Pack B — Fluency

Mathematics

Year 8 · Ratio and Proportion

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

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

BronzeQuestions 1–30

1.
Simplify the ratio $18:24$ fully. State the HCF you used.
2.
Simplify the ratio $12:18:30$ fully.
3.
Write 9 hours out of 30 hours as a percentage.
4.
In a bag there are 8 strawberry sweets and 12 orange sweets. Write the ratio of strawberry to orange in its simplest form.
5.
Find 40% of 500 g.
6.
Write $\dfrac{{num}}{{den}}$ as a percentage.
7.
8 apples cost 2.40 chf. Find the cost of 1 apple.
8.
Fill in the missing value so that $2:7$ is equivalent to $\square:28$ .
9.
Increase 80 chf by 15%.
10.
Decrease 90 chf by 20%.
11.
The cost of 8 pens is 12 chf. Find the cost of 1 pen.
12.
4 pencils cost 1.20 chf. How much do 15 pencils cost?
13.
Test whether the table $(1, 3), (2, 9), (3, 27)$ represents a proportional relationship.
14.
For the proportional relationship $y = kx$ , find $k$ given that $y = 35$ when $x = 5$ .
15.
Using $y = 4x$ , find $y$ when $x = 12$ .
16.
Using $y = 5x$ , find $x$ when $y = 40$ .
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 : 150 m. Find the real distance for 6 cm on the map.
22.
A scale is 1 : 100. A real length of 40 m is represented by what length on the drawing?
23.
A scale model is 1 : 24. A model car is 20 cm long. Find the length of the real car in metres.
24.
Two figures are similar with linear scale factor 2.5. The smaller has length 6 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 14 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 210 mm : 14 cm : 35 cm.
32.
Share 80 chf in the ratio $3:5$ .
33.
A bag of 25 sweets contains 15 raspberry sweets; the rest are pineapple. (a) Write the ratio raspberry : pineapple in simplest form. (b) What percentage of the sweets are pineapple?
34.
8 chocolate bars cost 2.40 chf. How much would 20 chocolate bars cost at the same rate?
35.
8 chocolate bars cost 2.40 chf. How many bars can be bought for 6.00 chf?
36.
A car travels 360 km in 4 hours. Find its average speed in km/h and km/min.
37.
A water tank fills at 8 L/min. How many litres in (a) 12 minutes? (b) 2 hours?
38.
Find the unit rate: 24 chf for 8 kg of fruit.
39.
An item that cost 200 chf is now 150 chf. Find the percentage decrease.
40.
A salary of 5500 chf is increased by 6%. Find the new salary.
41.
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.
42.
A printer prints 24 pages in 3 minutes. How long to print 60 pages?
43.
A school bus uses 15 L of fuel to travel 180 km. How far can it travel on 25 L at the same rate?
44.
A cake recipe uses 2 eggs to make 6 cakes. How many eggs are needed for 15 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 $y = 3x$ , 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.
4 kg of apples costs 10 chf. How much do 7 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 0.4. State the equation and find $y$ when $x = 15$ .
51.
On a map with scale 3 cm : 600 m, a rectangle measures 4 cm by 7 cm. What are its real dimensions in metres?
52.
Two similar triangles have a side ratio of 4 cm : 10 cm corresponding sides. The smaller has another side 7 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 28% of the course, and René ran the rest. What percentage did René run?
62.
A bike is on sale for 480 chf. This is a 20% discount on the original price. What was the original price? Give your answer to 2 decimal places.
63.
If $x:y = 2:3$ and $x:z = 4:5$ , find the ratio $y:z$ in its simplest form.
64.
Out of 1200 competitors, 900 finished the race. The ratio of female : male finishers is $5:4$ . Find the number of male finishers and the number of female finishers.
65.
You earn 200 chf a week. Your boss gives you a 20% pay increase. The following week your new pay is cut by 10%. How much do you earn now?
66.
Shop A: 10 bars for 2.50 chf. Shop B: 12 bars for 2.40 chf, plus 3 free bars. Which shop offers better value per bar?
67.
A submarine is at $-120$ m. It rises by 20% of its current depth (towards the surface). What is its new depth?
68.
Which of these ratios are equivalent to $2:5$ ? Pick all that apply.\nA: $6:15$ , B: $10:20$ , C: $8:20$ .
69.
A car travels 144 km in 2 hours. (a) Find the speed in km/h. (b) Convert to m/s.
70.
A price of 80 chf is increased by 50% then decreased by 20%. What is the final price?
71.
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 $y = 91$ .
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 4 cm and 5 cm; the larger has corresponding sides 12 cm and $x$ cm. Find $x$ .
82.
On a floor plan at 1 cm : 100 cm, a room measures 5 cm by 4 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 6 front buttons and 1 cuff buttons; a short-sleeve shirt has 4 front buttons and no cuff buttons. The factory uses 8 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 50% and then decreased by 20%. Find the single percentage change equivalent to these two changes.
93.
A jacket is reduced by 30% in a January sale, then by a further 10% in February. After both reductions it costs 63 chf. Find the original price.
94.
In a class of 60 pupils, the ratio of pupils who walk to school to those who cycle is $4:1$ . $\dfrac{1}{4}$ of the class take the bus and 5 pupils take the train. How many walk?
95.
In a school the ratio of Year 7 : Year 8 : Year 9 is $7:6:5$ . Year 7 has 140 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 £30000 new, find its value after 3 years (to nearest £).
97.
Currency: 1 chf = 0.98 EUR. A shopkeeper charges 2% commission on euro purchases. If a tourist exchanges 300 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 £1800 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 1000 m by 1500 m. Rectangle R2 has real dimensions 400 m by 600 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.