Ratio and Proportion

We often compare quantities, such as the heights of two friends or the costs of two items. Comparing by subtraction tells how much more one is, but comparing by division tells how many times one is of the other. The comparison of two quantities of the same kind by division is called a ratio, written a : b. In this chapter you will learn that both quantities must be in the same unit, so 50 paise : Rs 1 becomes 50 : 100, or 1 : 2 in simplest form. You will form equivalent ratios by multiplying or dividing both terms by the same number. The chapter then explains proportion: four quantities are in proportion when the first two have the same ratio as the last two, as in 3 : 5 :: 9 : 15, where the product of the extremes equals the product of the means. Finally, you will use the unitary method, finding the value of one unit first, to solve everyday problems.

Detailed Worksheet: Ratio and Proportion Section A - Definitions (10 marks) 1. What is a ratio? Why must the two quantities in a ratio be in the same unit? (2 marks) 2. What are equivalent ratios? Write two ratios equivalent to 4 : 7. (2 marks) 3. Define proportion. In 2 : 3 :: 8 : 12, name the extreme terms and the middle terms. (2 marks) 4. What is the unitary method? Give one example where it is used in daily life. (2 marks) 5. When is a ratio said to be in its simplest form? Write 24 : 36 in simplest form. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the ratio in simplest form: (i) 81 to 108 (ii) 98 to 63 (iii) 33 km to 121 km (iv) 30 minutes to 1.5 hours (v) 40 cm to 1.5 m (vi) 55 paise to Rs 1. (3 marks) 7. Determine whether the following are in proportion: (i) 15, 45, 40, 120 (ii) 33, 121, 9, 96 (iii) 24, 28, 36, 48. Show the working. (3 marks) 8. Divide Rs 60 between Seema and Sudha in the ratio 1 : 2. Also divide 20 pens between Sheela and Sangeeta in the ratio 3 : 2. (3 marks) 9. If the cost of 7 m of cloth is Rs 1470, find the cost of 5 m of cloth. Also find how much cloth can be bought for Rs 2730. (3 marks) 10. A car travels 90 km in 2.5 hours. (i) How far will it travel in 5 hours at the same speed? (ii) How long will it take to travel 252 km? (iii) Find the ratio of distance travelled in 5 hours to that in 2.5 hours. (3 marks) Section C - Diagrams (10 marks) 11. Draw a bar diagram to show that 3 : 4 and 6 : 8 are equivalent ratios. Use it to explain how equivalent ratios are formed, and write one more ratio equivalent to them. (4 marks) 12. Draw a rectangle with length 12 cm and breadth 4 cm (or a scaled sketch). Find the ratio of length to breadth and of breadth to perimeter, labelling the sides. (3 marks) 13. Make a neat chart showing the cost of 1, 2, 3, 4 and 5 kg of rice when 3 kg costs Rs 180. Use it to show that the cost increases in the same ratio as the quantity. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. In a class of 30 students, 12 are girls. (i) Find the ratio of girls to boys. (ii) Find the ratio of boys to total students. (iii) If 6 new girls join the class, find the new ratio of girls to boys. (iv) Explain why 12 : 18 and 2 : 3 express the same comparison. (5 marks) 15. Rahim says, "Since 5 - 3 = 2 and 7 - 5 = 2, the ratios 3 : 5 and 5 : 7 are equal." (i) Check by finding the ratios as fractions. (ii) Is Rahim right? (iii) Explain the difference between comparing by subtraction and by division. (iv) Give an example where comparing by division is more meaningful. (5 marks) 16. Three different shops sell the same brand of biscuits: Shop P sells 3 packets for Rs 75, Shop Q sells 5 packets for Rs 120 and Shop R sells 4 packets for Rs 104. (i) Find the cost of one packet at each shop. (ii) Which shop gives the best value? (iii) How much would you save on 20 packets by buying from the cheapest instead of the costliest shop? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Convert quantities to the same unit before forming a ratio and write every ratio in its simplest form.