**Chapter 9 - Direct Proportion and Inverse Proportion**

**Practice Set 37**

**Practice Set 37**

**1. If 7 kg onions cost 140 rupees, how much must we pay for 12 kg onions?**

**Solution:**

**7 kg onions cost ₹140 **

**∴ 12 kg onions will cost ₹ ?**

** **

**Let the cost of 12 kg onions be ₹x**

** **

**The quantity of onions and their cost are in direct proportion**

** **

**$${\therefore\;\frac7{140}\;=\;\frac{12}x\\\therefore\;7x\;=\;12\;\times\;140\;\\\therefore\;x\;=\;\frac{(12\;\times\;140)}7\\\therefore\;x\;=\;\frac{1680}7\\\therefore\;x\;=\;240}$$**

** **

**Answer:** For 12 kg onions we have to pay ₹240.

** **

**2. If 600 rupees buy 15 bunches of feed, how many will 1280 rupees buy?**

**Solution:**

**600 rupees buy 15 bunches of feed**

**∴ 1280 rupees will buy ? bunches of feed**

** **

**Let the bunches of feed bought for Rs 1280 be x**

** **

**The quantity of feed bought and their cost are in direct proportion**

** **

**$${\therefore\;\frac{600}{15}\;=\;\frac{1280}x\\\therefore\;600x\;=\;1280\;\times\;15\;\\\therefore\;x\;=\;\frac{(1280\;\times\;15)}{600}\\\therefore\;x\;=\;\frac{19200}{600}\\\therefore\;x\;=\;32}$$**

** **

**Answer:** 32 bunches of feed can be bought for Rs 1280.

** **

**3. For 9 cows, 13 kg 500 g of food supplement are required every day. In the same proportion, how much will be needed for 12 cows? **

**Solution**:

**9 cows require 13 kg 500 g of food supplement**

**∴ 12 cows will require ? food supplement**

** **

**Let the food supplement required for 12 cows be x kg**

** **

**The quantity of food supplement required and the number of cows are in direct proportion**

** **

**13kg 500g = 13kg + 0.5kg = 13.5kg**

** **

**$${\therefore\;\frac9{13.5}\;=\;\frac{12}x\\\therefore\;9x\;=\;12\;\times\;13.5\;\\\therefore\;x\;=\;\frac{(12\;\times\;13.5)}9\\\therefore\;x\;=\;\frac{162}9\\\therefore\;x\;=\;18}$$**

** **

**Answer:** The food supplement required for 12 cows is 18 kg.

** **

**4. The cost of 12 quintals of soyabean is 36,000 rupees. How much will 8 quintals cost?**

**Solution:**

**Cost of 12 quintals of soyabean is ₹36,000**

**Cost of 8 quintals of soyabean will be ₹ ?**

** **

**Let the cost of 8 quintals of soyabean be ₹x**

** **

**The quantity of soyabeans and their cost are in direct proportion.**

** **

**$${\therefore\;\frac{12}{36000}\;=\;\frac8x\\\therefore\;12x\;=\;36000\;\times\;8\;\\\therefore\;x\;=\;\frac{(36000\;\times\;8)}{12}\\\therefore\;x\;=\;\frac{288000}{12}\\\therefore\;x\;=\;24000}$$**

** **

**Answer**: The cost of 8 quintals of soyabean is Rs 24000.

** **

**5. Two mobiles cost 16,000 rupees. How much money will be required to buy 13 such mobiles ?**

**Solution:**

**2 mobiles cost ₹16,000**

**13 mobiles will cost ₹ ?**

** **

**Let the cost of 13 mobiles be ₹ x**

** **

**The quantity of mobiles and their cost are in direct proportion**

** **

**$${\therefore\;\frac2{16000}\;=\;\frac{13}x\\\therefore\;2x\;=\;16000\;\times\;13\;\\\therefore\;x\;=\;\frac{(16000\;\times\;13)}2\\\therefore\;x\;=\;\frac{208000}2\\\therefore\;x\;=\;104000}$$**

** **

**Answer:** ₹104000 will be required to buy 13 mobiles.

**Practice Set 38**

**Practice Set 38****1. Five workers take 12 days to weed a field. How many days would 6 workers take? How many would 15 take?**

**Solution:**

**To weed a field;**

**5 workers take 12 days**

**6 workers will take ? days**

** **

**Let 6 workers take x days **

** **

**The number of workers and the time required to weed the field are in inverse proportion**

** **

**∴ 6 × x = 5 × 12**

**∴ x = \( \frac{5 × 12}{6} \) **

**∴ x = \( \frac{60}{6} \) **

**∴ x = 10 days**

** **

**Also, **

**15 workers take ? days**

** **

**Let 15 workers take y days **

** **

**15 × y = 5 × 12**

**∴ y = \( \frac{5 × 12}{15} \) **

**∴ y = \( \frac{60}{15} \) **

**∴ y = 4 days**

** **

**Answer:** 6 workers will take 10 days and 15 workers will take 4 days to weed the field.

** **

**2. Mohanrao took 10 days to finish a book, reading 40 pages every day. How many pages must he read in a day to finish it in 8 days?**

**Solution:**

**To finish a book in 10 days, he reads 40 pages every day**

**To finish a book in 8 days, he should read ? pages every day**

** **

**Let Mohanrao read x pages every day to finish the book in 8 days.**

** **

**The number of pages read per day and the days required to finish the book are in inverse proportion**

** **

**∴ 8 × x = 10 × 40**

**∴ x = \( \frac{10 × 40}{8} \) **

**∴ x = \( \frac{400}{8} \) **

**∴ x = 50**

** **

**Answer:** Mohanrao will have to read 50 pages every day to finish the book in 8 days.

** **

**3. Mary cycles at 6 km per hour. How long will she take to reach her Aunt’s house which is 12 km away? If she cycles at a speed of 4 km/hr, how long would she take ?**

**Solution:**

**Mary cycles at 6 km in 1 hour**

**∴ She cycles 12 km in ? hour**

** **

**Let the time required to cycle 12 km be x hours. **

** **

**The speed of the cycle and the time required to travel the same distance are in inverse proportion**

** **

**∴ 6 × x = 12 × 1**

**∴ x = \( \frac{12}{6} \) **

**∴ x = 2 hours**

** **

**Mary cycles at 6 km/hr speed and reach in 2 hours**

**∴ If she cycles at 4 km/hr speed, shell reach in ? hours**

** **

**Let the time required to cycle 12 km be x hours.**

** **

**4 × x = 6 × 2**

**∴ x = \( \frac{6 × 2}{4} \) **

**∴ x = \( \frac{12}{4} \) **

**∴ x = 3 hours**

**Answer: **Mary will require 2 hours if she is cycling at 6 km/hr and 3 hours if she is cycling at 4 km/hr to reach her Aunt’s house.

** **

**4. The stock of grain in a government warehouse lasts 30 days for 4000 people. How many days will it last for 6000 people ?**

**Solution:**

**For 4000 people, the stock lasts for 30 days**

**∴ For 6000 people, the stock will last for ? days**

** **

**Let the stock of grain last for x days for 6000 people.**

** **

**The number of people and the days for which stock will last are in inverse proportion**

** **

**∴ 6000 × x = 4000 × 30**

**∴ x = \( \frac{4000 × 30}{6000} \) **

**∴ x = \( \frac{120000}{6000} \) **

**∴ x = \( \frac{120}{6} \) **

**∴ x = 20**

** **

**Answer: **The stock of grain will last for 20 days for 6000 people.

**Practice Set 39**

**Practice Set 39****1. Suresh and Ramesh together invested 144000 rupees in the ratio 4:5 and bought a plot of land. After some years they sold it at a profit of 20%. What is the profit each of them got?**

**Solution:**

**Total investment = Rs 144000**

**Profit earned = 20%**

** **

**∴ Total profit = 20% of 144000 **

**∴ Total profit = \( \frac{20}{100} \) × 144000 **

**∴ Total profit = Rs 28800**

** **

**Proportion of investment of Suresh and Ramesh = 4:5**

** **

**Let the profit of Suresh be Rs 4x and that of Ramesh be Rs 5x.**

** **

**4x + 5x = 28800**

**∴ 9x = 28800**

**∴ x = \( \frac{28800}{9} \)**

**∴ x = 3200**

** **

**∴ Suresh’s profit = 4x = 4 × 3200 **

**∴ Suresh’s profit = Rs 12800**

** **

**And,**

**Ramesh’s profit = 5x = 5 × 3200**

**∴ Ramesh’s profit = Rs 16000**

** **

**Answer:** The profit earned by Suresh and Ramesh are Rs 12800 and Rs 16000 respectively.

** **

**2. Virat and Samrat together invested 50000 and 120000 rupees to start a business. They suffered a loss of 20%. How much loss did each of them incur?**

**Solution:**

**Total investment = Rs 50000 + Rs 120000 **

**∴ Total investment = Rs 170000**

**Loss incurred = 20%**

** **

**∴ Total loss = 20% of 170000**

**∴ Total loss = \( \frac{20}{100} \) × 170000 **

**∴ Total loss = Rs 34000**

** **

**Proportion of investment = 50000 : 120000**

**∴ Proportion of investment = 5 : 12**

** **

**Let the loss incurred by Virat be Rs 5x and that by Samrat be Rs 12x.**

** **

**5x + 12x = 34000**

**∴ 17x = 34000**

**∴ x = \( \frac{34000}{17} \)**

**∴ x = 200**

** **

**∴ Virat’s loss = 5x = 5 × 2000 **

**∴ Virat’s loss = Rs 10000**

** **

**And,**

**Samrat’s loss = 12x = 12 × 2000**

**∴ Samrat’s loss = Rs 24000**

** **

**Answer:** The loss incurred by Virat and Samrat are Rs 10000 and Rs 24000 respectively.

** **

**3. Shweta, Piyush and Nachiket together invested 80000 rupees and started a business of selling sheets and towels from Solapur. Shweta’s share of the capital was 30000 rupees and Piyush’s 12000. At the end of the year they had made a profit of 24%. What was Nachiket’s investment and what was his share of the profit?**

**Solution:**

**Total investment = Rs 80000**

** **

**Nachiket’s investment = Total investment – Shweta’s investment – Piyush’s investment**

**∴ Nachiket’s investment = 80000 – 30000 – 12000**

**∴ Nachiket’s investment = 50000 – 12000 **

**∴ Nachiket’s investment = Rs 38000**

** **

**Profit earned = 24%**

**∴ Total profit = 24% of 80000 **

**∴ Total profit = \( \frac{24}{100} \) x 80000 **

**∴ Total profit = Rs 19200**

** **

**Proportion of investment = 30000 : 12000 : 38000**

**∴ Proportion of investment = 15 : 6 : 19**

** **

**Let the profit of Shweta, Piyush and Nachiket be Rs 15x, Rs 6x and Rs 19x respectively**

**15x + 6x + 19x = 19200**

**∴ 40x = 19200**

**∴ x = \( \frac{19200}{40} \)**

**∴ x = 480**

** **

**∴ Nachiket’s profit = 19x = 19 × 480 = Rs 9120**

** **

**Answer:** Nachiket’s investment is Rs 38000 and his profit is Rs 9120.

** **

**4. A and B shared a profit of 24500 rupees in the proportion 3:7. Each of them gave 2% of his share of the profit to the Soldiers’ Welfare Fund. What was the actual amount given to the Fund by each of them?**

**Solution:**

**Proportion of share of A and B = 3:7**

**Let the profits of A and B be Rs 3x and Rs 7x respectively.**

** **

**3x + 7x = 24500**

**∴ 10x = 24500**

**∴ x = \( \frac{24500}{10} \)**

**∴ x = 2450**

** **

**Profit earned by A = 3x = 3 × 2450 **

**∴ Profit earned by A = Rs 7350**

** **

**Amount given by A = 2% of his profit**

**∴ Amount given by A = \( \frac{2}{100} \) × 7350 **

**∴ Amount given by A = Rs 147**

** **

**Profit earned by B = 7x = 7 × 2450 **

**∴ Profit earned by B = Rs 17150**

** **

**Amount given by B = 2% of his profit**

**∴ Amount given by B = \( \frac{2}{100} \) × 17150 **

**∴ Amount given by B = Rs 343**

** **

**Answer:** The amount given by A and B to the Soldiers’ Welfare Fund are Rs 147 and Rs 343 respectively.

** **

**5 . Jaya, Seema, Nikhil and Neelesh put in altogether 360000 rupees to form a partnership, with their investments being in the proportion 3:4:7:6. What was Jaya’s actual share in the capital ? They made a profit of 12%. How much profit did Nikhil make ?**

**Solution:**

**Total investment = Rs 360000**

**Profit earned = 12%**

**∴ Total profit = 12% of 360000**

**∴ Total profit = \( \frac{12}{100} \) × 360000 **

**∴ Total profit = 43200**

** **

**Proportion of investment = 3 : 4 : 7 : 6**

**Let the investment of Jaya, Seema, Nikhil and Neelesh be Rs 3x, Rs 4x, Rs 7x and Rs 6x respectively.**

**3x + 4x + 7x + 6x = 360000**

**∴ 20x = 360000**

**∴ x = \( \frac{360000}{20} \)**

**∴ x = 18000**

** **

**∴ Jaya’s investment = 3x = 3 x 18000 **

**∴ Jaya’s investment = Rs 54000**

** **

**Also, profit made by them is Rs 43200**

**∴ 3x + 4x + 7x + 6x = 43200**

**∴ 20x = 43200**

**∴ x = \( \frac{43200}{20} \)**

**∴ x = 2160**

** **

**∴ Nikhil’s profit = 7x = 7 x 2160 **

**∴ Nikhil’s profit = Rs 15120**

** **

**Answer: **Jaya’s share in the capital was Rs 54000 and the profit made by Nikhil was Rs 15120.