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GCSE · Algebra - Proportion

A8 Direct and Inverse Proportion

AQA 8365 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Which of the following equations shows y directly proportional to x?

A) y = x + 5
B) y = 3x
C) y = 5/x
D) y = x²

(Total for Question 1 is 1 mark)

Which of the following equations shows y inversely proportional to x²?

A) y = kx²
B) y = k/x²
C) y = kx
D) y = k/x

(Total for Question 2 is 1 mark)

y is directly proportional to x. When x = 6, y = 21. Find the value of y when x = 10.

(Total for Question 3 is 2 marks)

y is inversely proportional to x. When x = 5, y = 18. Find the value of y when x = 9.

(Total for Question 4 is 2 marks)

p is directly proportional to q². When q = 4, p = 32. Find the value of p when q = 7.

(Total for Question 5 is 3 marks)

m is inversely proportional to n². When n = 3, m = 8. Find the value of m when n = 6.

(Total for Question 6 is 3 marks)

y is directly proportional to √x. When x = 9, y = 15.

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

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

(Total for Question 7 is 4 marks)

The time, T hours, taken to fill a water tank is inversely proportional to the number of pumps, n, used. Using 4 pumps takes 5 hours.

(a)Find a formula for T in terms of n.(2)

(b)Find the time taken to fill the tank using 8 pumps.(1)

(c)State one assumption needed for this model to be valid.(1)

(Total for Question 8 is 4 marks)

y is directly proportional to x³. When x = 2, y = 40.

(a)Find the value of y when x = 5.(3)

(b)Find the value of x when y = 1080.(2)

(Total for Question 9 is 5 marks)

The energy, E joules, stored in a stretched spring is directly proportional to the square of its extension, x cm. When x = 3, E = 27.

(a)Find a formula connecting E and x.(2)

(b)Find E when x = 5.(1)

(c)Find the extension when E = 108.(2)

(Total for Question 10 is 5 marks)

The force, F newtons, between two magnets is inversely proportional to the square of the distance, d cm, between them. When d = 2, F = 45.

(a)Find F in terms of d.(2)

(b)Find F when d = 6.(1)

(c)Find the distance when F = 20.(2)

(d)State what happens to F as d increases.(1)

(Total for Question 11 is 6 marks)

z is directly proportional to x and inversely proportional to y². When x = 8 and y = 2, z = 12.

(a)Find a formula connecting x, y and z.(2)

(b)Find z when x = 10 and y = 5.(2)

(c)Find the value of y when x = 27 and z = 2.(2)

(Total for Question 12 is 6 marks)

y is directly proportional to x². Show that if x is doubled, y increases by 300%.

(Total for Question 13 is 3 marks)

p is inversely proportional to q³. Show that if q is trebled, p is divided by 27.

(Total for Question 14 is 3 marks)

y is directly proportional to x. When x = 3, y = 12. Find, in exact surd form, the values of x for which y = x² - 7.

(Total for Question 15 is 4 marks)

y is directly proportional to x². When x = 2, y = 20.

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

(b)Separately, y is inversely proportional to w. When x = 2 (so y = 20), it is also known that w = 25 at this point. Find a formula for y in terms of w.(2)

(c)Hence find the value of w when x = 5.(3)

(Total for Question 16 is 7 marks)

The volume, V cm³, of a certain type of crystal is directly proportional to the square of its width, w cm, and directly proportional to its height, h cm. A crystal with width 4 cm and height 10 cm has volume 320 cm³.

(a)Find a formula for V in terms of w and h.(2)

(b)Find the volume of a crystal with width 5 cm and height 6 cm.(2)

(c)A crystal has height 8 cm and volume 144 cm³. Find its width, w cm.(2)

(Total for Question 17 is 6 marks)

y is directly proportional to x³ for x ≥ 0. The graph of y against x³ is a straight line through the origin. When x = 2, y = 56.

(a)State the gradient of the graph of y against x³.(1)

(b)Find the value of y when x = 1/2.(2)

(c)A second variable, z, is such that z is inversely proportional to x, and z = 4 when x = 7. Find the value of x (other than x = 0) for which y = z, giving your answer correct to 3 significant figures.(4)

(Total for Question 18 is 7 marks)

Mark scheme · A8 Direct and Inverse Proportion

Question 1

B1 B cao
Answer: B) y = 3x

Question 2

B1 B cao
Answer: B) y = k/x²

Question 3

M1 finds the constant of proportionality, k = 3.5 (from 21 = 6k)
A1 y = 35 cao
Answer: y = 35

Question 4

M1 finds the constant of proportionality, k = 90 (from k = 5 x 18)
A1 y = 10 cao
Answer: y = 10

Question 5

M1 finds k = 2 (from 32 = k(4)², i.e. 32 = 16k)
M1 substitutes q = 7 into p = 2q²
A1 p = 98 cao
Answer: p = 98

Question 6

M1 finds k = 72 (from k = 8 x 3²)
M1 substitutes n = 6 into m = 72/n²
A1 m = 2 cao
Answer: m = 2

Question 7

(a) M1 finds k = 5 (from 15 = k √9, i.e. 15 = 3k)
(a) A1 y = 5 √x cao
(a) Answer: y = 5 √x
(b) M1 substitutes x = 25 into their formula from part (a)
(b) A1 y = 25 cao
(b) Answer: y = 25

Question 8

(a) M1 finds k = 20 (from k = 5 x 4)
(a) A1 T = 20/n cao
(a) Answer: T = 20/n
(b) B1 T = 2.5 hours cao
(b) Answer: T = 2.5 hours
(c) B1 valid assumption, e.g. all pumps work at the same constant rate and do not interfere with each other oe
(c) Answer: All pumps work at the same constant rate

Question 9

(a) M1 finds k = 5 (from 40 = k(2)³, i.e. 40 = 8k)
(a) M1 substitutes x = 5 into y = 5x³
(a) A1 y = 625 cao
(a) Answer: y = 625
(b) M1 sets up 1080 = 5x³ and rearranges to x³ = 216
(b) A1 x = 6 cao
(b) Answer: x = 6

Question 10

(a) M1 finds k = 3 (from 27 = k(3)², i.e. 27 = 9k)
(a) A1 E = 3x² cao
(a) Answer: E = 3x²
(b) B1 E = 75 cao
(b) Answer: E = 75
(c) M1 sets up 108 = 3x² and solves x² = 36
(c) A1 x = 6 cm cao (rejects x = -6 since extension is positive)
(c) Answer: x = 6 cm

Question 11

(a) M1 finds k = 180 (from k = 45 x 2²)
(a) A1 F = 180/d² cao
(a) Answer: F = 180/d²
(b) B1 F = 5 cao
(b) Answer: F = 5
(c) M1 sets up 20 = 180/d² and solves d² = 9
(c) A1 d = 3 cm cao
(c) Answer: d = 3 cm
(d) B1 F decreases (towards zero) oe, since F is inversely proportional to d²
(d) Answer: F decreases towards zero as d increases

Question 12

(a) M1 finds k = 6 (from 12 = k(8)/2², i.e. 12 = 2k)
(a) A1 z = 6x/y² cao
(a) Answer: z = 6x/y²
(b) M1 substitutes x = 10, y = 5 into their formula
(b) A1 z = 2.4 cao
(b) Answer: z = 2.4
(c) M1 sets up 2 = 6(27)/y² and solves y² = 81
(c) A1 y = 9 cao
(c) Answer: y = 9

Question 13

M1 lets the original y = kx²
M1 finds the new y when x is doubled: k(2x)² = 4kx² = 4y
A1 correct conclusion: the increase is 4y - y = 3y, which is 300% of the original value, as required
Answer: y increases by 300% (shown)

Question 14

M1 lets the original p = k/q³
M1 finds the new p when q is trebled: k/(3q)³ = k/(27q³) = p/27
A1 correct conclusion: p is divided by 27, as required
Answer: p is divided by 27 (shown)

Question 15

M1 finds k = 4 (from 12 = 3k), so y = 4x
M1 sets up 4x = x² - 7 and rearranges to x² - 4x - 7 = 0
M1 applies the quadratic formula with a = 1, b = -4, c = -7, giving x = (4 ± √16 + 28)/2
A1 x = 2 + √11 or x = 2 - √11 cao (both values, exact surd form)
Answer: x = 2 + √11 or x = 2 - √11

Question 16

(a) M1 finds k = 5 (from 20 = k(2)², i.e. 20 = 4k)
(a) A1 y = 5x² cao
(a) Answer: y = 5x²
(b) M1 finds the second constant of proportionality, k2 = 500 (from k2 = 20 x 25)
(b) A1 y = 500/w cao
(b) Answer: y = 500/w
(c) M1 finds y = 125 using their formula from part (a) (y = 5(5)²)
(c) M1 substitutes their y = 125 into their formula from part (b) to form 125 = 500/w
(c) A1 w = 4 cao
(c) Answer: w = 4

Question 17

(a) M1 finds k = 2 (from 320 = k(4)²(10), i.e. 320 = 160k)
(a) A1 V = 2w² h cao
(a) Answer: V = 2w² h
(b) M1 substitutes w = 5, h = 6 into their formula
(b) A1 V = 300 cm³ cao
(b) Answer: V = 300 cm³
(c) M1 sets up 144 = 2w²(8) and solves w² = 9
(c) A1 w = 3 cm cao
(c) Answer: w = 3 cm

Question 18

(a) B1 gradient = 7 cao
(a) Answer: Gradient = 7
(b) M1 substitutes x = 1/2 into y = 7x³
(b) A1 y = 7/8 (or 0.875) cao
(b) Answer: y = 7/8
(c) M1 finds the constant for z, k2 = 28 (from k2 = 4 x 7), so z = 28/x
(c) M1 sets up 7x³ = 28/x
(c) M1 multiplies both sides by x and divides by 7 to obtain x⁴ = 4
(c) A1 x = 1.41 (3sf) cao (exact value √2)
(c) Answer: x = √2 = 1.41 (3sf)

Direct and Inverse Proportion - Worksheets, Questions and Revision