Pack A — Fluency

Mathematics

Year 8 · Ratio and Proportion

Pack A · Fluency

Name: _________________________________

Date: _________________ Class: ___________

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

BronzeQuestions 1–30

1.
Simplify the ratio $16:36$ fully. State the HCF you used.
2.
Simplify the ratio $21:14:35$ fully.
3.
Write 6 hours out of 24 hours as a percentage.
4.
In a bag there are 5 strawberry sweets and 15 orange sweets. Write the ratio of strawberry to orange in its simplest form.
5.
Find 60% of 600 g.
6.
Write $\dfrac{{num}}{{den}}$ as a percentage.
7.
12 apples cost 3.00 chf. Find the cost of 1 apple.
8.
Fill in the missing value so that $3:5$ is equivalent to $\square:20$ .
9.
Increase 100 chf by 25%.
10.
Decrease 125 chf by 25%.
11.
The cost of 5 pens is 7.5 chf. Find the cost of 1 pen.
12.
4 pencils cost 1.20 chf. How much do 10 pencils cost?
13.
Test whether the table $(1, 4), (2, 8), (3, 12)$ represents a proportional relationship.
14.
For the proportional relationship $y = kx$ , find $k$ given that $y = 20$ when $x = 4$ .
15.
Using $y = 6x$ , find $y$ when $x = 9$ .
16.
Using $y = 3x$ , find $x$ when $y = 24$ .
17.
On a graph of $y = 2x$ , find the $y$ -coordinate when $x = -3$ .
18.
A graph of $y$ against $x$ passes through the origin and the point $(2, 10)$ . State the proportionality equation.
19.
In a direct-proportion graph, the line is straight and passes through which point always?
20.
Decide which equation represents direct proportion: $y = 4x$ , $y = x + 4$ , $y = x^2$ .
21.
A map has scale 1 cm : 200 m. Find the real distance for 4 cm on the map.
22.
A scale is 1 : 50. A real length of 25 m is represented by what length on the drawing?
23.
A scale model is 1 : 18. A model car is 25 cm long. Find the length of the real car in metres.
24.
Two figures are similar with linear scale factor 3. The smaller has length 7 cm. Find the length on the larger figure.
25.
Two similar triangles. The smaller has shortest side 6 cm; the larger has shortest side 18 cm. Find the linear scale factor.
26.
On a map, 2 cm represents 1 km. Find the real distance for 8 cm.
27.
Are these triangles definitely similar? They have angles 50°, 60°, 70° and 50°, 60°, 70°. Justify.
28.
A rectangle on a 1 : 50 plan measures 6 cm by 4 cm. Find the real dimensions in metres.
29.
A scale drawing uses 1 cm : 5 m. The real garden is 35 m wide. Find the drawing width.
30.
Two figures are similar. State two properties of similar figures (give a sentence each).

SilverQuestions 31–60

31.
Simplify the ratio 270 mm : 36 cm : 45 cm.
32.
Share 60 chf in the ratio $2:3$ .
33.
A bag of 20 sweets contains 13 raspberry sweets; the rest are pineapple. (a) Write the ratio raspberry : pineapple in simplest form. (b) What percentage of the sweets are pineapple?
34.
12 chocolate bars cost 3.00 chf. How much would 30 chocolate bars cost at the same rate?
35.
12 chocolate bars cost 3.00 chf. How many bars can be bought for 4.50 chf?
36.
A car travels 240 km in 3 hours. Find its average speed in km/h and km/min.
37.
A water tank fills at 6 L/min. How many litres in (a) 12 minutes? (b) 2 hours?
38.
Find the unit rate: 15 chf for 6 kg of fruit.
39.
An item that cost 80 chf is now 60 chf. Find the percentage decrease.
40.
A salary of 4000 chf is increased by 8%. Find the new salary.
41.
Distance is directly proportional to time. In 2 minutes a rollercoaster travels 36 m. Find the distance travelled in 3 minutes and in 5 minutes.
42.
A printer prints 30 pages in 2 minutes. How long to print 75 pages?
43.
A school bus uses 12 L of fuel to travel 96 km. How far can it travel on 18 L at the same rate?
44.
A cake recipe uses 3 eggs to make 6 cakes. How many eggs are needed for 14 cakes?
45.
Test whether $y$ is directly proportional to $x$ given the table: x: 2, 5, 8; y: 6, 15, 24.
46.
On the graph of $y = 7x$ , what is the gradient of the line? Describe how the gradient relates to $k$ .
47.
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 $t = 7$ .
48.
3 kg of apples costs 7.5 chf. How much do 5 kg cost?
49.
Two quantities are directly proportional. When $a = 6$ , $b = 15$ . Find $b$ when $a = 22$ .
50.
The graph of $y$ against $x$ is a straight line through the origin with gradient 2.5. State the equation and find $y$ when $x = 6$ .
51.
On a map with scale 3 cm : 600 m, a rectangle measures 6 cm by 9 cm. What are its real dimensions in metres?
52.
Two similar triangles have a side ratio of 3 cm : 6 cm corresponding sides. The smaller has another side 5 cm. Find the corresponding side of the larger triangle.
53.
A drawing is at 1 : 25. Two parallel lines on the drawing are 4 cm apart. Find the real distance.
54.
Two triangles are similar. The smaller has sides 3, 4, 5 cm. The largest side of the larger triangle is 25 cm. Find the other two sides.
55.
A photo is enlarged by a linear scale factor of 1.5. The original is 10 cm by 15 cm. Find the new dimensions.
56.
Convert the scale "1 cm : 5 m" to the form "1 : k".
57.
Convert the scale "1 : 50 000" to the form "1 cm : ? km".
58.
A model dinosaur is 1 : 25 of life-size. Its leg in the model is 8 cm. Find the real leg length in m.
59.
Two similar rectangles have lengths 4 cm and 6 cm. Their widths are 3 cm and $w$ cm. Find $w$ .
60.
A 30 m tall flagpole casts a 12 m shadow. A nearby tree casts a 5 m shadow. Find the tree height (using similar triangles).

GoldQuestions 61–90

61.
In a 3-leg relay, Stefania ran $\dfrac{2}{5}$ of the course, Jan ran 35% of the course, and René ran the rest. What percentage did René run?
62.
A bike is on sale for 590 chf. This is a 26% discount on the original price. What was the original price? Give your answer to 2 decimal places.
63.
If $x:y = 4:5$ and $x:z = 3:7$ , find the ratio $y:z$ in its simplest form.
64.
Out of 2000 competitors, 1650 finished the race. The ratio of female : male finishers is $20:13$ . Find the number of male finishers and the number of female finishers.
65.
You earn 100 chf a week. Your boss gives you a 25% pay increase. The following week your new pay is cut by 25%. How much do you earn now?
66.
Shop A: 12 bars for 3.00 chf. Shop B: 12 bars for 3.50 chf, plus 3 free bars. Which shop offers better value per bar?
67.
A submarine is at $-80$ m. It rises by 25% of its current depth (towards the surface). What is its new depth?
68.
Which of these ratios are equivalent to $3:4$ ? Pick all that apply.\nA: $9:12$ , B: $15:18$ , C: $21:28$ .
69.
A car travels 270 km in 3 hours. (a) Find the speed in km/h. (b) Convert to m/s.
70.
A price of 100 chf is increased by 25% then decreased by 20%. What is the final price?
71.
For two quantities $x$ and $y$ , $y$ is directly proportional to $x$ . When $x = 4$ , $y = 18$ . (a) Write an equation linking $x$ and $y$ . (b) Find $y$ when $x = 10$ . (c) Find $x$ when $y = 90$ .
72.
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.
73.
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.
74.
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.
75.
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?
76.
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.
77.
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.
78.
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.
79.
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 $y = 30$ .
80.
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?
81.
Two triangles are similar. The smaller has sides 3 cm and 4 cm; the larger has corresponding sides 6 cm and $x$ cm. Find $x$ .
82.
On a floor plan at 1 cm : 50 cm, a room measures 8 cm by 6 cm on the plan. (a) Find the real dimensions. (b) Find the real floor area in $\text{m}^2$ .
83.
Two similar shapes have areas in the ratio 9 : 25. Find the linear scale factor.
84.
Two similar rectangles have a linear scale factor of 2. The smaller has area 30 cm². Find the area of the larger rectangle.
85.
A map has scale 1 : 25 000. A path on the map is 6 cm long. Convert the real path length to (a) m, (b) km.
86.
A flagpole and a nearby tree both cast shadows in sunlight. The pole is 12 m tall and casts a 4 m shadow. The tree casts a 6 m shadow. Find the tree height. Why are the triangles similar?
87.
A drawing at 1 : 200 shows a circle of diameter 5 cm. Find (a) the real diameter; (b) the real area of the circle.
88.
A triangle on a 1 : 50 plan has a base of 8 cm and height 6 cm. (a) Find the real base and height. (b) Find the real area in m².
89.
A picture frame is similar to another with linear scale factor $\tfrac{3}{4}$ . The smaller (with scale factor $\tfrac{3}{4}$ ) has area 36 cm². Find the area of the larger.
90.
On the same map (scale 1 : 50 000), three towns A, B, C are at positions A(0, 0), B(12 cm, 0), C(0, 9 cm). (a) Find the real distance AB and the real distance AC. (b) Find the real distance BC.

PlatinumQuestions 91–120

91.
A long-sleeve shirt has 8 front buttons and 2 cuff buttons; a short-sleeve shirt has 6 front buttons and no cuff buttons. The factory uses 10 times as many front buttons as cuff buttons in total. Find the ratio of long-sleeve to short-sleeve shirts produced.
92.
A price is increased by 20% and then decreased by 25%. Find the single percentage change equivalent to these two changes.
93.
A jacket is reduced by 20% in a January sale, then by a further 25% in February. After both reductions it costs 60 chf. Find the original price.
94.
In a class of 40 pupils, the ratio of pupils who walk to school to those who cycle is $8:5$ . $\dfrac{1}{4}$ of the class take the bus and 4 pupils take the train. How many walk?
95.
In a school the ratio of Year 7 : Year 8 : Year 9 is $5:4:3$ . Year 7 has 100 pupils. (a) How many pupils are in Year 9? (b) What percentage of the school is in Year 8? Give your answer to 1 d.p.
96.
A car depreciates by 15% in its first year and by 10% each subsequent year. If the car cost £20000 new, find its value after 3 years (to nearest £).
97.
Currency: 1 chf = 1.04 EUR. A shopkeeper charges 2% commission on euro purchases. If a tourist exchanges 250 chf, how many euros do they receive?
98.
A 10-litre solution is 60% acid. How much pure water must be added to dilute it to 40% acid?
99.
Three friends share an inheritance in the ratio $2:3:5$ . The eldest receives £1200 more than the youngest. Find each share.
100.
Two prices fell by the same percentage. The first fell from £400 to £320; the second fell from £600 to $x$ . Find $x$ .
101.
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.
102.
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.
103.
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.
104.
Investigate: if $y \propto x$ and $x \propto z$ , show that $y \propto z$ and find the combined constant.
105.
On a graph of $y = kx$ , two points are $(3, p)$ and $(8, p + 20)$ . Find $k$ and $p$ .
106.
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.
107.
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.
108.
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 $y$ .
109.
Show by counter-example that a relationship may have constant ratio at two points but not be proportional overall.
110.
Investigate: is the perimeter of a square proportional to its side length? Is the area? Justify.
111.
Rectangle R1 has real dimensions 1200 m by 1800 m. Rectangle R2 has real dimensions 600 m by 900 m. (a) Find the area scale factor $\text{area}(R1):\text{area}(R2)$ in simplest form. (b) Is this the square of the length scale factor? Justify.
112.
Two similar circles have radii a and b, and circumferences 9.4 cm and c cm. (a) Express c in terms of a and b. (b) Show that their areas are in ratio $a^2 : b^2$ .
113.
Two similar shapes have linear scale factor $k$ . (a) Show that their area ratio is $k^2$ . (b) Two similar glaciers have areas 2.16 km² and 0.24 km². Find the linear scale factor.
114.
A photograph is to be enlarged so the area is doubled. Find the linear scale factor required, exact and to 3 d.p.
115.
A 1 : 200 scale model of a building uses 12 m² of card. Find the surface area of the real building in m².
116.
Triangles ABC and DEF are similar with $A \leftrightarrow D$ etc. Given $AB = 6$ , $AC = 8$ , $DE = 9$ . Find $DF$ .
117.
A 1 : 1000 model city has buildings that take up 50 cm² of model area in total. Find the real area in km².
118.
A triangle has vertices $(0,0)$ , $(6, 0)$ , $(0, 4)$ . Enlarge by factor 2 from the origin. Give the image vertices, and check that the area is multiplied by 4.
119.
The map of a city has scale 1 : 12 500. On the map, a park has area 36 cm². Find the real area of the park in m² and in km².
120.
Two similar buildings: the smaller has volume 8 m³ and surface area 24 m². The larger has linear scale factor 3 relative to the smaller. Find (a) the larger volume, (b) the larger surface area.