Mathematics questions, chapter 4

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Ratios, Rates and Proportions

40 marks. Use Practice to check each answer as you go, or Exam to answer everything first. You may use a calculator. Show your working for Section B.

Section A: objective questions

12 marks

Choose the correct answer.

Question 1

Which of the following represents 2 weeks to 10 days to 1 week in the form a : b : c?

2 : 10 : 1
7 : 5 : 14
14 : 10 : 7
14 : 10 : 1

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Change everything to days first: 2 weeks = 14 days and 1 week = 7 days, so the ratio is 14 : 10 : 7. 2 : 10 : 1 mixes weeks and days.

For the teacher: Remembering, low difficulty, standard 4.1.1

Textbook p. 76

Question 2

Diagram 1 shows a rectangle divided into 15 equal squares. Which ratio is equivalent to the ratio of the number of shaded squares to the number of unshaded squares?

A rectangle of 5 by 3 equal squares; 6 squares are shaded and 9 are unshaded — Diagram 1

4 : 6
2 : 5
3 : 2
6 : 15

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6 shaded and 9 unshaded: 6 : 9 = 2 : 3 = 4 : 6. 6 : 15 (= 2 : 5) compares shaded squares with all the squares, and 3 : 2 is the wrong way round.

For the teacher: Understanding, low difficulty, standard 4.1.2

Textbook p. 77

Question 3

Express 800 g : 2 kg in its simplest form.

400 : 1
2 : 5
4 : 10
5 : 2

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2 kg = 2 000 g. 800 : 2 000 = 2 : 5 (divide by the HCF, 400). 400 : 1 forgets to change kg to g, and 4 : 10 is not in simplest form.

For the teacher: Applying, low difficulty, standard 4.1.3

Textbook p. 78

Question 4

Express 23 : 56 in its simplest form.

2 : 5
4 : 5
5 : 4
3 : 6

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Multiply both parts by 6, the LCM of 3 and 6: 23 × 6 : 56 × 6 = 4 : 5. 2 : 5 just takes the numerators, which is wrong because the denominators are different.

For the teacher: Applying, medium difficulty, standard 4.1.3

Textbook p. 79

Question 5

Which of the following is a rate?

3 kg to 5 kg
15 cm : 20 cm
4 boys to 5 girls
RM12 for 3 kg

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A rate compares two quantities with different units. RM12 for 3 kg compares money (RM) with mass (kg). The others compare quantities of the same kind, so they are ratios.

For the teacher: Remembering, low difficulty, standard 4.2.1

Textbook p. 81

Question 6

A cheetah runs at 20 m/s. What is this speed in km/h?

5.56 km/h
1 200 km/h
72 km/h
7.2 km/h

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In 1 hour (3 600 s) it runs 20 × 3 600 = 72 000 m = 72 km, so 72 km/h. 5.56 comes from dividing the wrong way, and 1 200 uses 60 s instead of 3 600 s.

For the teacher: Applying, medium difficulty, standard 4.2.1

Textbook p. 82

Question 7

4 pens cost RM6. At the same rate, what is the cost of 10 pens?

RM15
RM12
RM2.40
RM60

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1 pen costs RM6 ÷ 4 = RM1.50, so 10 pens cost 10 × RM1.50 = RM15. RM2.40 comes from 4 × 6 ÷ 10 (the fraction upside down).

For the teacher: Applying, low difficulty, standard 4.3.2

Textbook p. 85

Question 8

Given p : q = 3 : 4 and q : r = 6 : 5, find p : q : r.

3 : 4 : 5
3 : 6 : 5
18 : 24 : 15
9 : 12 : 10

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Make q the same: the LCM of 4 and 6 is 12. 3 : 4 = 9 : 12 and 6 : 5 = 12 : 10, so p : q : r = 9 : 12 : 10. 3 : 4 : 5 just joins the ratios without making q equal.

For the teacher: Analysing, medium difficulty, standard 4.4.1

Textbook p. 87

Question 9

A bag has red, blue and yellow beads in the ratio 2 : 3 : 4. There are 24 blue beads. How many beads are there altogether?

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Blue is 3 parts: 1 part = 24 ÷ 3 = 8. Total = (2 + 3 + 4) × 8 = 9 × 8 = 72. 216 = 24 × 9 forgets to find 1 part; 32 is only the yellow beads.

For the teacher: Applying, medium difficulty, standard 4.4.2

Textbook p. 88

Question 10

A machine prints 450 pages in 6 minutes. At the same rate, how many pages does it print in 20 minutes?

9 000
1 500
135
470

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Rate = 450 ÷ 6 = 75 pages per minute, so 75 × 20 = 1 500 pages. 135 comes from 450 × 6 ÷ 20 (the wrong way round).

For the teacher: Applying, medium difficulty, standard 4.4.3

Textbook p. 90

Question 11

Diagram 2 shows 20 black and white counters. What percentage of the counters are white?

20 counters in 4 rows of 5: 8 are black and 12 are white — Diagram 2

40%
66.7%
12%
60%

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There are 12 white counters out of 20: 1220 × 100% = 60%. 40% is the black counters, and 66.7% compares black with white (8 : 12) instead of with the whole.

For the teacher: Understanding, medium difficulty, standard 4.5.1

Textbook p. 94

Question 12

The original price of a shirt is RM80. It is sold for RM60. What is the percentage discount?

25%
75%
20%
33.3%

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6080 = p100 gives p = 75, so the selling price is 75% of the original price. Discount = 100% − 75% = 25%. 75% is the selling price, and 33.3% divides RM20 by the selling price instead of the original price.

For the teacher: Applying, high difficulty, standard 4.5.2

Textbook p. 96

Section B: structured questions

28 marks

Answer all questions. Show your working.

Question 13

Diagram 3 shows three squares, A, B and C.

Three squares: A has sides of 4 cm, B has sides of 6 cm and C has sides of 10 cm — Diagram 3

[1 mark] (a)Write the ratio of the side length of A to B to C in its simplest form.

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4 : 6 : 10 = 2 : 3 : 5 [1]1

Tick each point you got.

[1 mark] (b)Write the ratio of the perimeter of A to B to C in its simplest form.

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16 : 24 : 40 = 2 : 3 : 5 [1]1

Tick each point you got.

[2 marks] (c)Write the ratio of the area of A to B to C in its simplest form.

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16 : 36 : 100 [1]1
= 4 : 9 : 25 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 4.1.3

Textbook p. 79

Question 14

Diagram 4 shows the prices of the same kind of grapes at two shops.

Price tags: Shop P, RM9.60 for 1.2 kg; Shop Q, RM3.90 per 500 g — Diagram 4

[1 mark] (a)Find the price per kg at Shop P.

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RM9.60 ÷ 1.2 = RM8.00 per kg [1]1

Tick each point you got.

[1 mark] (b)Find the price per kg at Shop Q.

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1 kg = 2 × 500 g: RM3.90 × 2 = RM7.80 per kg [1]1

Tick each point you got.

[1 mark] (c)Which shop sells the grapes at a cheaper price? Give a reason.

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Shop Q, because RM7.80 per kg is less than RM8.00 per kg [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 4.2.1

Textbook p. 82

Question 15

Marine biologists catch 40 seahorses in a river estuary, tag them and release them. A month later they catch 35 seahorses, and 7 of them are tagged.

Photo of a seahorse, Hippocampus kuda — Diagram 5

[1 mark] (a)Using x for the number of seahorses in the estuary, write a proportion for this situation.

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735 = 40x (or an equivalent proportion) [1]1

Tick each point you got.

[2 marks] (b)Estimate the number of seahorses in the estuary.

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7 × x = 40 × 35, or 15 = 40x [1]1
x = 200 seahorses [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 4.4.4

Textbook p. 91

Question 16

Diagram 6 shows a wire PS of length 30 cm. It is cut into three parts at Q and R. Given that PQ : QR = 4 : 3 and RS = 9 cm.

Wire P, Q, R, S of total length 30 cm; RS is 9 cm — Diagram 6

[1 mark] (a)Find the length of PR.

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PR = 30 − 9 = 21 cm [1]1

Tick each point you got.

[2 marks] (b)Find the lengths of PQ and QR.

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1 part = 21 ÷ (4 + 3) = 3 cm [1]1
PQ = 4 × 3 = 12 cm and QR = 3 × 3 = 9 cm [1]1

Tick each point you got.

[1 mark] (c)Find PQ : QR : RS in its simplest form.

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12 : 9 : 9 = 4 : 3 : 3 [1]1

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 4.4.2

Textbook p. 89

Question 17

Diagram 7 shows the mass of paper collected for recycling by 1 Amanah and 1 Bestari. The ratio of the mass collected by 1 Bestari to the mass collected by 1 Cekap is 4 : 5.

Bar chart: 1 Amanah collected 24 kg and 1 Bestari collected 36 kg of paper — Diagram 7

[1 mark] (a)Write the ratio of the mass collected by 1 Amanah to 1 Bestari in its simplest form.

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24 : 36 = 2 : 3 [1]1

Tick each point you got.

[2 marks] (b)Find the ratio of the mass collected by 1 Amanah to 1 Bestari to 1 Cekap.

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LCM of 3 and 4 is 12: 2 : 3 = 8 : 12 and 4 : 5 = 12 : 15 [1]1
8 : 12 : 15 [1]1

Tick each point you got.

[1 mark] (c)Find the mass of paper collected by 1 Cekap.

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36 ÷ 4 × 5 = 45 kg [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 4.4.1

Textbook p. 87

Question 18

Diagram 8 shows an empty water tank in the shape of a cuboid. Water flows into the tank at a rate of 3.2 litres per minute. (1 litre = 1 000 cm³)

Cuboid water tank 80 cm long, 50 cm wide and 60 cm high — Diagram 8

[2 marks] (a)Find the volume of the tank in litres.

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80 × 50 × 60 = 240 000 cm³ [1]1
240 000 ÷ 1 000 = 240 litres [1]1

Tick each point you got.

[1 mark] (b)How long does it take to fill the tank completely?

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240 ÷ 3.2 = 75 minutes [1]1

Tick each point you got.

[1 mark] (c)The tap is turned on at 0715 hours. At what time is the tank full?

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75 minutes = 1 hour 15 minutes; 0715 + 1 h 15 min = 0830 hours [1]1

Tick each point you got.

For the teacher: Analysing, high difficulty, standard 4.4.3

Textbook p. 101

Question 19

The ratio of the number of boys to the number of girls in a choir is 3 : 5. There are 24 boys. Aina wants to find the number of girls. Diagram 9 shows her working.

Aina's working. Line 1: Total parts = 3 + 5 = 8. Line 2: 1 part = 24 ÷ 8 = 3. Line 3: Girls = 5 × 3 = 15 — Diagram 9

[1 mark] (a)Aina's answer is wrong. In which line did she make a mistake? Explain the mistake.

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Line 2 (or Line 1 with the reason that only the boys' 3 parts should be used): the 24 boys are 3 parts, not 8 parts, so she should divide 24 by 3 [1]1

Tick each point you got.

[2 marks] (b)Find the correct number of girls.

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1 part = 24 ÷ 3 = 8 [1]1
Girls = 5 × 8 = 40 [1]1

Tick each point you got.

For the teacher: Evaluating, high difficulty, standard 4.4.2

Textbook p. 88

Question 20

Puan Siti wants to make 1.5 litres of fruit punch from orange juice, mango juice and soda. Her conditions are: orange juice must be exactly 50% of the punch, and there must be more mango juice than soda.

[1 mark] (a)Suggest a ratio of orange juice to mango juice to soda that meets both conditions. Explain why orange juice is 50% of the punch.

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A ratio a : b : c with a = b + c and b > c, e.g. 3 : 2 : 1, because orange is 3 of the 6 parts = 36 = 50% [1]1

Tick each point you got.

[2 marks] (b)Using your ratio, calculate the volume of each drink, in ml, needed for 1.5 litres of punch.

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Other ratios are accepted, e.g. 5 : 3 : 2 gives 1 part = 150 ml: orange 750 ml, mango 450 ml, soda 300 ml. Orange juice is always 750 ml, half of 1 500 ml.

For 3 : 2 : 1: 1 part = 1 500 ml ÷ 6 = 250 ml [1]1
Orange 750 ml, mango 500 ml, soda 250 ml [1]1

Tick each point you got.

For the teacher: Creating, high difficulty, standard 4.5.3

Textbook p. 97

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