Chapter 9 - Direct Proportion and Inverse Proportion
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
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
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.