y is directly proportional to x. When x = 5, y = 15.
(a)Work out the value of y when x = 8.(2)
(b)Work out the value of x when y = 39.(2)
(Total for Question 1 is 4 marks)
2
A shop sells pens in packs. The cost of pens is directly proportional to the number of pens bought. 6 pens cost £4.20.
(a)Work out the cost of 10 pens.(2)
(b)Rhys has £6. Work out the greatest number of pens he can buy.(2)
(Total for Question 2 is 4 marks)
3
On a certain day, the amount received in pounds sterling, P, when exchanging euros is directly proportional to the number of euros exchanged, E. Grace exchanges 250 euros and receives £215.
(a)Work out the exchange rate, in pounds sterling received per euro, giving your answer to 3 significant figures.(2)
(b)Prisha exchanges 460 euros. Work out how many pounds sterling she receives.(2)
(Total for Question 3 is 4 marks)
4
The time, T hours, taken by a train to travel a fixed distance is inversely proportional to its average speed, S mph. When S = 50, T = 4.
(a)Find a formula for T in terms of S.(2)
(b)Work out T when S = 80.(2)
(Total for Question 4 is 4 marks)
5
A cake recipe for 8 people uses 240 g of caster sugar and 200 g of butter. The amounts of each ingredient needed are directly proportional to the number of people.
(a)Work out the mass of caster sugar needed to make the recipe for 6 people.(2)
(b)Nadia has 620 g of butter. Work out the maximum number of people she can make the recipe for.(3)
(Total for Question 5 is 5 marks)
6
y is directly proportional to x2. When x = 4, y = 32.
(a)Find a formula for y in terms of x.(2)
(b)Work out the value of y when x = 5.(2)
(c)Work out the positive value of x when y = 98.(2)
(Total for Question 6 is 6 marks)
7
y is inversely proportional to x. When x = 4, y = 9.
(a)Find a formula for y in terms of x.(2)
(b)Work out the value of x when y = 6.(2)
(Total for Question 7 is 4 marks)
8
The table shows some corresponding values of x and y. x: 1, 2, 4 y: 48, 24, 12
(a)Show that y is inversely proportional to x.(2)
(b)Write down a formula for y in terms of x.(1)
(Total for Question 8 is 3 marks)
9
The table shows values of x and y, where y is directly proportional to x2. x: 1, 3, 5 y: 4, 36, 100
(a)Find the value of k in y = kx2.(1)
(b)Hence find the value of y when x = 10.(2)
(Total for Question 9 is 3 marks)
10
y is directly proportional to √x. When x = 9, y = 15.
(a)Find a formula for y in terms of x.(2)
(b)Work out the value of y when x = 16.(2)
(c)Work out the value of x when y = 17.5.(2)
(Total for Question 10 is 6 marks)
11
y is inversely proportional to x2. When x = 3, y = 10.
(a)Find a formula for y in terms of x.(2)
(b)Work out the value of y when x = 5, giving your answer as a decimal.(2)
(Total for Question 11 is 4 marks)
12
The pressure, P pascals, of a fixed mass of gas at constant temperature is inversely proportional to its volume, V cm3. When the volume is 150 cm3, the pressure is 240 pascals.
(a)Find a formula for P in terms of V.(2)
(b)Work out the pressure when the volume is decreased to 100 cm3.(2)
(c)The volume is increased so that the pressure drops to 90 pascals. Work out the new volume.(2)
(Total for Question 12 is 6 marks)
13
It takes 5 taps 22 minutes to fill a tank, with all taps flowing at the same constant rate. Work out how many minutes it would take 7 taps, flowing at the same rate, to fill an identical tank. Give your answer correct to 1 decimal place.
(Total for Question 13 is 3 marks)
14
y is directly proportional to x and inversely proportional to z. When x = 4 and z = 2, y = 10.
(a)Find a formula for y in terms of x and z.(3)
(b)Work out the value of y when x = 9 and z = 3.(2)
(Total for Question 14 is 5 marks)
15
The intensity of light, I lux, from a lamp is inversely proportional to the square of the distance, d metres, from the lamp. At a distance of 3 m, the intensity is 20 lux.
(a)Find a formula for I in terms of d.(2)
(b)Work out the intensity at a distance of 5 m. Give your answer to 3 significant figures.(2)
(c)For safe reading, the intensity must be at least 8 lux. Find the maximum distance, to 1 decimal place, at which this is possible.(3)
(Total for Question 15 is 7 marks)
16
The braking distance, D metres, of a car is directly proportional to the square of its speed, v mph. A car travelling at 30 mph has a braking distance of 13.5 m.
(a)Find a formula for D in terms of v.(2)
(b)A car is travelling at 50 mph. Investigators find that its actual braking distance is 3 times the value predicted by the formula in part (a), due to a wet road surface. Work out the actual braking distance of the car at 50 mph.(3)
(Total for Question 16 is 5 marks)
17
V is directly proportional to L3. When L = 2, V = 40.
(a)Find a formula for V in terms of L.(2)
(b)Work out V when L = 5.(2)
(c)Work out the value of L when V = 200, giving your answer to 3 significant figures.(3)
(Total for Question 17 is 7 marks)
18
y is directly proportional to x2. Work out the percentage increase in y when x is increased by 20%.
(Total for Question 18 is 3 marks)
19
y is inversely proportional to x3. Work out the percentage decrease in y when x is increased by 50%.
(Total for Question 19 is 3 marks)
20
y is directly proportional to x2. z is inversely proportional to y. When x = 2, y = 12 and z = 3.
(a)Find a formula for y in terms of x.(2)
(b)Find a formula for z in terms of x.(3)
(c)Find the positive value of x for which y = z.(3)
(Total for Question 20 is 8 marks)
Mark scheme · 5.2D Direct and Inverse Proportion: Fluency and Exam Drill
Question 1
(a) M1 finds k = 15/5 = 3, or sets up y = 3x oe
(a) A1 y = 24 cao
(a) Answer: y = 24
(b) M1 sets up 39 = 3x oe (ft their k)
(b) A1 x = 13 cao
(b) Answer: x = 13
Question 2
(a) M1 finds unit cost = 4.20/6 = 0.70 oe, or a correct scaling method
(a) A1 £7.00 cao
(a) Answer: £7.00
(b) M1 6/0.70 (= 8.57...) oe, ft their unit cost
(b) A1 8 (pens) cao, rounded down to a whole number of pens
(b) Answer: 8 pens
Question 3
(a) M1 215/250 oe
(a) A1 0.860 awrt 3sf
(a) Answer: 0.860 pounds sterling per euro
(b) M1 460 x 0.86 oe (ft their rate)
(b) A1 £395.60 cao
(b) Answer: £395.60
Question 4
(a) M1 finds k = ST = 50 x 4 = 200 oe
(a) A1 T = 200/S oe
(a) Answer: T = 200/S
(b) M1 substitutes S = 80 into their formula
(b) A1 T = 2.5 hours cao
(b) Answer: T = 2.5 hours
Question 5
(a) M1 finds sugar per person = 240/8 = 30 (g) oe
(a) A1 180 g cao
(a) Answer: 180 g
(b) M1 finds butter per person = 200/8 = 25 (g) oe
(b) dM1 620/25 (= 24.8) oe, dependent on the first M1
(b) A1 24 cao, rounded down to a whole number of people
(b) Answer: 24 people
Question 6
(a) M1 finds k = 32/42 = 32/16 = 2 oe
(a) A1 y = 2x2 oe
(a) Answer: y = 2x2
(b) M1 substitutes x = 5 into their formula
(b) A1 y = 50 cao
(b) Answer: y = 50
(c) M1 98 = 2x2 leading to x2 = 49 oe
(c) A1 x = 7 cao (positive root only)
(c) Answer: x = 7
Question 7
(a) M1 finds k = 4 x 9 = 36 oe
(a) A1 y = 36/x oe
(a) Answer: y = 36/x
(b) M1 sets up 6 = 36/x oe
(b) A1 x = 6 cao
(b) Answer: x = 6
Question 8
(a) M1 finds xy for at least two pairs, e.g. 1 x 48 = 48, 2 x 24 = 48
(a) A1 all three products equal 48, so xy is constant, confirming y is inversely proportional to x (dep on showing the constant product)
(a) Answer: xy = 48 for every pair, confirming inverse proportion
(b) B1 y = 48/x oe (ft their constant from part a)
(b) Answer: y = 48/x
Question 9
(a) B1 k = 4 (from e.g. 4/12, or 36/32, or 100/52)
(a) Answer: k = 4
(b) M1 substitutes x = 10 into y = 4x2 (ft their k)
(b) A1 y = 400 cao
(b) Answer: y = 400
Question 10
(a) M1 finds k = 15/√9 = 15/3 = 5 oe
(a) A1 y = 5sqrt(x) oe
(a) Answer: y = 5sqrt(x)
(b) M1 substitutes x = 16: y = 5sqrt(16)
(b) A1 y = 20 cao
(b) Answer: y = 20
(c) M1 sets up 17.5 = 5sqrt(x) leading to √x = 3.5 oe
(c) A1 x = 12.25 cao
(c) Answer: x = 12.25
Question 11
(a) M1 finds k = 10 x 32 = 90 oe
(a) A1 y = 90/x2 oe
(a) Answer: y = 90/x2
(b) M1 substitutes x = 5 into their formula
(b) A1 y = 3.6 cao
(b) Answer: y = 3.6
Question 12
(a) M1 finds k = 150 x 240 = 36000 oe
(a) A1 P = 36000/V oe
(a) Answer: P = 36000/V
(b) M1 substitutes V = 100 into their formula
(b) A1 360 pascals cao
(b) Answer: 360 pascals
(c) M1 sets up 90 = 36000/V oe
(c) A1 400 cm3 cao
(c) Answer: 400 cm3
Question 13
M1 finds total tap-minutes = 5 x 22 (= 110) oe
dM1 110/7 oe, dependent on the first M1
A1 15.7 (minutes) awrt 1 d.p.
Answer: 15.7 minutes (1 d.p.)
Question 14
(a) M1 y = kx/z oe seen
(a) dM1 10 = k x 4/2 (= 2k) solved, dependent on the first M1
(a) A1 y = 5x/z oe
(a) Answer: y = 5x/z
(b) M1 substitutes x = 9, z = 3 into their formula
(b) A1 y = 15 cao
(b) Answer: y = 15
Question 15
(a) M1 finds k = 20 x 32 = 180 oe
(a) A1 I = 180/d2 oe
(a) Answer: I = 180/d2
(b) M1 substitutes d = 5 into their formula
(b) A1 7.20 lux awrt 3sf
(b) Answer: 7.20 lux (3sf)
(c) M1 sets up 8 = 180/d2 oe
(c) dM1 d2 = 180/8 (= 22.5) oe, dependent on the first M1
(c) A1 4.7 (m) awrt 1 d.p.
(c) Answer: 4.7 m (1 d.p.)
Question 16
(a) M1 finds k = 13.5/302 = 0.015 oe
(a) A1 D = 0.015v2 oe
(a) Answer: D = 0.015v2
(b) M1 substitutes v = 50 into their formula to find the predicted D (= 37.5) oe
(b) dM1 multiplies their predicted D by 3, dependent on the first M1
(b) A1 112.5 m cao
(b) Answer: 112.5 m
Question 17
(a) M1 finds k = 40/23 = 40/8 = 5 oe
(a) A1 V = 5L3 oe
(a) Answer: V = 5L3
(b) M1 substitutes L = 5 into their formula
(b) A1 V = 625 cao
(b) Answer: V = 625
(c) M1 sets up 200 = 5L3 leading to L3 = 40 oe
(c) dM1 takes the cube root of their 40, dependent on the first M1
(c) A1 3.42 awrt 3sf
(c) Answer: L = 3.42 (3sf)
Question 18
M1 identifies the new x as 1.2x oe
M1 finds new y = k(1.2x)2 = 1.44kx2 oe
A1 44% increase cao
Answer: 44% increase
Question 19
M1 identifies the new x as 1.5x oe
M1 finds new y = k/(1.5x)3 = (8/27)k/x3 oe (1.53 = 3.375 seen)
A1 70.4% decrease awrt 3sf (accept exact 19/27 as a percentage)
Answer: 70.4% decrease (3sf)
Question 20
(a) M1 finds k = 12/22 = 3 oe
(a) A1 y = 3x2 oe
(a) Answer: y = 3x2
(b) M1 finds the constant for z: z x y = 3 x 12 = 36 oe
(b) dM1 writes z = 36/y and substitutes y = 3x2 (ft their formula from part a), dependent on the first M1
(b) A1 z = 12/x2 oe
(b) Answer: z = 12/x2
(c) M1 sets up 3x2 = 12/x2 oe (ft their formulae)
(c) dM1 rearranges to x4 = 4, dependent on the first M1