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.
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Answer key: Ratios, fractions and linear functions worksheet — GCSE Foundation
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- (c) 2:3 — Divide both parts by the common factor 3a: 6a ÷ 3a = 2 and 9a ÷ 3a = 3, giving the ratio 2 : 3.
- (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.
- (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) 3 : 4 — Method: a fraction compares a part with the whole, while this ratio compares one part with the other part, so find the fraction that is black before writing the ratio. Working: if 3/7 are blue then the black pens make up 7/7 − 3/7 = 4/7 of the box, so out of every 7 pens 3 are blue and 4 are black, and blue : black = 3 : 4. Answer: 3 : 4. The distractors: 3 : 7 comes from reading the numerator and the denominator of 3/7 straight off as the two parts, which compares the blue pens with the whole box rather than with the black pens; 4 : 3 comes from writing the black pens before the blue pens, reversing the order asked for; 4 : 7 is the same numerator-and-denominator reading applied to the black fraction 4/7, again comparing a part with the whole box.
- (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.
- (c) 2.5 — x : y = 2 : 5 means that for every matching pair of values, y ÷ x = 5 ÷ 2 = 2.5. So y = 2.5x, and comparing with y = kx gives k = 2.5. Dividing the other way round, 2 ÷ 5 = 0.4, gives x in terms of y — that is the constant for x = 0.4y, not for y = kx. Taking the y-part of the ratio on its own, 5, reads one number off the ratio instead of dividing the y-part by the x-part; 5 would only be right if the x-part were 1. Subtracting the two parts, 5 − 2 = 3, treats the ratio as a difference, but a ratio compares two quantities by multiplication, not by subtraction. The constant is k = 2.5.
- (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.
- (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) 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.
- (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) 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) 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) Yes, correct, because 24 ÷ 8 × 3 = 9. — Method: divide the count you know by its ratio part to find the value of one part, then multiply by the part you want. Working: one part = 24 ÷ 8 = 3 counters, so green = 3 × 3 = 9 counters, and the student is correct. Dividing 24 by 3 instead of 8 gives 24 ÷ 3 × 8 = 64, the ratio parts the wrong way round. Stopping after 24 ÷ 8 = 3 gives 3, forgetting to multiply by the green ratio part. Adding 24 + 3 = 27 confuses adding a ratio part with scaling by it.
- (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.
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