P is directly proportional to Q. When Q = 5, P = 30. Find the value of P when Q = 9.
(Total for Question 1 is 2 marks)
2
m is directly proportional to n. When n = 8, m = 20.
(a)Find a formula for m in terms of n.(2)
(b)Find the value of n when m = 45.(1)
(Total for Question 2 is 3 marks)
3
The cost, C pounds, of a length of ribbon is directly proportional to its length, L metres. 3 metres of ribbon costs 4.50 pounds.
(a)Work out the cost of 8 metres of the same ribbon.(2)
(b)Sarah has 10.50 pounds to spend on this ribbon. Work out the maximum length of ribbon she can buy.(2)
(Total for Question 3 is 4 marks)
4
y is inversely proportional to x. When x = 4, y = 15. Find the value of y when x = 10.
(Total for Question 4 is 3 marks)
5
The table shows some values of x and y, where y is directly proportional to x. x: 2, 5, 8 y: 6, ?, 24
(a)Find the missing value of y in the table.(1)
(b)Find the value of x for which y = 30.(2)
(Total for Question 5 is 3 marks)
6
The time, T hours, taken to build a wall is inversely proportional to the number of workers, w. It takes 6 workers 8 hours to build the wall.
(a)Find a formula for T in terms of w.(2)
(b)How long would it take 4 workers to build the same wall?(2)
(Total for Question 6 is 4 marks)
7
P is inversely proportional to Q. When Q = 3, P = 8. Show that when Q = 6, P = 4.
(Total for Question 7 is 2 marks)
8
For each equation, state whether y is directly proportional to x, inversely proportional to x, or neither. Give a reason for each answer.
(a)y = 5x(1)
(b)y = 7/x(1)
(c)y = x + 3(1)
(Total for Question 8 is 3 marks)
9
Aisha is exchanging pounds sterling for euros before a trip to France. The number of euros, E, she receives is directly proportional to the number of pounds, P, she exchanges. She is given 172.50 euros for 150 pounds.
(a)Work out how many euros she would receive for 220 pounds.(2)
(b)On her return she has 40 euros left. Assuming the same exchange rate, work out how many pounds this is worth. Give your answer to the nearest penny.(2)
(Total for Question 9 is 4 marks)
10
The tables show corresponding values of x and y.
(a)x: 2, 4, 6 y: 8, 32, 72 Determine, giving a reason, whether y could be directly proportional to x2.(2)
(b)x: 1, 2, 3 y: 5, 11, 19 Determine, giving a reason, whether y could be directly proportional to x2.(2)
(Total for Question 10 is 4 marks)
11
The energy, E joules, stored in a spring is directly proportional to the square of its extension, x cm. When x = 3, E = 45.
(a)Find a formula for E in terms of x.(2)
(b)Find the value of E when x = 6.(1)
(c)Find the value of x when E = 80.(1)
(Total for Question 11 is 4 marks)
12
y is directly proportional to √x. When x = 16, y = 20.
(a)Find a formula for y in terms of x.(2)
(b)Find the value of y when x = 25.(1)
(c)Find the value of x when y = 30.(1)
(Total for Question 12 is 4 marks)
13
y is inversely proportional to x2. When x = 2, y = 12.
(a)Find a formula for y in terms of x.(2)
(b)Find the value of y when x = 4.(1)
(c)Find the value of x when y = 0.75.(1)
(Total for Question 13 is 4 marks)
14
F is inversely proportional to d3. When d = 5, F = 8.
(a)Find a formula for F in terms of d.(2)
(b)Find the value of F when d = 10.(1)
(c)Find the value of d when F = 125.(1)
(Total for Question 14 is 4 marks)
15
Using 3 identical pumps, it takes 100 minutes to fill a swimming pool. The time taken, T minutes, is inversely proportional to the number of pumps used, n. Work out the minimum number of pumps needed to fill the pool in no more than 40 minutes.
(Total for Question 15 is 3 marks)
16
y is directly proportional to x2.
(a)x is increased by 20%. Find the percentage increase in y.(3)
(b)y is now inversely proportional to x2 instead. x is increased by 20%. Find the percentage decrease in y, correct to 1 decimal place.(3)
(Total for Question 16 is 6 marks)
17
P is directly proportional to Q and inversely proportional to R. When Q = 8 and R = 4, P = 6.
(a)Find a formula for P in terms of Q and R.(3)
(b)Work out the value of P when Q = 12 and R = 2.(2)
(c)Work out the value of R when P = 9 and Q = 15.(2)
(Total for Question 17 is 7 marks)
18
For candles of the same height, the volume of wax used, V cm3, is directly proportional to the square of the radius, r cm. A candle of radius 3 cm uses 108 cm3 of wax.
(a)Find a formula for V in terms of r.(2)
(b)A candle of the same height uses 320 cm3 of wax. Find its radius, correct to 1 decimal place.(2)
(c)The radius of a candle is increased by 50%, with the height unchanged. Find the percentage increase in the volume of wax used.(2)
(Total for Question 18 is 6 marks)
19
The resistance, R ohms, of a wire is directly proportional to its length, L metres, and inversely proportional to the square of its diameter, d millimetres. A wire of length 2 m and diameter 0.5 mm has a resistance of 8 ohms.
(a)Find a formula for R in terms of L and d.(3)
(b)Find the resistance of a wire of the same material with length 5 m and diameter 1 mm.(2)
(Total for Question 19 is 5 marks)
20
y is directly proportional to xn, where n is a positive integer. When x = 2, y = 12. When x = 8, y = 768. Find the value of n. Hence find the value of y when x = 32.
(Total for Question 20 is 5 marks)
Mark scheme · 7.3 Direct and Inverse Proportion
Question 1
M1 finds the constant of proportionality k = 30/5 (=6), or uses a valid scale factor method
A1 P = 54 cao
Answer: P = 54
Question 2
(a) M1 k = 20/8 (=2.5)
(a) A1 m = 2.5n oe (accept m = 5n/2)
(a) Answer: m = 2.5n
(b) B1 n = 18 (ft from their formula in part a) cao
(b) Answer: n = 18
Question 3
(a) M1 finds cost per metre: 4.50/3 (=1.50), or a correct scale factor method
(a) A1 12.00 pounds cao
(a) Answer: 12 pounds (GBP 12.00)
(b) M1 L = 10.50/1.50 (ft cost per metre from part a)
(b) A1 7 metres cao
(b) Answer: 7 metres
Question 4
M1 finds the constant of proportionality k = xy = 4 x 15 (=60)
M1 correct method y = k/x, substituting x = 10 (ft)
A1 y = 6 cao
Answer: y = 6
Question 5
(a) B1 y = 15 cao
(a) Answer: 15
(b) M1 uses k = 3 (ft) to set up 30 = 3x, or equivalent scale factor method
(b) A1 x = 10 cao
(b) Answer: 10
Question 6
(a) M1 k = wT = 6 x 8 (=48)
(a) A1 T = 48/w oe
(a) Answer: T = 48/w
(b) M1 T = 48/4 (ft from part a)
(b) A1 12 hours cao
(b) Answer: 12 hours
Question 7
M1 finds the constant of proportionality k = PQ = 3 x 8 (=24)
A1 P = 24/6 = 4 cso
Answer: P = 4 (shown)
Question 8
(a) B1 directly proportional, since it is of the form y = kx with k = 5
(a) Answer: Directly proportional (k = 5)
(b) B1 inversely proportional, since it is of the form y = k/x with k = 7
(b) Answer: Inversely proportional (k = 7)
(c) B1 neither, since it is not of the form y = kx or y = k/x (does not pass through the origin) oe
(c) Answer: Neither
Question 9
(a) M1 finds the constant of proportionality k = 172.5/150 (=1.15)
(a) A1 253 euros cao
(a) Answer: 253 euros
(b) M1 P = 40/1.15 (ft rate from part a)
(b) A1 34.78 pounds (awrt 34.78)
(b) Answer: 34.78 pounds (awrt)
Question 10
(a) M1 finds y/x2 for at least two pairs: 8/4 = 2, 32/16 = 2, 72/36 = 2
(a) A1 yes, y is directly proportional to x2 since y/x2 is constant (=2) oe
(a) Answer: Yes (y = 2x2)
(b) M1 finds y/x2 for at least two pairs: 5/1 = 5, 11/4 = 2.75
(b) A1 no, y is not directly proportional to x2 since y/x2 is not constant oe
(b) Answer: No
Question 11
(a) M1 k = 45/32 (=5)
(a) A1 E = 5x2 oe
(a) Answer: E = 5x2
(b) B1 E = 180 (ft)
(b) Answer: 180
(c) B1 x = 4 cao
(c) Answer: 4
Question 12
(a) M1 k = 20/√16 (=20/4=5)
(a) A1 y = 5sqrt(x) oe
(a) Answer: y = 5sqrt(x)
(b) B1 y = 25 cao
(b) Answer: 25
(c) B1 x = 36 cao
(c) Answer: 36
Question 13
(a) M1 k = 12 x 22 (=48)
(a) A1 y = 48/x2 oe
(a) Answer: y = 48/x2
(b) B1 y = 3 cao
(b) Answer: 3
(c) B1 x = 8 cao
(c) Answer: 8
Question 14
(a) M1 k = 8 x 53 (=1000)
(a) A1 F = 1000/d3 oe
(a) Answer: F = 1000/d3
(b) B1 F = 1 cao
(b) Answer: 1
(c) B1 d = 2 cao
(c) Answer: 2
Question 15
M1 finds the constant of proportionality k = nT = 3 x 100 (=300)
M1 n = 300/40 (=7.5)
A1 8 pumps, with reasoning that the number of pumps must be a whole number and 7 pumps would take longer than 40 minutes
Answer: 8 pumps
Question 16
(a) M1 new x = 1.2x oe
(a) M1 new y = k(1.2x)2 = 1.44kx2 oe, showing the multiplier 1.44
(a) A1 44% increase oe
(a) Answer: 44% increase
(b) M1 new x = 1.2x oe
(b) M1 new y = k/(1.2x)2 = k/(1.44x2) oe, showing the multiplier 1/1.44
(b) A1 30.6% decrease (awrt 30.6%)
(b) Answer: 30.6% decrease (awrt)
Question 17
(a) M1 writes P = kQ/R oe
(a) M1 substitutes Q = 8, R = 4, P = 6 to form an equation for k, e.g. 6 = k x 8/4
(a) A1 k = 3, so P = 3Q/R oe cao
(a) Answer: P = 3Q/R
(b) M1 substitutes into P = 3Q/R (ft): P = 3 x 12/2
(b) A1 P = 18 cao
(b) Answer: 18
(c) M1 rearranges and substitutes (ft): R = 3 x 15/9
(c) A1 R = 5 cao
(c) Answer: 5
Question 18
(a) M1 k = 108/32 (=12)
(a) A1 V = 12r2 oe cao
(a) Answer: V = 12r2
(b) M1 r2 = 320/12 (ft, =26.67 to 2dp)
(b) A1 r = 5.2 cm (awrt 5.2)
(b) Answer: 5.2 cm (awrt)
(c) M1 new r = 1.5r, new V = 12(1.5r)2 = 12 x 2.25r2 oe, showing the multiplier 2.25
(c) A1 125% increase
(c) Answer: 125% increase
Question 19
(a) M1 writes R = kL/d2 oe
(a) M1 substitutes L = 2, d = 0.5, R = 8 to form an equation for k, e.g. 8 = k x 2/0.25
(a) A1 k = 1, so R = L/d2 oe cao
(a) Answer: R = L/d2
(b) M1 substitutes L = 5, d = 1 into R = L/d2 (ft)
(b) A1 R = 5 ohms cao
(b) Answer: 5 ohms
Question 20
M1 forms the ratio 768/12 (=64) and expresses in terms of powers of 2, e.g. 23n/2n = 22n oe (accept use of logarithms)