Printable · GCSE Foundation · ages 14-16
Equations of direct and inverse proportion worksheet — GCSE Foundation
Fifteen questions on "equations of direct and inverse proportion" — DfE statement R13. 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: Equations of direct and inverse proportion worksheet — GCSE Foundation
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- (a) 12 — Speed × time is constant: k = 20 × 15 = 300. At 25 pages per minute, the time is 300 ÷ 25 = 12 minutes. Getting 18.75 comes from treating speed and time as directly proportional and working out 15 × 25 ÷ 20 instead of dividing k by the new speed. Getting 20 comes from adding the increase in speed (25 − 20 = 5) onto the time (15 + 5 = 20). Getting 10 comes from subtracting that same increase in speed from the time (15 − 5 = 10).
- (b) They are in inverse proportion, because x × y = 60 for both pairs. — Testing inverse proportion means checking that x × y is the same for every pair: 4 × 15 = 60 and 6 × 10 = 60, so the quantities are in inverse proportion. Saying they are not in inverse proportion because x + y differs uses addition, which is not the correct test. Saying they are not in inverse proportion because y ÷ x differs uses the test for direct proportion, and finding that it differs tells us nothing about inverse proportion. Saying x × y = 40 for both pairs is an arithmetic slip: 4 × 15 = 60, not 40.
- (d) 35 — Method: in direct proportion the ratio y : x is the same for every pair, so find the constant and substitute the new value of x. Working: k = 20 ÷ 8 = 2.5, so y = 2.5x; when x = 14, y = 2.5 × 14 = 35. Answer: 35. The distractors: 26 comes from additive thinking — x rises by 6, so 6 is added to y — which would keep the difference constant rather than the ratio; 28 comes from rounding the constant 2.5 down to 2 and working out 2 × 14, which loses the half in the constant; 5.6 comes from using the constant upside down, 8 ÷ 20 = 0.4, and working out 0.4 × 14.
- (c) They are in direct proportion, because y ÷ x = 2.5 for both pairs. — Testing direct proportion means checking that y ÷ x is the same for every pair: 15 ÷ 6 = 2.5 and 25 ÷ 10 = 2.5, so the quantities are in direct proportion. Saying they are not in proportion because x + y differs uses addition, which is not the correct test for proportion. Saying they are not in proportion because y − x differs also uses the wrong test — subtraction, not division. Saying they are in proportion because x × y is 90 and 250 uses multiplication, which is the test for inverse proportion, and the two products are not even equal to each other, so this option also contradicts itself.
- (a) 3 hours — Method: for a fixed pool the rate of flow multiplied by the time taken is constant, so multiplying the rate by a factor divides the time by that same factor. Working: tap B's rate is 2 times tap A's rate, so tap B's time is 6 ÷ 2 = 3 hours. Answer: 3 hours. The distractors: 12 hours comes from multiplying the time by 2 as well, which treats the time as directly proportional to the rate and has the faster tap taking longer; 4 hours comes from reading ‘twice as fast’ additively, as two hours quicker, and working out 6 − 2 instead of scaling the time by a factor of 2; 1.5 hours comes from applying the factor of 2 twice, halving 6 to 3 and then halving again.
- (b) 16 — Method: substitute x into each equation separately, then subtract the smaller value from the larger. Working: y = 3 × 4 = 12 and z = 7 × 4 = 28, so the difference is 28 − 12 = 16. Answer: 16. The distractors: 4 comes from subtracting the constants, 7 − 3, which is the difference between the two gradients rather than the difference between the values at x = 4; 40 comes from adding the two values, 12 + 28, instead of subtracting them; 28 is the value of z on its own, given instead of being compared with the value of y.
- (a) 6 — Method: for inverse proportion the product xy is the same for every pair, so find that product and use it to work back to the missing value. Working: xy = 2 × 15 = 30, so when x = 5 the equation 5y = 30 gives y = 30 ÷ 5 = 6. Answer: 6. The distractors: 37.5 comes from treating the pair as direct proportion and scaling y up with x, 15 × 5 ÷ 2, although in inverse proportion y falls as x rises; 30 is the constant product itself, given as a value of y rather than used to find one; 12 comes from additive thinking — x rises by 3, so 3 is taken off y — which would make the two quantities differ by a constant instead of multiplying to one.
- (d) b = k ÷ g — Inverse proportion means b × g stays constant (equal to k), so b = k ÷ g. 'b = kg' is the equation for direct proportion, not inverse. 'b = k + g' treats the relationship as additive, which is not proportion at all. 'b = g ÷ k' is direct proportion in disguise: rearranging it gives g = kb, so b still grows as g grows. For inverse proportion the g must be underneath the constant, not above it.
- (c) y is trebled — In direct proportion, y = kx, so multiplying x by 3 multiplies y by 3 as well: y is trebled. Dividing y by 3 is what would happen for inverse proportion, not direct. Saying y stays the same ignores the proportional relationship entirely. Saying y increases by 3 mistakes multiplying by 3 for adding 3.
- (d) 8 — The product of price and number of tickets is constant: k = 4 × 12 = 48. At £6 per ticket, the number of tickets is 48 ÷ 6 = 8. Getting 18 comes from treating price and tickets as directly proportional and working out 12 × 6 ÷ 4 instead of dividing k by the new price. Getting 12 assumes the number of tickets does not change when the price changes. Getting 6 comes from writing down the new price instead of working out the number of tickets.
- (a) 4 — Method: find the constant of proportionality from the pair given, write the equation, then substitute the new value of y and solve. Working: k = 21 ÷ 7 = 3, so y = 3x; putting y = 12 gives 12 = 3x, and x = 12 ÷ 3 = 4. Answer: 4. The distractors: 36 comes from multiplying by the constant instead of dividing by it, 12 × 3, which is the proportion set up upside down; 84 comes from multiplying 12 by the 7 from the first pair, using a value of x as though it were the constant; 9 comes from working out 12 − 3, treating the equation as y = x + 3 rather than y = 3x.
- (c) 3 hours — Method: inverse proportion means speed × time is constant for the journey, so find that constant and divide it by the new speed. Working: 60 × 2 = 120, which is the distance in kilometres; at 40 km/h the time is 120 ÷ 40 = 3 hours. Answer: 3 hours. The distractors: 1.5 hours is the ratio of the speeds, 60 ÷ 40, given as a time instead of being used to scale the original 2 hours; 1 hour 20 minutes comes from treating time as directly proportional to speed, 2 × 40 ÷ 60, which has the slower train arriving sooner; 2 hours comes from finding the constant 120 and then dividing it by the original 60 km/h again, so the time never changes.
- (d) 5 — Method: in an equation of inverse proportion the constant is divided by the value of x. Working: y = 20 ÷ x with x = 4 gives y = 20 ÷ 4 = 5. Answer: 5. The distractors: 80 comes from multiplying 20 by 4 instead of dividing, which is direct proportion applied to an inverse relationship; 16 comes from working out 20 − 4, treating the constant as something to subtract from; 0.2 comes from dividing the wrong way round, 4 ÷ 20.
- (b) 12 — Method: in an equation of direct proportion the value of x is substituted and multiplied by the constant. Working: y = 3x with x = 4 gives y = 3 × 4 = 12. Answer: 12. The distractors: 7 comes from adding the 3 and the 4 instead of multiplying them, reading 3x as 3 + x; 34 comes from writing the 3 and the 4 side by side, treating 3x as the digits of a two-digit number rather than as a product; 1 comes from working out 4 − 3, which turns the constant into an amount to be taken away.
- (d) 5 — Method: rearrange y = kx to make the constant the subject, then substitute the pair of values given. Working: k = y ÷ x, so k = 10 ÷ 2 = 5. Answer: 5. The distractors: 20 comes from multiplying 10 by 2 instead of dividing, which is the rearrangement done the wrong way round; 12 comes from adding the pair, 10 + 2, treating the relationship as y = x + k; 8 comes from working out 10 − 2, the same additive reading with the operation reversed.
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