- 3.A straight-line graph passes through the origin and the point (4, 10). The line represents y = kx. Write down the ratio x : y in its simplest form.
- 4.Two quantities are connected by the formula y = 0.8x. Write down the ratio y : x in its simplest form.y = 0.8x
- 5.A prize fund of £240 is shared between two winners in the ratio 3 : 5. Write the smaller winner's share as a fraction of the whole prize fund.
- 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.Write the ratio 4 : 9 in the form 1 : n.
- 8.The ratio of y to x is always 3 : 4. Write down which equation could represent this relationship.
- 9.A recipe uses flour and sugar in the ratio 4 : 1 by mass. The mass of sugar is s grams and the mass of flour is f grams. Write down a formula for f in terms of s.
- 10.A table shows three pairs of values: when x = 2, y = 6; when x = 5, y = 15; when x = 8, y = 24. In every pair the ratio y : x is the same. Write down a formula for y in terms of x.
- 11.Two straight-line graphs represent y = 2x and y = 5x. For the same value of x, where x ≠ 0, write down the ratio of the y-value on the first graph to the y-value on the second graph, in its simplest form.y = 2xy = 5x
- 12.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.
- 13.Write the ratio 6a : 9a in its simplest form, where a is a positive number.
- 14.The ratio of two lengths is 0.6 : 1.5. Write this ratio in its simplest whole-number form.
- 15.A concrete mix is made from gravel and cement in the ratio 5 : 1 by mass. The mass of cement is c kg and the total mass of the mix is m kg. Write down a formula for m in terms of c.
Answer key
- (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) 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%.
- (c) 2 : 5 — The point (4, 10) gives x = 4, y = 10, so x : y = 4 : 10. Dividing both parts by their highest common factor, 2, gives 2 : 5 in simplest form. Inverting the whole ratio gives 5 : 2, which is y : x instead of x : y. Dividing only the x-part by 2 and leaving the y-part as 10 gives 2 : 10, but scaling one part on its own changes the ratio: 2 : 10 is the same as 1 : 5, not 4 : 10. Dividing only the y-part by 2 and leaving the x-part as 4 gives 4 : 5, the same one-sided mistake made on the other part of the ratio.
- (c) 4 : 5 — Method: a formula of the form y = kx says that for every 1 of x there are k of y, so the ratio y : x is k : 1, which is then written with whole numbers and cancelled. Working: here k = 0.8, so y : x = 0.8 : 1; multiplying both parts by 10 gives 8 : 10, and dividing both parts by their highest common factor 2 gives 4 : 5. Answer: 4 : 5. The distractors: 5 : 4 comes from writing x : y, reversing the order asked for; 8 : 10 has the decimal cleared but the ratio left unsimplified, and the question asks for the simplest form; 8 : 1 comes from multiplying only the left-hand part by 10 and leaving the 1 unchanged, which changes the relationship instead of keeping it.
- (a) 3/8 — The whole prize fund is split into 3 + 5 = 8 equal parts. The smaller winner receives 3 of these parts, so their share is 3/8 of the whole fund. Taking the larger number of parts, 5, as the numerator instead gives 5/8, the larger winner's share. Writing the ratio numbers directly as a fraction without adding them, 3/5, treats the ratio as a fraction of the OTHER share rather than of the whole. Inverting the ratio gives 5/3, which is not even a valid fraction of a whole, since it is greater than 1.
- (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) 1 : 2.25 — Method: to write a ratio in the form 1 : n, divide both parts by the first part. Working: 4 ÷ 4 = 1 and 9 ÷ 4 = 2.25, so 4 : 9 = 1 : 2.25. Working out 9 ÷ 4 = 2.25 correctly but then writing it as the first part gives 2.25 : 1, the two parts the wrong way round. Subtracting 9 − 4 = 5 gives 1 : 5, confusing the difference between the parts with the ratio. Multiplying 4 × 9 = 36 gives 1 : 36, confusing the product of the parts with the ratio.
- (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).
- (a) f = 4s — Method: in the ratio 4 : 1 the sugar is 1 part, so one part weighs s grams, and the flour is 4 of those same parts. Working: one part is s, so four parts are 4 × s, giving f = 4s; as a check, if s = 3 then the flour is 4 × 3 = 12 g, and 12 : 3 does simplify to 4 : 1. Answer: f = 4s. The distractors: f = s/4 uses the ratio the wrong way round, as though the flour were 1 part and the sugar 4; f = s + 3 comes from reading the ratio as a difference, 4 − 1 = 3, and adding that difference instead of multiplying; f = 5s uses 4 + 1 = 5, the total number of parts, as the multiplier, but 5 parts is the whole mixture and not the flour on its own.
- (c) y = 3x — Method: if the ratio y : x is the same in every pair then y is always the same multiple of x, and that multiple is found by dividing a y value by its own x value. Working: 6 ÷ 2 = 3, 15 ÷ 5 = 3 and 24 ÷ 8 = 3, so every y is 3 lots of its x, which gives y = 3x. Answer: y = 3x. The distractors: y = x + 4 comes from subtracting instead of dividing on the first pair, 6 − 2 = 4, and testing it no further; y = x + 16 comes from the same subtraction on the last pair, 24 − 8 = 16; y = x/3 comes from dividing the x value by the y value, 2 ÷ 6, which gives the ratio x : y and not y : x.
- (d) 2:5 — The ratio of the y-values equals the ratio of the coefficients of x, since x cancels: 2x : 5x = 2 : 5.
- (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.
- (c) 2:3 — Divide both parts by the common factor 3a: 6a ÷ 3a = 2 and 9a ÷ 3a = 3, giving the ratio 2 : 3.
- (d) 2:5 — Multiply both parts by 10 to clear the decimals: 6 : 15. Divide both parts by their common factor 3: 2 : 5.
- (c) m = 6c — Method: the whole is the sum of the parts in the ratio, and the cement is 1 part, so one part weighs c kg. Working: the mix has 5 + 1 = 6 parts, each of mass c kg, so the total mass is 6 × c, giving m = 6c. Answer: m = 6c. The distractors: m = 5c uses the 5 gravel parts as the multiplier and forgets that the cement is in the mix too, so it gives the mass of the gravel and not the total; m = c + 5 comes from reading the ratio as '5 more than' and adding, which treats a number of parts as a mass in kilograms; m = c/6 turns the relationship upside down, as though the total were shared into the cement rather than the cement multiplied up to the total.
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