Proportion (harder) Quiz (Grade 7)

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Topic:

Score: 0 / 10

Proportion (harder) — Notes & Worked Examples Show / Hide

Explanation:

Translate the proportion statement into an algebraic equation using a constant of proportionality k: if y is directly proportional to xⁿ, write y = kxⁿ; if y is inversely proportional to xⁿ, write y = kxⁿ. Substitute the given pair of values to find k and write the full formula before finding any unknown values.

For compound three-variable problems (where y depends on x and x depends on z), substitute the second formula directly into the first and simplify the powers and constants carefully. Remember to take both roots or the positive physical root when solving for x² or √x.

Example 1: y is inversely proportional to x². Given y = 5 when x = 6, find x when y = 20 (x > 0).

Solution:

y = kx² ⇒ 5 = k6²

k = 5 × 36 = 180, so y = 180x²

When y = 20: 20 = 180x² ⇒ x² = 18020 = 9

Since x > 0, x = √9 = 3

Answer: x = 3

Example 2: P = 3Q² and Q is inversely proportional to √R. Given Q = 4 when R = 9, write P in terms of R.

Solution:

Q = k√R ⇒ 4 = k√9 ⇒ k = 4 × 3 = 12

Substitute Q = 12√R into P = 3Q²:

P = 3 × (12√R)² = 3 × 144R

Answer: P = 432R

One Page Guide (PDF) Proportion (harder) Worksheets

Question 1

[1 mark]

y is directly proportional to x².
Given that y = 45 when x = 3, find the value of y when x = 6.

y =

Worked Solution & Answer

y = kx² ⇒ 45 = k(3²) ⇒ k = 5
When x = 6, y = 5(6²) = 180. B1

Answer: y = y = 180

Question 2

[1 mark]

4 identical machines take 10 hours to print a batch of leaflets
How long would it take 5 of these machines?

hours

Worked Solution & Answer

1) Multiply the number of workers/machines by the time taken to find the total work required for 1 worker/machine
2) Divide that total by the new number of workers/machines (or new time)
8 hours

Answer: 8 hours hours

Question 3

[2 marks]

Work out
(a) Sketch graph for y is directly proportional to x (1) (b) Sketch graph for y is inversely proportional to x (1)

(a) Using the axes below, sketch a graph to represent the statement
y is directly proportional to x [1 mark]

(b) Using the axes below, sketch a graph to represent the statement
y is inversely proportional to x [1 mark]

Worked Solution & Answer

(a)

y = kx is a straight line through (0, 0) B1

Answer (a): Straight line with positive gradient passing through the origin O

(b)

y = k/x is a decreasing hyperbola in quadrant 1 B1

Answer (b): Reciprocal curve y = k/x in the first quadrant asymptotic to both axes

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

[3 marks]

p is inversely proportional to t

Complete the table of values.

Worked Solution & Answer

p × t = 100 × 1 = 100 ⇒ when t = 25, p = 4 M1
when p = 5, t = 20 M1
when t = 2, p = 50 A1

Answer: t = 20; p = 4, 50

Question Paper (PDF) • Mark Scheme (PDF)

Question 7 OCR Jun 2023 1F Q22

[4 marks]

Hiro and Taylor are both electricians.
Hiro **does not** charge to visit a house but charges a fixed rate per hour for the work needed.
This graph shows the relationship between the hours worked and the total charge made by Hiro.

(a) Explain how this graph shows that Hiro’s total charge is directly proportional to the hours worked. [2 marks]

(b) Taylor **does** charge to visit a house and charges the same fixed rate per hour as Hiro for the work needed.
On the axes above, draw a graph to show the relationship between the hours worked and the total charge made by Taylor. [2 marks]

Worked Solution & Answer

(a)

A graph showing direct proportion must be a straight line through the origin. M1 C1

Answer (a): It is a straight line that passes through the origin (0, 0).

(b)

Same hourly rate ⇒ same gradient (parallel line) M1
call-out charge ⇒ positive y-intercept. A1

Answer (b): Straight line parallel to Hiro’s line with a positive vertical intercept.

Question Paper (PDF) • Mark Scheme (PDF)

Question 8

[4 marks]

The table shows a set of values for x and y.

(a) Find an equation for y in terms of x. [2 marks]

y =

(b) Find the positive value of x when y = 16 [2 marks]

Worked Solution & Answer

(a)

y = k/x² M1
Substitute x = 1, y = 9 ⇒ k = 9, so y = 9/x² A1

Answer (a): y = y = 9x²

(b)

16 = 9/x² M1
⇒ x² = 9/16
⇒ x = 3/4 A1

Answer (b): 34

Question Paper (PDF) • Mark Scheme (PDF)

Question 9 Specimen Set 2 3F Q27

[4 marks]

At a depth of x metres, the temperature of the water in an ocean is T °C.
At depths below 900 metres, T is inversely proportional to x.
T is given by
T = 4500x

(a) Work out the difference in the temperature of the water at a depth of 1200 metres and the temperature of the water at a depth of 2500 metres. [3 marks]

°C

(b) One of the graphs could show that T is inversely proportional to x.
Write down the letter of this graph. [1 mark]

Worked Solution & Answer

(a)

At x = 1200, T = 4500 ÷ 1200 = 3.75 °C M1
at x = 2500, T = 4500 ÷ 2500 = 1.8 °C M1
Difference = 3.75 – 1.8 = 1.95 °C A1

Answer (a): 1.95 °C

(b)

Graph D is the reciprocal curve y = k/x B1

Answer (b): D

Question Paper (PDF) • Mark Scheme (PDF)

Question 10

[5 marks]

Here is a table of values of x and y.

(a) Is Nadia correct?
You must give a reason for your answer. [1 mark]

(b) Calculate the value of w when t = 7.84 [3 marks]

w =

Worked Solution & Answer

(a)

No, because y is not a constant multiple of x (e.g. when x = 2, y = 0 so the graph does not pass through (0, 0) / 4 ÷ 4 ≠ 8 ÷ 6)

Answer (a): No, because y is not a constant multiple of x (e.g. when x = 2, y = 0 so the graph does not pass through (0, 0) / 4 ÷ 4 ≠ 8 ÷ 6)

(b)

w = k√t ⇒ 140 = k√64 = 8k ⇒ k = 17.5 M2
w = 17.5 × √7.84 = 17.5 × 2.8 = 49 A1

Answer (b): w = 49

(c)

Curve starting at (0, 0) with positive, decreasing gradient (w = k√t)

Answer (c): Curve starting at (0, 0) with positive, decreasing gradient (w = k√t)

Question Paper (PDF) • Mark Scheme (PDF)

Score: 0 / 10 (0 / 28 marks)

Proportion (harder) Worksheets Next: Algebraic fractions →

Grade 7 Quizzes

19 Topics

7A - Surds	7B - Bounds
7C - Proportion (harder)	7D - Algebraic fractions
7E - The quadratic formula	7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs	7H - Rearranging (harder)
7J - Functions (harder)	7K - Iteration
7L - Area of any triangle	7M - The sine rule
7N - The cosine rule	7P - Congruent triangles
7Q - 3D pythagoras / trig.	7R - Histograms
7S - Conditional probability	7T - Exact trig. values (harder)
7U - Area of a segment