Secondary 2 Maths sample questions with answers

Map Scales, Direct and Inverse Proportion

1. 1On a map, 5 cm represents 2 km. Express the scale of the map in the form 1 : n.

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   Answer: 1 : 40 000

   1. 2 km = 200 000 cm
   2. 5 : 200 000 = 1 : 40 000

2. 2A map is drawn to a scale of 1 : 60 000. The distance between two schools on the map is 7.5 cm. Find the actual distance between the schools, in kilometres.

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   Answer: 4.5 km

   1. 7.5 × 60 000 = 450 000 cm
   2. 450 000 cm = 4.5 km

3. 3A map has a scale of 1 : 150 000. The actual distance between two towns is 18 km. Find the distance between the towns on the map.

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   Answer: 12 cm

   1. 1 cm represents 150 000 cm = 1.5 km
   2. 18 ÷ 1.5 = 12 cm

4. 4It is given that y is directly proportional to x, and that y = 21 when x = 6. Find the value of y when x = 10.

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   Answer: 35

   1. y = kx, so 21 = 6k and k = 3.5
   2. y = 3.5x
   3. When x = 10, y = 3.5 × 10 = 35

5. 5It is given that y is inversely proportional to x, and that y = 9 when x = 4. Find the value of x when y = 12.

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   Answer: 3

   1. y = k/x, so 9 = k/4 and k = 36
   2. xy = 36
   3. When y = 12, x = 36 ÷ 12 = 3

6. 6A stall sells 4 bottles of juice for $5.60 and 10 bottles of the same juice for $13.50. Is the cost directly proportional to the number of bottles bought? Justify your answer.

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   Answer: No. The cost per bottle is not the same.

   1. Cost of 1 bottle: $5.60 ÷ 4 = $1.40
   2. Cost of 1 bottle: $13.50 ÷ 10 = $1.35
   3. The cost per bottle is not the same, so the cost is not directly proportional to the number of bottles.

7. 7It takes 6 cleaners 15 hours to clean an exhibition hall. How many hours would 9 cleaners take to clean the same hall, if they all work at the same rate?

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   Answer: 10 hours

   1. 6 × 15 = 90 cleaner-hours
   2. 90 ÷ 9 = 10 hours

8. 8It is given that y is inversely proportional to x^2, and that y = 6 when x = 4. Find the value of y when x = 3.

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   Answer: 32/3 = 10 2/3

   1. y = k/x^2, so 6 = k/16 and k = 96
   2. When x = 3, y = 96/9 = 32/3 = 10 2/3

9. 9On a map, 6 cm represents 1.5 km. (a) Express the scale of the map in the form 1 : n. (b) The distance between two bus interchanges is 4.2 km. Find the distance between them on the map. (c) A canal is 9.4 cm long on the map. Find its actual length in kilometres.

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   Answer: (a) 1 : 25 000 (b) 16.8 cm (c) 2.35 km

   1. (a) 1.5 km = 1.5 × 100 000 = 150 000 cm
   2. 6 : 150 000 = 1 : 25 000 (divide both by 6)
   3. (b) 1 cm represents 25 000 cm = 0.25 km
   4. Map distance = 4.2 ÷ 0.25 = 16.8 cm
   5. (c) Actual length = 9.4 × 0.25 = 2.35 km

10. 10A map has a scale of 1 : 80 000. Find the actual distance, in kilometres, represented by 5.5 cm on the map.

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    Answer: 4.4 km

    1. 5.5 × 80 000 = 440 000 cm
    2. 440 000 cm = 4.4 km

11. 11On a map, 2 cm^2 represents 18 km^2. Express the scale of the map in the form 1 : n.

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    Answer: 1 : 300 000

    1. 1 cm^2 represents 18 ÷ 2 = 9 km^2
    2. 1 cm represents sqrt(9) = 3 km = 300 000 cm
    3. The scale is 1 : 300 000

12. 12A model of a building is made to a scale of 1 : 200. The roof of the model has an area of 150 cm^2. Find the actual area of the roof in m^2.

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    Answer: 600 m^2

    1. 1 cm represents 200 cm = 2 m, so 1 cm^2 represents 4 m^2
    2. Actual area = 150 × 4 = 600 m^2