Direct Proportion

Question 1:

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

ANSWER:

Let us suppose the cost of 12 kg onions is x rupees.
The number of onions and their cost vary in direct proportion.
∴7140=12x⇒x=12×1407$$\begin{gathered} \therefore \frac{7}{140}=\frac{12}{x} \\ \Rightarrow x=\frac{12\times 140}{7} \end{gathered}$$
= 240 rupees
Hence, the cost of 12 kg onions is 240 rupees.

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Question 2:

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

ANSWER:

Let us suppose x bunches of feed can be bought in 1280 rupees.
The number of bunches of feed and their cost vary in direct proportion.
∴15600=x1280⇒x=15×1280600$$\begin{gathered} \therefore \frac{15}{600}=\frac{x}{1280} \\ \Rightarrow x=\frac{15\times 1280}{600} \end{gathered}$$
= 32
Hence, 32 bunches of feed can be bought in 1280 rupees.

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Question 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?

ANSWER:

Let us suppose x kg of food supplement required for 12 cows.
The quantity of food supplement and the number of cows vary in direct proportion.
∴913.5=12x⇒x=12×13.59$$\begin{gathered} \therefore \frac{9}{13.5}=\frac{12}{x} \\ \Rightarrow x=\frac{12\times 13.5}{9} \end{gathered}$$
= 18 kg
Hence, 18 kg of food supplement required for 12 cows.

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Question 4:

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

ANSWER:

Let us suppose the cost of 8 quintals of soyabean is x rupees.
The quantity of soyabeans and their cost vary in direct proportion. 
∴123600=8x⇒x=8×360012$$\begin{gathered} \therefore \frac{12}{3600}=\frac{8}{x} \\ \Rightarrow x=\frac{8\times 3600}{12} \end{gathered}$$
= 24000 rupees
Hence, the cost of 8 quintals of soyabean is 24000 rupees.

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Question 5:

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

ANSWER:

Let us suppose the cost of 13 mobiles is x rupees.
The number of mobiles and their cost vary in direct proportion.
∴216000=13x⇒x=13×160002$$\begin{gathered} \therefore \frac{2}{16000}=\frac{13}{x} \\ \Rightarrow x=\frac{13\times 16000}{2} \end{gathered}$$
= 104000 rupees
Hence, the cost of 13 mobiles is 104000 rupees.

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Question 1:

Five workers take 12 days to weed a field. How many days would 6 workers take ? How many would 15 take ?

ANSWER:

Let us suppose 6 workers will take x days to weed a field. 
As the number of workers increases, the number of days decreases.
So, the number of workers and number of days are in inverse proportion. 
∴ 5 × 12 = 6 × x
⇒x=606$\Rightarrow x=\frac{60}{6}$
⇒ x = 10 days
Let us suppose 15 workers will take y days to weed a field. 
∴ 5 × 12 = 15 × y
⇒y=6015$\Rightarrow y=\frac{60}{15}$
⇒ y = 4 days
Hence, 6 workers will take 10 days, while 15 workers will take 4 days to weed a field.

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Question 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?

ANSWER:

Let us suppose Mohanrao will have to read x pages every day to finish the book in 8 days.
As the number of days decreases, the number of pages increases.
So, the number of days and number of pages are in inverse proportion. 
∴ 10 × 40 = 8 × x
⇒x=400/8$\Rightarrow x=\frac{\mathrm{400/}}{8}$
⇒ x = 50 pages
Hence, Mohanrao will have to read 50 pages every day to finish the book in 8 days

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Question 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 ?

ANSWER:

Given:
Case -1:
Speed = 6 km/hr
Distance = 12 km
∴Time=DistanceSpeed=126$$\begin{gathered} \therefore \mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}} \\ =\frac{12}{6} \end{gathered}$$
= 2 hours
Case -2:
​Speed = 4 km/hr
Distance = 12 km
∴Time=DistanceSpeed=124$$\begin{gathered} \therefore \mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}} \\ =\frac{12}{4} \end{gathered}$$
= 3 hours
Hence, if the speed of cycle is 6 km/hr then, Marry will take 2 hours and if the speed of cycle is 4 km/hr then, she will take 3 hours to reach her Aunt’s house.

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Question 4:

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

ANSWER:

Let us suppose the stock of grain in a government warehouse lasts x days for 6000 people.
As the number of people increases, the number of days decreases.
So, the number of days and number of people are in inverse proportion. 
∴ 30 × 4000 = 6000 × x
⇒x=1200006000$\Rightarrow x=\frac{120000}{6000}$
⇒ x = 20 days
Hence, the stock of grain in a government warehouse lasts 20 days for 6000 people.

PARTNERSHIP 

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Question 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?

ANSWER:

The proportion of Suresh's and Ramesh’s investment is 4:5.
The profit is shared in the same proportion as the investment, hence, proportion of profit is 4:5.
Now, profit = 20100×144000$\frac{20}{100}\times 144000$
= 28800 rupees
Therefore, the profit of Suresh and Ramesh  is given by
Suresh's profit = 49×28800$\frac{4}{9}\times 28800$
= 12800 rupees
Ramesh's profit = 59×28800$\frac{5}{9}\times 28800$
= 16000 rupees
Hence, Suresh and Ramesh got the profit of 12800 and 16000 rupees respectively.

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Question 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 ?

ANSWER:

The proportion of Virat's and Samrat’s investment is given by
50000:120000 = 5:12
The loss is shared in the same proportion as the investment, hence, proportion of profit is 5:12.
Now, Loss = 20100×(50000+120000)$\frac{20}{100}\times \left(50000+120000\right)$
20100×(170000)$\frac{20}{100}\times \left(170000\right)$
= 34000 rupees
Therefore, the loss incurred by Virat and Samrat is given by
Virat's loss = 517×34000$\frac{5}{17}\times 34000$
= 10000 rupees
Samrat's loss = 1217×34000$\frac{12}{17}\times 34000$
= 24000 rupees
Hence, Virat and Samrat incurred the loss of ​10000 and 24000 rupees respectively.

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Question 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?

ANSWER:

Nachiket’s investement = Total investment − (Shweta’s investment + Piyush’s investment)
= 80000 − (30000 + 12000)
= 80000 − 42000
= 38000 rupees
The proportion of Shweta's, Piyush's and Nachiket's investment is given by
30000:12000:38000 = 15:6:19
The profit is shared in the same proportion as the investment, hence, proportion of profit is 15:6:19.
Now, Profit = 24100×(80000)$\frac{24}{100}\times \left(80000\right)$
= 19200 rupees
Therefore, Nachiket’s share of the profit is given by
= 1940×19200$\frac{19}{40}\times 19200$
= 9120 rupees
Hence, Nachiket’s investment and his share of the profit are ​38000 and 9120 rupees respectively.

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Question 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?

ANSWER:

Amount of share to the Soldiers’ Welfare Fund = 2% of 24500
= 490 rupees
The profit is shared in the proportion 3:7.
Therefore, A’s share to the Fund is given by
= 310×490$\frac{3}{10}\times 490$
= 147 rupees
Therefore, B’s share to the Fund is given by
= ​710×490$\frac{7}{10}\times 490$
= 343 rupees
Hence, A's and B's share to the fund are 147 and 343 rupees respectively.

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Question 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 ?

ANSWER:

Total investment = 360000 rupees
Total profit = 12100×360000$\frac{12}{100}\times 360000$
= 43200 rupees
The profit is shared in the same proportion as the investment, hence, proportion of profit is 3:4:7:6.
Jaya's share is given by
320×360000$\frac{3}{20}\times 360000$
= 54000 rupees
Nikhil's share in the profit is given by
720×43200$\frac{7}{20}\times 43200$
= 15120 rupees
Hence, Jaya's share and Nikhil's profit are 54000 and 15120 rupees respectively.