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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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.In a fruit drink, 3/8 of the total volume is orange juice and the rest is apple juice. Write down the ratio of orange juice to apple juice, in its simplest form.
- 3.In a bag, the ratio of green counters to yellow counters is 3 : 8. There are 24 yellow counters. A student says, 'There must be 9 green counters.' Is the student correct? Choose the correct verdict and reason.
- 4.Write the ratio 2/3 : 1/2 as a ratio of whole numbers, in its simplest form.
- 5.Given that a is 40% of b, write the ratio a : b in its simplest form.
- 6.A vegetable patch is divided between carrots, potatoes and onions in the ratio 2 : 3 : 5. Write down the fraction of the patch used for potatoes.
- 7.Two quantities x and y are in the ratio x : y = 2 : 5, and y = kx for a constant k. Work out the value of k.
- 8.Two quantities are connected by the formula y = 0.8x. Write down the ratio y : x in its simplest form.y = 0.8x
- 9.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.
- 10.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
- 11.A recipe requires flour and butter in the ratio 3 : 2. Write down which of these amounts could be used in the recipe.
- 12.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.
- 13.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.
- 14.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.
- 15.In a box of pens, 3/7 of the pens are blue and the rest are black. Write down the ratio of the number of blue pens to the number of black pens, in its simplest form.
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.
- (c) 3 : 5 — If orange juice is 3/8 of the total, apple juice is the remaining 1 − 3/8 = 5/8. The ratio of orange to apple is therefore 3 : 5. Inverting gives 5 : 3, apple to orange instead of orange to apple. Using the denominator 8 as the second part of the ratio, 3 : 8, compares orange juice to the whole drink rather than to the apple juice alone. Pairing the total 8 with the apple fraction's numerator 5 gives 8 : 5, which mixes a whole-total figure with a part figure.
- (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.
- (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.
- (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.
- (b) 3/10 — Total parts = 2 + 3 + 5 = 10. Potatoes make up 3 parts, so the fraction is 3/10.
- (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) 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.
- (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.
- (d) 2:5 — The ratio of the y-values equals the ratio of the coefficients of x, since x cancels: 2x : 5x = 2 : 5.
- (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 — 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.
- (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.
- (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.
- (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.
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