Direct and inverse proportion Quiz (Grade 5)

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Direct and inverse proportion — Notes & Worked Examples Show / Hide

Explanation:

If y is directly proportional to x (written y ∝ x), then y = kx. If y is inversely proportional to x (written y ∝ 1x), then y = kx. Read the question carefully for powers or roots such as x², x³, or √x.

Always follow three steps: (1) write the proportion equation with constant k, (2) substitute the given pair of values to find k and write the full formula, and (3) substitute the new value to find the unknown.

Example 1: y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.

Solution:

Write equation: y = kx²

Substitute x = 3, y = 45: 45 = k × 3² → 45 = 9k → k = 5

Formula: y = 5x²

When x = 5: y = 5 × 5² = 5 × 25 = 125

Answer: y = 125

Example 2: y is inversely proportional to x. When x = 4, y = 15. Find x when y = 12.

Solution:

Write equation: y = kx

Substitute x = 4, y = 15: 15 = k4 → k = 15 × 4 = 60

Formula: y = 60x

When y = 12: 12 = 60x → x = 60 ÷ 12 = 5

Answer: x = 5

One Page Guide (PDF) Direct and inverse proportion Worksheets

Question 1

[1 mark]

Sarah wants to make 30 cookies using this recipe:
@@TABULAR₀@@
How much of each ingredient does she need?

Worked Solution & Answer

1) Divide to find the value for 1 unit (the unitary method)
2) Multiply by the number of units needed
1800g chocolate chips 2400ml milk

Answer: 1800g chocolate chips 2400ml milk

Question 2

[1 mark]

@@TABULAR₀@@
How much of each ingredient is needed for 12 people?

Worked Solution & Answer

1) Divide to find the value for 1 unit (the unitary method)
2) Multiply by the number of units needed
240g flour 360g sugar 120g butter

Answer: 240g flour 360g sugar 120g butter

Question 3 Specimen Set 1 1H Q16

[2 marks]

These graphs show four different proportionality relationships between y and x.

Match each proportionality graph (A, B, C, D) with a statement in the table.

Worked Solution & Answer

y ∝ x → Graph D M1
y ∝ 1/x → Graph A
y ∝ x² → Graph B
y ∝ 1/x² → Graph C A1

Answer: D, A, B, C

Question Paper (PDF) • Mark Scheme (PDF)

Question 4 OCR Nov 2023 4H Q8

[2 marks]

Here are sketches of six graphs, labelled **A** to **F**.

(a) y is directly proportional to x. [1 mark]

(b) y is inversely proportional to x. [1 mark]

Worked Solution & Answer

(a)

Direct proportion y = kx is a straight line through the origin ⇒ Graph C. B1

Answer (a): C

(b)

Inverse proportion y = k/x is a reciprocal curve in the first quadrant ⇒ Graph E. B1

Answer (b): E

Question Paper (PDF) • Mark Scheme (PDF)

Question 5 OCR Nov 2018 3F Q24 / 6H Q9

[2 marks]

These graphs show different relationships between the variables x and y.

(a) Identify the graph which shows the following.
y is directly proportional to x. [1 mark]

(b) y is inversely proportional to x. [1 mark]

Worked Solution & Answer

(a)

Direct proportion y = kx is a straight line through the origin ⇒ Graph A. B1

Answer (a): A

(b)

Inverse proportion y = k/x is a reciprocal hyperbola ⇒ Graph C. B1

Answer (b): C

Question Paper (PDF) • Mark Scheme (PDF)

Question 6 AQA Nov 2024 1F Q27 / 1H Q8

[3 marks]

(a) 10 people complete a job in 9 hours. How many hours for 15 people? [2 marks]
8 (b) 6 work slower and 9 work faster. Effect on hours? [1 mark]

(a) In this part, assume that each person works at the same rate.
10 people can complete a job in 9 hours.
If 15 people work on the same job, how many hours will it take to complete the job? [2 marks]

Answer hours

(b) In fact, of the 15 people
[INDENT] 6 work at a slower rate
[INDENT] 9 work at a faster rate.
What does this mean about the number of hours it will take to complete the job?
Tick **one** box. [1 mark]

Worked Solution & Answer

(a)

10 × 9 = 90 person-hours. 90 ÷ 15 = 6 hours M1 A1

Answer (a): Answer 6 hours

(b)

Since the exact slower and faster rates are unknown, it is not possible to say B1

Answer (b): It is not possible to say

Question Paper (PDF) • Mark Scheme (PDF)

Question 7 OCR Nov 2020 1F Q21 / 4H Q3

[4 marks]

Three people take 212 hours to deliver leaflets to 270 houses.
Assuming all people deliver leaflets at the same rate, how long will it take five people to deliver leaflets to 405 houses?
Give your answer in hours and minutes.

Worked Solution & Answer

3 people deliver to 270 houses in 2.5 hours ⇒ 1 person delivers to 90 houses in 2.5 hours ⇒ 36 houses per person per hour. M2
5 people deliver 5 × 36 = 180 houses per hour M1
Time for 405 houses = 405 ÷ 180 = 2.25 hours = 2 hours 15 minutes. A1

Answer: 2 hours 15 mins

Question Paper (PDF) • Mark Scheme (PDF)

Question 8 IGCSE Jun 2025 2F Q17

[4 marks]

Here is a list of ingredients to make spaghetti carbonara for 4 people.

(a) Work out how much cheese he needs. [2 marks]

(b) Work out how many people Ruhi makes the spaghetti carbonara for. [2 marks]

Worked Solution & Answer

(a)

(12 ÷ 4) × 50 = 3 × 50 = 150 g M1 A1

Answer (a): 150 g

(b)

(180 ÷ 40) × 4 = 4.5 × 4 = 18 people M1 A1

Answer (b): 18

Question Paper (PDF) • Mark Scheme (PDF)

Question 9 OCR Jun 2023 6H Q4

[4 marks]

Work out
(a) The time taken to complete a journey halves as the speed doubles.
On the axes below, sketch a graph to show this relationship.
(b) It takes 40 minutes to fill a garden pond using water from 5 identical hose pipes.
Assuming the rate of flow of water from each hose pipe is the same, work out how many minutes it would take to fill the same garden pond using 2 of these hose pipes.

(a) The time taken to complete a journey halves as the speed doubles.
On the axes below, sketch a graph to show this relationship. [2 marks]

(b) It takes 40 minutes to fill a garden pond using water from 5 identical hose pipes.
Assuming the rate of flow of water from each hose pipe is the same, work out how many minutes it would take to fill the same garden pond using 2 of these hose pipes. [2 marks]

minutes

Worked Solution & Answer

(a)

Time is inversely proportional to speed, giving a decreasing reciprocal curve in the first quadrant that does not touch either axis. M1 A1

Answer (a): Reciprocal curve (y = k/x) in the first quadrant approaching both axes as asymptotes without touching them.

(b)

1 hose pipe takes 40 × 5 = 200 minutes. M1
2 hose pipes take 200 ÷ 2 = 100 minutes. A1

Answer (b): 100 minutes

Question Paper (PDF) • Mark Scheme (PDF)

Question 10 IGCSE Jun 2018 2H Q16

[5 marks]

R is proportional to t²
The graph shows the relationship between R and t for 0 ≤ t ≤ 4

(a) Find a formula for R in terms of t. [3 marks]

R =

(b) show that t is inversely proportional to √(x) for t > 0 [2 marks]

Worked Solution & Answer

(a)

R = kt² M2
at (2, 10), 10 = 4k ⇒ k = 2.5 ⇒ R = 2.5t² A1

Answer (a): R = R = 2.5t²

(b)

(5/2)t² = 8/(5x) M1
⇒ t² = 16/(25x)
⇒ t = (4/5)/√x = 0.8/√x C1

Answer (b): 2.5t² = 8/(5x) ⇒ t² = 16/(25x) ⇒ t = 0.8/√x, so t is inversely proportional to √x

Question Paper (PDF) • Mark Scheme (PDF)

Score: 0 / 10 (0 / 28 marks)

Direct and inverse proportion Worksheets Next: Reverse percentages →

Grade 5 Quizzes

25 Topics

5A - Combining ratios	5B - Direct and inverse proportion
5C - Reverse percentages	5D - Standard form
5E - Speed	5F - Changing the subject
5G - Quadratic expressions	5H - Solving quadratic equations
5J - Drawing quadratic graphs	5K - Cubic/reciprocal graphs
5L - Simultaneous equations	5M - Sector areas and arc lengths
5N - Sim. equ. graphically	5P - Midpoint of a line segment
5Q - Gradient of a line	5R - Equation of a line
5S - Spheres and cones	5T - Similar shapes (lengths)
5U - Trigonometry (SOHCAHTOA)	5V - Exact trig. values
5W - Probability trees	5X - Vectors
5Y - Venn diagrams	5Z - Density
5AA - Pressure