Pack B — Fluency

Mathematics

Year 8 · 8.3 Proportions

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

1.
The cost of 8 pens is 12 chf. Find the cost of 1 pen.
2.
4 pencils cost 1.20 chf. How much do 15 pencils cost?
3.
Test whether the table $(1, 3), (2, 9), (3, 27)$ represents a proportional relationship.
4.
For the proportional relationship $y = kx$ , find $k$ given that $y = 35$ when $x = 5$ .
5.
Using $y = 4x$ , find $y$ when $x = 12$ .
6.
Using $y = 5x$ , find $x$ when $y = 40$ .
7.
On a graph of $y = 2x$ , find the $y$ -coordinate when $x = -3$ .
8.
A graph of $y$ against $x$ passes through the origin and the point $(2, 10)$ . State the proportionality equation.
9.
In a direct-proportion graph, the line is straight and passes through which point always?
10.
Decide which equation represents direct proportion: $y = 4x$ , $y = x + 4$ , $y = x^2$ .

SilverQuestions 11–20

11.
Distance is directly proportional to time. In 2 minutes a rollercoaster travels 50 m. Find the distance travelled in 3 minutes and in 7 minutes.
12.
A printer prints 24 pages in 3 minutes. How long to print 60 pages?
13.
A school bus uses 15 L of fuel to travel 180 km. How far can it travel on 25 L at the same rate?
14.
A cake recipe uses 2 eggs to make 6 cakes. How many eggs are needed for 15 cakes?
15.
Test whether $y$ is directly proportional to $x$ given the table: x: 2, 5, 8; y: 6, 15, 24.
16.
On the graph of $y = 3x$ , what is the gradient of the line? Describe how the gradient relates to $k$ .
17.
A tap fills a bucket at a constant rate. After 3 minutes there are 12 L in the bucket. (a) Find the rate. (b) Write $V = kt$ . (c) Find $V$ at $t = 7$ .
18.
4 kg of apples costs 10 chf. How much do 7 kg cost?
19.
Two quantities are directly proportional. When $a = 6$ , $b = 15$ . Find $b$ when $a = 22$ .
20.
The graph of $y$ against $x$ is a straight line through the origin with gradient 0.4. State the equation and find $y$ when $x = 15$ .

GoldQuestions 21–30

21.
For two quantities $x$ and $y$ , $y$ is directly proportional to $x$ . When $x = 5$ , $y = 35$ . (a) Write an equation linking $x$ and $y$ . (b) Find $y$ when $x = 12$ . (c) Find $x$ when $y = 91$ .
22.
Decide whether each table represents direct proportion. Justify. Table A: x = 1, 2, 4, y = 3, 6, 12. Table B: x = 1, 2, 4, y = 3, 6, 8.
23.
In a sale, the cost $C$ of buying $n$ tickets is $C = 12.50n$ . (a) State the unit cost. (b) How many tickets can you buy with 100 chf? (c) Is this a proportional relationship? Justify.
24.
The cost of running a car for $d$ km is $C = 0.18d$ chf. (a) Find the cost for 250 km. (b) Find $d$ for a budget of 90 chf. (c) Sketch the graph and label the gradient.
25.
Two friends share a rate. Alex earns 40 chf for 5 hours of work. Brigit earns 60 chf for 8 hours. Whose hourly rate is higher?
26.
A scale model is a proportional copy of a real building. The model door is 6 cm tall and the real door is 2 m tall. (a) Find the scale factor. (b) The real building is 24 m tall. Find the height of the model.
27.
The mass of a uniform metal bar is directly proportional to its length. A 50 cm bar has mass 120 g. (a) Write an equation. (b) Find the mass of a 75 cm bar. (c) Find the length of a 240 g bar.
28.
On a fitness tracker, calories burned $C$ is roughly proportional to steps $s$ : $C = 0.04s$ . (a) How many calories for 8000 steps? (b) How many steps to burn 200 calories? (c) Is the relationship truly proportional in real life? Justify briefly.
29.
A graph of $y$ versus $x$ shows a straight line through the origin. From the graph, $y = 12$ when $x = 8$ . (a) Find $k$ . (b) Find $y$ when $x = 20$ . (c) Find $x$ when $y = 30$ .
30.
A car travels $d$ km on $\ell$ litres of fuel: $d = 14\ell$ . (a) State the unit rate (km/L). (b) Find $d$ when $\ell = 28$ . (c) The driver only has 600 chf and fuel costs 2 chf/L. What is the maximum distance she can drive?

PlatinumQuestions 31–40

31.
A tap fills a bath proportionally to time. After 5 min the bath has 30 L. (a) Write a model $V = kt$ . (b) The bath holds 200 L. How long to fill from empty? (c) The drain leaks at 2 L/min (constant). Modify the model to account for the leak, and find the time to fill.
32.
A rectangular field has length proportional to width: $\ell = 2w$ . The perimeter is 60 m. (a) Find $\ell$ and $w$ . (b) Find the area. (c) If the perimeter doubles, what happens to the area? Justify.
33.
Compare two paint-mixing schemes. Scheme A: $y = 2x$ (yellow per blue). Scheme B: $y = 3x - 4$ . Identify which is proportional and find values that give the same shade in both.
34.
Investigate: if $y \propto x$ and $x \propto z$ , show that $y \propto z$ and find the combined constant.
35.
On a graph of $y = kx$ , two points are $(3, p)$ and $(8, p + 20)$ . Find $k$ and $p$ .
36.
Distance is directly proportional to time. A cyclist covers 12 km in 30 min. (a) Find the speed. (b) Write a model with units. (c) Determine how far in 1 h 45 min. (d) Sketch the d–t graph for the first 2 hours.
37.
A rope is divided into 3 lengths in the ratio $2:3:5$ . The longest is 60 cm longer than the shortest. (a) Find each length. (b) If the lengths are doubled, are the ratios still $2:3:5$ ? Justify.
38.
Two quantities are proportional: $y$ doubles every time $x$ doubles. The point $(2, 8)$ lies on the graph. (a) Find the constant of proportionality. (b) Find $y$ when $x = 7$ . (c) Sketch the graph and explain why doubling $x$ doubles $y$ .
39.
Show by counter-example that a relationship may have constant ratio at two points but not be proportional overall.
40.
Investigate: is the perimeter of a square proportional to its side length? Is the area? Justify.