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GCSE · Ratio, Proportion and Rates of Change

7.3D Direct and Inverse Proportion: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 85 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

y is directly proportional to x.

(Total for Question 1 is 1 mark)

y is inversely proportional to x.

(Total for Question 2 is 1 mark)

y is directly proportional to x. When x = 6, y = 18.

(a)Find the value of k in the formula y = kx.(2)

(b)Work out the value of y when x = 10.(2)

(Total for Question 3 is 4 marks)

y is directly proportional to x^2. When x = 3, y = 45.

(a)Find a formula for y in terms of x.(2)

(b)Work out the value of y when x = 6.(2)

(Total for Question 4 is 4 marks)

y is directly proportional to x^3. When x = 2, y = 40.

(a)Find a formula for y in terms of x.(2)

(b)Work out the value of y when x = 4.(2)

(Total for Question 5 is 4 marks)

y is directly proportional to √x. When x = 16, y = 12.

(a)Find a formula for y in terms of x.(2)

(b)Work out the value of y when x = 25.(2)

(Total for Question 6 is 4 marks)

y is inversely proportional to x. When x = 4, y = 15.

(a)Find a formula for y in terms of x.(1)

(b)Work out the value of y when x = 12.(1)

(Total for Question 7 is 2 marks)

y is inversely proportional to x^2. When x = 2, y = 18.

(a)Find a formula for y in terms of x.(2)

(b)Work out the positive value of x when y = 8.(3)

(Total for Question 8 is 5 marks)

p is inversely proportional to q^3. When q = 2, p = 1.

(a)Find a formula for p in terms of q.(2)

(b)Work out the value of p when q = 4.(2)

(Total for Question 9 is 4 marks)

The table shows some values of x and y, where y is directly proportional to x^2.
x: 2, 4, 6
y: 12, 48, ?

(a)Find the value of k in y = kx^2.(2)

(b)Complete the table by finding the missing value of y when x = 6.(2)

(Total for Question 10 is 4 marks)

The cost, C pounds, of a length of ribbon is directly proportional to its length, L metres. 5 metres of ribbon costs 6.50 pounds.

(a)Find the value of k in the formula C = kL.(1)

(b)Work out the cost of 12 metres of the same ribbon.(2)

(Total for Question 11 is 3 marks)

It takes 6 builders 15 days to build a wall, all working at the same constant rate. The number of days, D, needed is inversely proportional to the number of builders, n.

(a)Find the value of k in the formula D = k/n.(1)

(b)Work out how many days it would take 9 builders, working at the same rate, to build the same wall.(2)

(c)Give one reason why this inverse proportion model might not give a realistic answer if a very large number of builders were used.(1)

(Total for Question 12 is 4 marks)

Three graphs, A, B and C, each show y plotted against x for x > 0. Graph A is a straight line through the origin. Graph B is a curve that starts high up on the y-axis, decreases steeply, and gets closer and closer to the x-axis without ever touching it, and never touches the y-axis either. Graph C is a straight line that crosses the y-axis above the origin (it does not pass through the origin).

(a)State which graph, A, B or C, could show y directly proportional to x.(1)

(b)State which graph, A, B or C, could show y inversely proportional to x.(1)

(c)Explain why graph C cannot show y directly proportional to x, or y inversely proportional to x.(1)

(Total for Question 13 is 3 marks)

y is directly proportional to x^2. Find the effect on the value of y when the value of x is doubled.

(Total for Question 14 is 2 marks)

y is directly proportional to x^2. y = 32 when x = 4.

(a)Find a formula for y in terms of x.(2)

(b)Work out the value of y when x = 10.(1)

(c)Work out the positive value of x when y = 50.(2)

(Total for Question 15 is 5 marks)

The time, T hours, taken for a car journey is inversely proportional to its average speed, S mph. When S = 40, T = 3.

(a)Find a formula for T in terms of S.(1)

(b)Work out the time taken for the same journey at an average speed of 60 mph.(2)

(c)The journey is instead driven so that it takes 2.4 hours. Work out the average speed required.(2)

(Total for Question 16 is 5 marks)

y is directly proportional to x^3. The value of x is increased by 50%. Work out the percentage increase in the value of y.

(Total for Question 17 is 3 marks)

p is inversely proportional to q^2. The value of q is decreased by 50%. Work out the percentage change in the value of p, stating whether it is an increase or a decrease.

(Total for Question 18 is 3 marks)

A company claims that the cooking time, T minutes, needed for a joint of meat of mass m kg is directly proportional to m. The company's own cooking instructions state that a 2 kg joint needs 50 minutes, and that a 5 kg joint needs 110 minutes.

(a)Using the data for the 2 kg joint, and assuming T is directly proportional to m, find the value of k in T = km.(1)

(b)Use your value of k to predict the cooking time needed for a 5 kg joint.(1)

(c)Hence, giving a reason, state whether the company's claim that T is directly proportional to m is correct.(2)

(Total for Question 19 is 4 marks)

The braking distance, d metres, of a car is directly proportional to the square of its speed, s mph. A car travelling at 30 mph has a braking distance of 18 m.

(a)Find a formula for d in terms of s.(2)

(b)Work out the braking distance of a car travelling at 50 mph.(2)

(c)A different car needs a braking distance of 72 m to come to a stop. Assuming the same relationship, find the speed of this car.(3)

(Total for Question 20 is 7 marks)

The signal strength, S bars, received from a phone mast is inversely proportional to the square of the distance, d km, from the mast. When d = 2, S = 63.

(a)Find a formula for S in terms of d.(2)

(b)Work out the signal strength at a distance of 6 km.(2)

(c)Priya says: 'If you treble the distance from the mast, the signal strength becomes a third of what it was.' Using your answers to parts (a) and (b), or otherwise, explain whether Priya is correct.(2)

(Total for Question 21 is 6 marks)

A is directly proportional to B² and inversely proportional to C. When B = 3 and C = 4, A = 18.

(a)Find a formula for A in terms of B and C.(3)

(b)Work out the value of A when B = 6 and C = 8.(2)

(Total for Question 22 is 5 marks)

y is inversely proportional to x^2. Prove algebraically that when x is multiplied by 3, y is divided by 9.

(Total for Question 23 is 3 marks)

Mark scheme · 7.3D Direct and Inverse Proportion: Fluency and Exam Drill

Question 1

B1 y = kx oe
Answer: y = kx

Question 2

B1 y = k/x oe
Answer: y = k/x

Question 3

(a) M1 sets up 18 = k*6 oe
(a) A1 k = 3 cao
(a) Answer: k = 3
(b) M1 substitutes x = 10 into y = 3x (ft their k)
(b) A1 y = 30 cao
(b) Answer: y = 30

Question 4

(a) M1 finds k = 45/3² (= 45/9) oe
(a) A1 y = 5x² oe
(a) Answer: y = 5x²
(b) M1 substitutes x = 6 into their formula
(b) A1 y = 180 cao
(b) Answer: y = 180

Question 5

(a) M1 finds k = 40/2³ (= 40/8) oe
(a) A1 y = 5x³ oe
(a) Answer: y = 5x³
(b) M1 substitutes x = 4 into their formula
(b) A1 y = 320 cao
(b) Answer: y = 320

Question 6

(a) M1 finds k = 12/√16 (= 12/4) oe
(a) A1 y = 3sqrt(x) oe
(a) Answer: y = 3sqrt(x)
(b) M1 substitutes x = 25: y = 3sqrt(25)
(b) A1 y = 15 cao
(b) Answer: y = 15

Question 7

(a) B1 y = 60/x oe (k = 60)
(a) Answer: y = 60/x
(b) B1 y = 5 cao (ft their formula)
(b) Answer: y = 5

Question 8

(a) M1 finds k = 18*2² (= 18*4) oe
(a) A1 y = 72/x² oe
(a) Answer: y = 72/x²
(b) M1 sets up 8 = 72/x² oe (ft their formula)
(b) M1 rearranges to x² = 9 oe
(b) A1 x = 3 cao (positive root only)
(b) Answer: x = 3

Question 9

(a) M1 finds k = 1*2³ (= 1*8) oe
(a) A1 p = 8/q³ oe
(a) Answer: p = 8/q³
(b) M1 substitutes q = 4 into their formula
(b) A1 p = 1/8 cao (oe 0.125)
(b) Answer: p = 1/8

Question 10

(a) M1 uses a pair of values, e.g. 12 = k*2², or 48 = k*4²
(a) A1 k = 3 cao
(a) Answer: k = 3
(b) M1 substitutes x = 6 into y = 3x² (ft their k)
(b) A1 y = 108 cao
(b) Answer: y = 108

Question 11

(a) B1 k = 1.3 oe
(a) Answer: k = 1.3
(b) M1 substitutes L = 12 into C = 1.3L (ft their k)
(b) A1 15.60 pounds cao
(b) Answer: 15.60 pounds

Question 12

(a) B1 k = 90
(a) Answer: k = 90
(b) M1 substitutes n = 9 into D = 90/n (ft their k)
(b) A1 10 days cao
(b) Answer: 10 days
(c) B1 any sensible practical limitation, e.g. there may not be enough space or tools for all the builders to work efficiently at the same time, so the time cannot keep decreasing indefinitely (oe)
(c) Answer: e.g. with very many builders there is not enough space or materials for them all to work efficiently, so productivity per builder falls and the model breaks down

Question 13

(a) B1 A
(a) Answer: Graph A
(b) B1 B
(b) Answer: Graph B
(c) B1 states that graph C does not pass through the origin, so it cannot be y = kx (oe); or states that it is a straight line, not a reciprocal curve, so it cannot be y = k/x (oe) - either correct reason is sufficient
(c) Answer: Graph C is a straight line not through the origin, so it cannot be direct proportion (y = kx must pass through the origin); and it is not a curve of the form y = k/x, so it cannot be inverse proportion either.

Question 14

M1 finds new y = k(2x)² = 4kx² oe
A1 y is multiplied by 4 (increases by a factor of 4) cao
Answer: y is multiplied by 4

Question 15

(a) M1 finds k = 32/4² (= 32/16) oe
(a) A1 y = 2x² oe
(a) Answer: y = 2x²
(b) B1 y = 200 (ft their formula)
(b) Answer: y = 200
(c) M1 sets up 50 = 2x² leading to x² = 25 oe
(c) A1 x = 5 cao
(c) Answer: x = 5

Question 16

(a) B1 T = 120/S oe (k = 120)
(a) Answer: T = 120/S
(b) M1 substitutes S = 60 into their formula
(b) A1 2 hours cao
(b) Answer: 2 hours
(c) M1 sets up 2.4 = 120/S oe
(c) A1 50 mph cao
(c) Answer: 50 mph

Question 17

M1 identifies the new x as 1.5x oe (or 3x/2)
M1 finds new y = k(1.5x)³ = 3.375kx³ oe (or 27kx³/8)
A1 237.5% increase cao
Answer: 237.5% increase

Question 18

M1 identifies the new q as 0.5q oe (or q/2)
M1 finds new p = k/(0.5q)² = 4k/q² oe (= 4p)
A1 300% increase cao (accept p becomes 4 times as large)
Answer: 300% increase

Question 19

(a) B1 k = 25
(a) Answer: k = 25
(b) B1 125 minutes (ft their k)
(b) Answer: 125 minutes
(c) B1 compares their predicted time (125 minutes) with the actual instructed time (110 minutes) and notes they are different (dep on a correct or ft predicted time)
(c) B1 concludes the claim is not correct, since direct proportion would require the predicted and actual times to match (oe)
(c) Answer: No, the claim is not correct, because the predicted time for the 5 kg joint (125 minutes) does not match the actual instructed time (110 minutes), so T is not directly proportional to m.

Question 20

(a) M1 finds k = 18/30² (= 18/900, = 1/50) oe
(a) A1 d = s²/50 oe (accept d = 0.02s²)
(a) Answer: d = s²/50
(b) M1 substitutes s = 50 into their formula
(b) A1 50 m cao
(b) Answer: 50 m
(c) M1 sets up 72 = s²/50 oe (ft their formula)
(c) M1 rearranges to s² = 3600 oe, dependent on the first M1
(c) A1 60 mph cao
(c) Answer: 60 mph

Question 21

(a) M1 finds k = 63*2² (= 63*4) oe
(a) A1 S = 252/d² oe
(a) Answer: S = 252/d²
(b) M1 substitutes d = 6 into their formula
(b) A1 7 (bars) cao
(b) Answer: 7 bars
(c) B1 compares S at d = 2 (63) with S at d = 6 (7) and notes that 7 is a ninth of 63, not a third (dep on a correct or ft value from part b)
(c) B1 concludes Priya is incorrect, because S is inversely proportional to d² (not d), so trebling d divides S by 9, not 3 (oe)
(c) Answer: Priya is incorrect. Trebling d from 2 to 6 changes S from 63 to 7, which is a ninth of the original value, not a third, because S is inversely proportional to d², not d.

Question 22

(a) M1 states A = kB²/C oe
(a) M1 substitutes to form 18 = k*3²/4 (= 9k/4), dependent on the first M1
(a) A1 k = 8, so A = 8B²/C oe
(a) Answer: A = 8B²/C
(b) M1 substitutes B = 6, C = 8 into their formula (ft their formula)
(b) A1 A = 36 cao
(b) Answer: A = 36

Question 23

M1 states y = k/x² (or equivalent starting relationship)
M1 substitutes 3x for x: y_new = k/(3x)² = k/(9x²), dependent on the first M1
A1 y_new = (1/9)*(k/x²) = y/9, so y is divided by 9 (must reach an explicit conclusion linking back to the original y) cao
Answer: y_new = y/9, so y is divided by 9, as required.

Direct and Inverse Proportion: Fluency and Exam Drill - Worksheets, Questions and Revision

7.3D Direct and Inverse Proportion: Fluency and Exam Drill

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