**(01) A and B rent pasture for 10 months. A puts 100 cows for 8 months. How many cows can B put for the remaining two months. If he pays 3/2 as much as A pays?**

**Sol**. Given: Time of pasture rent by A and B =10 months

A = 100 cows which can be treated as his capital for time = 8 months

To find: B = no. of cows I.e. His capital for time = 2 months

According to the question, the ratio of profit is 2/3

As we know that Capital x Time = Profit

Therefore, A/B = (100 x 8)/ (C2 x 2) = 2/3

C2 = (100 x 8 x 3)/ (2 x 2)

= 2400/ 4

= 600

**Ans. B has 600 cows for pasture.**

**(02) A & B started a business with Rs. 50,000 & 20,000. A is working partner and takes 20% of the total profit as his salary and remaining profit is divided according to their capital. If in this process A received Rs. 38000 more than B. Find the amount of total profit.**

**Sol. **Given: Capital of A = Rs. 50,000 and B = Rs. 20,000

Ratio of their Capitals = 50,000: 20,000 = 5: 2

A’s Salary = (1/ A’s ratio of capital) x (Sum of their Capital)

= (1/5) x 7

= 1.4

Profit for sharing = Sum of their Capital ratio- A’s Salary ratio

= 7 – 1.4

= 5.6

Now, A’s part = A’s salary ratio + (Ratio of A/ Sum of their ratio capital) x Profit for sharing

= 1.4 + [(5/7) x 5.6]

= 5.4

B’s part = (ratio of B/ Sum of ratio of their capital) = (2/7) x 5.6 = 1.6

According to the question, A’s part – B’s part = 38000

5.4 – 1.6 = 38000

3.8 = 38000

1 unit = 38000/ 3.8 = 10,000

Therefore, total profit = 10,000 x (Sum of their capital ratio)

= 10,000 x 7

= Rs. 70,000

**Ans. The total profit amount is Rs. 70,000**

**(03) A puts Rs. 375 more in a business than B. A invest for 8 months while B for 4 months. If the share of A is 75 more than that of B out of total profit of Rs. 125. Find the capital invested by A.**

**Sol. **Given: Time of A’s investment = 8 months while that of B’s = 4 months

Total profit = Rs.125

Out of A’s profit = Rs. 100

And B’s = Rs. 25 (As conditions according to the question)

Ratio of their Profit = 100: 25 = 4:1

Ratio of their time = 8:4 = 2:1

As we know that, C x T = P

Therefore, C = P/T

C1 = 4/ 2 = 2 and C2 = 1/1

Hence ratio of their Capital = 2

Since A puts 375 more than B, therefore (2 – 1) = 375

1 unit = 375

Capital invested by A = 375 x (Ratio of A’s capital) = 375 x 2 = Rs. 750

**Ans. A’s capital is 750**

**(04) A, B and C are three partners A received 5/8 part of total profit & remaining profit received by B and C equally. A’s income is increased by Rs. 450 when the profit rises from 4% to 9%. Find the capital invested by B & C each.**

**Sol. **Given: A’s profit = 5/8

Remaining profit = 1 – 5/8 = 3/8 (which is distributed between B and C equally)

Now, let us assume the total Capital to be Rs. 100

Profit of A = 4 % = 4 x (5/8) = 20/ 8

Profit of A = 9 % = 9 x (5/8) = 45/ 8

According to the question, 45/ 8 – 20/ 8 = 450

25/ 8 units = 450

1 unit = 450 x (8/ 25)

100 units = 450 x (8/ 25) x 100

Total profit = Rs. 14,400

Again given: Ratio of their profit = A : (B + C) = 5: 3

Therefore, B + C = (3/8) x 14400 = Rs. 5400

B = C = 5400/ 2 = Rs. 2700

**Ans. Therefore B and C invested Rs. 2700 each.**

**(05) A, B and C are three partners. A gets 2/7 part of total profit. B & C share the remaining profit equally. A’s income increases by Rs. 240 when profit rise from 10% to 15%. Find the capital invested by B & C.**

**Sol. **Given: A’s profit = 2/7

Remaining profit = 1 – 2/7 = 5/7 (which is distributed between B and C equally)

Now, let us assume the total Capital to be Rs. 100

Profit of A = 10 % = 10 x (2/7) = 20/7

Profit of A = 15% = 15 x (2/7) = 30/7

According to the question, 30/ 7 – 20/ 7 = 240

10/7 units = 240

1 unit = 240 x (7/ 10)

100 units = 240 x (7/ 10) x 100

Total profit = Rs. 16,800

Again given: Ratio of their profit = A : (B + C) = 2: 5

Therefore, B + C = (5/7) x 16800 = Rs. 12000

B = C = 5400/ 2 = Rs. 6000

**Ans. Therefore B and C invested Rs. 6000 each.**

**(06) A, B and C invested money in the ratio of 1/ 2 : 1/ 3: 1/ 5 in a business. After 4 months A doubled his investment and after 6 months B, half his investment. If the total profit at the end of year be 34650 then find the share of each in profit.**

**Sol. **Given: Ratio of capital invested by A, B and C = 1/ 2: 1/ 3: 1/ 5

Which can be simplified as 15: 10: 6 (solving by L.C.M. method)

Let the unit of unit capital invested by A is x

According to the question, total capital invested by A in 1 year = 15x. 4 + (15x.2).8

= 60x + 240x = 300x

Total capital invested by B in 1 year = 10x.6 + (10x/ 2).6 = 60x + 30x = 90x

Total capital invested by C in 1 year = 6x. 12 = 72x

Ratio of their profits = 300x : 90x : 72x (dividing by 6)

= 50x: 15x: 12x

Again according to the question, 50x + 15x + 12x = 34650

77x = 34650

x = 34650/ 77 = Rs. 450

** Ans Profit of A = 50x = 50. 450 = Rs. 22500**

** Profit of B = 15x = 15. 450 = Rs. 6750**

** Profit of C = 12x = 12. 450 = Rs. 5400**

**(07) A and B started a business by investing 36000 and 45000 respectively. After 4 months B withdraws 4/ 9 of his investment. its 5 months after she again invested 11/ 9 of its original investment. If the total earned profit at the end of the year, is 117240, then who will get more money as a share of profit and how much?**

**Sol.** Given: Capital of A = Rs. 36000 and that of B = Rs. 45000

Total capital invested by A at the end of year =36000 x 12 = Rs. 432000

Total capital invested by B at the end of year

= 45000 x 4 + (45000 – (4/ 9). 45000) x 5 + [(11/ 9). 45000 + (4/ 9). 45000 )] x 3

= 45000 x 4 + (45000 – 20000) x 5 + [55000 + 25000] x 3

= 180000 + 125000 + 240000

= 545000

Ratio of their profit = 432000: 545000 = 432 : 545

(432 + 545)x = 117240

977 x = 117240

1 x = 117240/ 977 = 120

Difference in their profit = (545 – 432) x 120 = 113 x 120 = 13560

**Ans. B will get Rs. 13560 more than A.**

**(08) A & B start A business, A invests 1/4 capital for 1/4th time and B invests 1/5 ^{th} capital for 1/2 time and C invests the remaining capital for full time. How should they divide the profit of Rs 1140?**

**Sol**. Let us consider the capital to be 20 if the time to be consider is 1 year then

Time for A = (1/ 4) x 12 = 3 months while capital = (1/ 4) x 20 = 5

Time for B = (1/ 2) x 12 = 6 months while capital = (1/ 5) x 20 = 4

Time for C = 12 months while capital = 20 – (5 + 4) = 11

As we know that capital x time = profit

Therefore their profit ratio = 5×3: 4×6: 11×12 = 15: 24:132

= 5: 8: 44 (dividing by 3)

**Ans. Hence they should divide the profit in the ratio of 5: 8: 44**

**(09) In a partnership business, B’s capital was half of A’s. If after 8 months. B withdrew half of his capital and after 2 months more A withdrew 1/4 ^{th }of his capital, then the profit ratio of A and B will be:**

**Sol**. Given: 2 x B’s capital = A’s capital

Therefore their capital ratio = A : B = 2: 1 (Let us consider the unit to be 100 I.e., 200: 100)

As we know that Capital x time = Profit

Then According to the question, Profit of A = 200 x 10 + (3/ 4) x 200 x 2 (1 – 1/ 4 = 3/ 4)

= 2000 + 300

= 2300

Profit of B = 100 x 8 + (1/ 2) x 100 x 4

= 800 + 200

= Rs. 1000

**Ans. The ratio of their profit will be 2300: 1000 = 23: 10 **

**(10) A and B invest in the ratio 3 : 5. After 6 months, C joins the business investing an amount equal to B’s. At the end of the year what will be the ratio of their profits?**

**Sol. **Given: Ratio of capital of A and B = 3: 5 (Let us consider the unit to be 100 I.e., 300: 500)

As we know that Capital x time = Profit

Then According to the question, Profit of A = 300 x 12 = Rs. 3600

Profit of B = 500 x 12 = Rs. 60000

Profit of C = 500 x 6 = Rs. 3000

Ratio of Profits = 3600: 60000 : 3000

= 6: 10: 5

**Ans. The required ratio is 6: 10: 5**