Direct and Inverse Proportions

When the number of pens bought increases, the total cost increases in the same ratio; when more workers are put on a job, the number of days needed decreases. In this chapter you will learn that two quantities x and y are in direct proportion if x/y = k stays constant, so that y increases when x increases in the same ratio. You will use this idea to solve problems on cost and quantity, distance and time at a uniform speed, map scales and models, and you will see that the graph of a direct proportion is a straight line through the origin. You will also learn that x and y are in inverse proportion if xy = k stays constant, so that when one quantity is doubled the other is halved. Speed and time for a fixed distance, workers and days for a fixed job, and the number of pipes and the time taken to fill a tank are typical examples. You will solve such problems by the unitary method.

Detailed Worksheet: Direct and Inverse Proportions Section A - Definitions (10 marks) 1. When are two quantities said to be in direct proportion? Give one example from daily life. (2 marks) 2. When are two quantities said to be in inverse proportion? Give one example from daily life. (2 marks) 3. What is meant by the constant of proportionality? Write the relation between x, y and k for direct and for inverse proportion. (2 marks) 4. State whether the following are in direct proportion, inverse proportion or neither: (i) the number of workers and the days taken to finish a job (ii) the distance travelled at a constant speed and the time taken (iii) the age of a child and its height. (2 marks) 5. If x and y are in direct proportion, what happens to y when x is doubled? What happens to y when x is doubled if they are in inverse proportion? (2 marks) Section B - Calculations and Applications (15 marks) 6. The cost of 5 m of a particular cloth is Rs 210. Find the cost of 2 m, 4 m, 10 m and 13 m of the same cloth. (3 marks) 7. On a map, 1 cm represents 18 km. (i) Two cities are 7.5 cm apart on the map; find the actual distance. (ii) Two towns are 81 km apart; how far apart will they be on the map? (3 marks) 8. Six pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only five pipes of the same type are used? (3 marks) 9. A car travelling at 60 km/h takes 2 hours to cover a distance. How long will it take to cover the same distance at 80 km/h? (3 marks) 10. In the pairs (x, y) = (3, 36), (5, 60), (7, ?) and (12, ?), x and y are in direct proportion. Find the missing values. Show that the pairs (2, 40), (4, 20) and (8, 10) are in inverse proportion. (3 marks) Section C - Diagrams (10 marks) 11. Pens cost Rs 12 each. Write the cost of 1, 2, 3, 4 and 5 pens and draw a graph of the number of pens against the cost. What shape is the graph and through which point does it pass? (4 marks) 12. A job takes 24 days for 1 worker. Find the days needed by 2, 3, 4, 6 and 8 workers, and plot the number of workers against the number of days. Describe the shape of the graph. (3 marks) 13. A classroom measures 6 m by 4 m. Make a scale drawing of the floor using the scale 1 cm represents 50 cm, and show the scale on your drawing. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A garrison of 120 soldiers has food for 30 days. After 5 days, 30 more soldiers join them. For how many more days will the remaining food last? Explain why this is a problem of inverse proportion. (5 marks) 15. Fifteen workers can build a wall in 48 hours. (i) How many workers are needed to build the same wall in 30 hours? (ii) If the wall were twice as long, how many hours would the workers found in (i) take? Explain which quantities are in direct and which in inverse proportion. (5 marks) 16. A vertical pole 5 m 60 cm high casts a shadow 3 m 20 cm long. At the same time, find (i) the length of the shadow cast by another pole 10 m 50 cm high (ii) the height of a pole that casts a shadow 5 m long. Why is it important that the shadows are measured at the same time? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. State clearly whether each problem involves direct or inverse proportion before solving it, and draw graphs on graph paper.