Proportional Reasoning 1 — Question 6

(a) We consider one set of patterns in the given wall. Number of grey bricks in one set of pattern = 2 + 3 + 4 = 9 Number of coloured bricks in one set of pattern = 3 + 2 + 1 = 6 ∴ Ratio of grey bricks to coloured bricks = 9 : 6 We have 9 : 6 = 3 : 2 ∴ Ratio in the simplest form = 3 : 2 (b) We use one set of patterns on the given wall Number of grey bricks in one set of pattern = 3 + 2 + 2 + 2 + 2 + 2 + 3 = 16 Number of coloured bricks in one set of pattern = 1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + 1 = 1 + 2 + 2 + 2 + 2 + 2 + 1 = 12 ∴ Ratio of grey bricks to coloured bricks = 16 : 12 We have 16 : 12 = 4 : 3 ∴ Ratio in the simplest form = 4 : 3. Trairasika – The Rule of Three Example 8. For the mid-day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted? The ratio of the number of students to the amount of rice needs to be proportional. So, 120 : 15 :: 80 : ? What is the factor of change in the first term? (Page 167) Solution: For 120 students, the rice required is 15 kg. Let x kg of rice be required for 80 students. ∴ Ratios 120 : 15 and 80 : x are in proportion. ∴ 120 : 15 :: 80 : x ⇒ \(\frac{120}{15}=\frac{80}{x}\) ⇒ x = 10 ∴ 10 kg of rice is required. Also, factor of change in the first term = \(\frac{80}{120}=\frac{2}{3}\) Alternative Method: Factor of change in second term = \(\frac {x}{15}\) Since the quality of food is the same, we have \(\frac{2}{3}=\frac{x}{15}\) ⇒ 3x = 30 ⇒ x = 10 ∴ 10 kg of rice is required. Example 9. (i) A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours? If it continues at the same speed, the ratio of the time taken should be proportional to the ratio of the distance covered. (ii) 150 : 90 :: 4 : x Is this the right way to formulate the question? (iii) How can you find the distance covered in 240 minutes? (Page 169) Solution: (i) We have, 4 hours = 4 × 60 = 240 minutes In 150 minutes, the distance covered = 90 km Let x km be covered in 4 hours, i.e., in 240 minutes. ∴ The ratios 150 : 90 and 240 : x are in proportion. ∴ 150 : 90 :: 240 : x (ii) Since units must be the same in comparing ratios, the given proportion 150 : 90 :: 4 : ? is meaningless. We have 4 hours = 240 minutes ∴ The proportion 150 : 90 :: 240 : ? is correct. (iii) We have 150 : 90 :: 240 : x ⇒ \(\frac{150}{90}=\frac{240}{x}\) ⇒ \(\frac{5}{3}=\frac{240}{x}\) ⇒ 5x = 3 × 240 ⇒ x = 144 ∴ Distance covered in 4 hours = 144 km. Example 10. A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹ 200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹ 800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive? Why? (Page 169) Solution: We have 1 kg = 1000 g In Himachal Pradesh, 200 g of tea costs ₹ 200. ∴ Ratio of weight to price in Himachal Prdesh = 200 : 200 = 1 : 1 In Meghalaya, 1000 g tea costs ₹ 800. ∴ Ratio of weight to price in Meghalaya = 1000 : 800 = 5 : 4 The ratios 1 : 1 and 5 : 4 are not proportional, because \(\frac{1}{1} \neq \frac{5}{4}\) Price of 200 g tea in Himachal Pradesh = ₹ 200 Price of 1000 g tea in Himachal Pradesh = \(\frac {200}{200}\) × 1000 = ₹ 1000 Since 1000 > 800, tea is more expensive in Himachal Pradesh. Figure It Out (Pages 170-171)