Partnership

Answer: Option B
Solution:
To determine the ratio of their investments, we need to use the concept that the profit-sharing ratio is based on the product of the amount of investment and the duration for which the investment is made.

Let the investments of the three partners be \(I_1\), \(I_2\), and \(I_3\) respectively, and the time durations for which they invested are 14 months, 8 months, and 7 months, respectively.

The formula for the share of each partner in the profit is:

\[
\text{Share of Partner} = \text{Investment} \times \text{Time duration}
\]

We are given that the profit-sharing ratio is 5:7:8, which means:

\[
\frac{I_1 \times 14}{I_2 \times 8} = \frac{5}{7} \quad \text{and} \quad \frac{I_2 \times 8}{I_3 \times 7} = \frac{7}{8}
\]

Step 1: Set up the proportion for the first pair (Partner 1 and Partner 2)

From the given ratio:

\[
\frac{I_1 \times 14}{I_2 \times 8} = \frac{5}{7}
\]

Cross-multiply to solve for \(I_1\) in terms of \(I_2\):

\[
7 \times (I_1 \times 14) = 5 \times (I_2 \times 8)
\]

\[
98 I_1 = 40 I_2
\]

\[
\frac{I_1}{I_2} = \frac{40}{98} = \frac{20}{49}
\]

Step 2: Set up the proportion for the second pair (Partner 2 and Partner 3)

Now, from the second ratio:

\[
\frac{I_2 \times 8}{I_3 \times 7} = \frac{7}{8}
\]

Cross-multiply to solve for \(I_2\) in terms of \(I_3\):

\[
8 \times (I_2 \times 8) = 7 \times (I_3 \times 7)
\]

\[
64 I_2 = 49 I_3
\]

\[
\frac{I_2}{I_3} = \frac{49}{64}
\]

Step 3: Combine the two ratios

Now, we know:

– \( \frac{I_1}{I_2} = \frac{20}{49} \)
– \( \frac{I_2}{I_3} = \frac{49}{64} \)

We can combine these to find the ratio of \(I_1\) to \(I_2\) to \(I_3\):

\[
\frac{I_1}{I_2} \times \frac{I_2}{I_3} = \frac{20}{49} \times \frac{49}{64}
\]

\[
\frac{I_1}{I_3} = \frac{20}{64} = \frac{5}{16}
\]

Thus, the overall ratio of \(I_1 : I_2 : I_3\) is:

\[
I_1 : I_2 : I_3 = 5 : 16 : 16
\]

Final Answer:
The ratio of their investments is 5 : 16 : 16.

Answer: Option C
Solution:
Let’s denote the capital of each partner as follows:

– Let \( A \)’s capital be \( x \).
– Let \( B \)’s capital be \( y \).
– Let \( C \)’s capital be \( z \).

From the given information:

1. **B’s capital is four times C’s capital**:
\[
y = 4z
\]

2. **Twice A’s capital is equal to thrice B’s capital**:
\[
2x = 3y
\] Substituting \( y = 4z \) into this equation:
\[
2x = 3 \times 4z = 12z
\] So, \( x = 6z \).

Step 1: Express the capitals in terms of \( z \)
Now, we have:

– \( A \)’s capital = \( x = 6z \),
– \( B \)’s capital = \( y = 4z \),
– \( C \)’s capital = \( z \).

Step 2: Calculate the total capital invested
The total capital invested by the three partners is:

\[
\text{Total capital} = x + y + z = 6z + 4z + z = 11z
\]

Step 3: Calculate B’s share in the profit
The profit is divided in proportion to the capital invested. Therefore, B’s share in the profit will be the ratio of B’s capital to the total capital:

\[
\text{B’s share} = \frac{B’s \, capital}{Total \, capital} = \frac{4z}{11z} = \frac{4}{11}
\]

Step 4: Calculate B’s actual share in the profit
The total profit is Rs 16500. B’s share of the profit is:

\[
\text{B’s share of profit} = \frac{4}{11} \times 16500 = \frac{66000}{11} = 6000
\]

Final Answer:
B’s share in the profit is Rs 6000.

Answer: Option A
Solution:
To solve this, we need to calculate the ratio of their investments and the duration of their investments, and then use that to determine Sameer’s share of the profit.

Step 1: Determine the capital and duration for each partner

– Kamal’s investment: Kamal invested Rs. 9000 for the entire year (12 months).
– Sameer’s investment: Sameer invested Rs. 8000, but he joined after 5 months, so his investment was for 7 months (12 months – 5 months).

Step 2: Calculate the “capital months” for each partner

The capital months are calculated by multiplying the amount of capital by the number of months it was invested.

– Kamal’s capital months: \( 9000 \times 12 = 108000 \) capital months.
– Sameer’s capital months: \( 8000 \times 7 = 56000 \) capital months.

Step 3: Calculate the ratio of their investments

The ratio of Kamal’s and Sameer’s investment in terms of capital months is:

\[
\text{Ratio of Kamal’s to Sameer’s investment} = \frac{108000}{56000} = \frac{108}{56} = \frac{27}{14}
\]

Step 4: Calculate Sameer’s share in the profit

The total profit is Rs. 6970. The total ratio of investment between Kamal and Sameer is \( 27:14 \).

Now, calculate Sameer’s share in the profit:

\[
\text{Sameer’s share} = \frac{14}{27 + 14} \times 6970 = \frac{14}{41} \times 6970
\]

\[
\text{Sameer’s share} = \frac{14 \times 6970}{41} = \frac{97580}{41} = 2380
\]

Final Answer:
Sameer’s share in the profit is Rs. 2380.

Answer: Option C
Solution:
The profit ratio between P and Q is given as 2:3. This means that for every 2 parts of profit that P receives, Q receives 3 parts.

Let the amount invested by Q be \( x \).

According to the ratio of profit and the fact that the profit is divided in proportion to the amounts invested, we can set up the following equation:

\[
\frac{\text{Investment by P}}{\text{Investment by Q}} = \frac{\text{Profit ratio of P}}{\text{Profit ratio of Q}}
\]

Substitute the given values:

\[
\frac{40000}{x} = \frac{2}{3}
\]

Now, solve for \( x \):

\[
40000 \times 3 = 2 \times x
\]

\[
120000 = 2x
\]

\[
x = \frac{120000}{2}
\]

\[
x = 60000
\]

So, the amount invested by Q is Rs 60,000.

Answer: Option D
Solution:
We are given the following information:

– 4 times P’s capital = 6 times Q’s capital = 10 times R’s capital
– Total profit = Rs. 4650

Let the capitals of P, Q, and R be represented as \( P \), \( Q \), and \( R \) respectively. Based on the given condition:

\[
4P = 6Q = 10R
\]

We can assume a common factor \( k \) such that:

\[
P = \frac{k}{4}, \quad Q = \frac{k}{6}, \quad R = \frac{k}{10}
\]

Now, to find the ratio of the capitals, we need to express the capitals in terms of a common multiple. We take the least common multiple (LCM) of 4, 6, and 10, which is 60.

So, let the actual capitals be:

\[
P = \frac{60}{4} = 15, \quad Q = \frac{60}{6} = 10, \quad R = \frac{60}{10} = 6
\]

The total capital ratio is:

\[
P : Q : R = 15 : 10 : 6
\]

Now, let’s calculate the total capital:

\[
15 + 10 + 6 = 31
\]

The total profit is Rs. 4650, and R’s share of the profit will be in the same ratio as R’s capital. The portion of the total profit that R will receive is:

\[
\text{R’s share} = \frac{6}{31} \times 4650
\]

Now, calculating this:

\[
\text{R’s share} = \frac{6 \times 4650}{31} = \frac{27900}{31} = 900
\]

So, R will receive Rs. 900 out of the total profit.

Answer: Option C
Solution:
To solve this, let’s calculate the total number of oxen-months contributed by each person. The oxen-months represent the total contribution of each person, considering the number of oxen they put and the duration of time they kept them in the pasture.

Step 1: Calculate the total oxen-months contributed by each person.

– A puts 10 oxen for 7 months, so A’s contribution is:
\[
10 \times 7 = 70 \text{ oxen-months}
\]

– B puts 12 oxen for 5 months, so B’s contribution is:
\[
12 \times 5 = 60 \text{ oxen-months}
\]

– C puts 15 oxen for 3 months, so C’s contribution is:
\[
15 \times 3 = 45 \text{ oxen-months}
\]

Step 2: Calculate the total oxen-months.

Now, add up the contributions from all three:

\[
70 + 60 + 45 = 175 \text{ oxen-months}
\]

Step 3: Calculate C’s share of the rent.

The total rent is Rs. 175, and C’s share of the total oxen-months is 45 out of 175. So, C’s share of the rent is:

\[
\text{C’s share} = \frac{45}{175} \times 175 = 45
\]

Thus, C must pay Rs. 45 as their share of the rent.

Answer: Option C
Solution:
We are given the following ratios:

– The ratio of A’s and C’s investment is 2:1.
– The ratio of A’s and B’s investment is 3:2.

We need to find how much B received out of the total profit of Rs. 157,300.

Step 1: Express the investments of A, B, and C in terms of a common variable.

Let’s assume that:

– The amount invested by A is \( 2x \) (since A and C’s investment ratio is 2:1, we can represent A’s investment as 2 parts and C’s investment as 1 part).
– The amount invested by C is \( x \).
– From the ratio of A’s and B’s investment (3:2), we can express B’s investment as \( \frac{2}{3} \times 2x = \frac{4x}{3} \).

Step 2: Calculate the total investment.

Now, we add the investments of A, B, and C:

\[
\text{Total investment} = 2x + \frac{4x}{3} + x
\]

To simplify, find a common denominator (which is 3):

\[
\text{Total investment} = \frac{6x}{3} + \frac{4x}{3} + \frac{3x}{3} = \frac{13x}{3}
\]

Step 3: Find the profit ratio.

The total profit is Rs. 157,300, and the profit will be distributed in the same ratio as the investments of A, B, and C. The ratio of their investments is:

\[
A : B : C = 2x : \frac{4x}{3} : x
\]

To simplify this, multiply each term by 3 to eliminate the fraction:

\[
A : B : C = 6x : 4x : 3x
\]

Thus, the ratio of their investments is 6:4:3.

Step 4: Calculate B’s share of the profit.

Now, the total profit of Rs. 157,300 will be divided in the ratio 6:4:3. The total parts are:

\[
6 + 4 + 3 = 13 \text{ parts}
\]

B’s share will be:

\[
\text{B’s share} = \frac{4}{13} \times 157300 = \frac{629200}{13} = 48400
\]

Thus, B will receive Rs. 48,400 from the total profit.

Answer: Option A
Solution:
We are given the following details:

– A’s investment = Rs. 20,000
– B’s investment = Rs. 15,000
– C’s investment = Rs. 20,000 (joined after 6 months)
– Total profit = Rs. 25,000
– The business runs for 2 years (24 months).

Step 1: Calculate the “capital-months” for each partner.

Capital-months represent the amount of money invested by each partner, multiplied by the number of months their money was invested.

A’s capital-months:
– A invested Rs. 20,000 for 24 months.
\[
A’s \, capital-months = 20,000 \times 24 = 480,000
\]

B’s capital-months:
– B invested Rs. 15,000 for 24 months.
\[
B’s \, capital-months = 15,000 \times 24 = 360,000
\]

C’s capital-months:
– C invested Rs. 20,000, but only after 6 months. So C invested for 18 months.
\[
C’s \, capital-months = 20,000 \times 18 = 360,000
\]

Step 2: Calculate the total capital-months.

The total capital-months is the sum of A’s, B’s, and C’s capital-months:

\[
\text{Total capital-months} = 480,000 + 360,000 + 360,000 = 1,200,000
\]

Step 3: Calculate B’s share of the profit.

The profit is divided in the ratio of the capital-months. The total capital-months is 1,200,000, and B’s share of the capital-months is 360,000. So, B’s share of the total profit is:

\[
B’s \, share = \frac{360,000}{1,200,000} \times 25,000
\]

\[
B’s \, share = \frac{360,000 \times 25,000}{1,200,000} = \frac{9,000,000,000}{1,200,000} = 7,500
\]

Thus, B’s share of the total profit is Rs. 7,500.

Answer: Option C
Solution:
To determine the profit-sharing ratio, we need to calculate the “capital-months” for each partner based on their investments and the duration for which they kept their money in the partnership.

Step 1: Calculate the capital-months for each partner.

P’s capital-months:
– P initially invests Rs. 25 lakh for 12 months (first year).
– P adds another Rs. 10 lakh after one year, making the total investment Rs. 35 lakh for the remaining 24 months (second and third years).

Thus, P’s total capital-months are:
\[
\text{P’s capital-months} = (25 \, \text{lakh} \times 12 \, \text{months}) + (35 \, \text{lakh} \times 24 \, \text{months})
\] \[
\text{P’s capital-months} = 300 \, \text{lakh-months} + 840 \, \text{lakh-months} = 1140 \, \text{lakh-months}
\]

Q’s capital-months:
– Q initially invests Rs. 35 lakh for 24 months (since Q withdrew Rs. 10 lakh after 2 years, this means Q kept Rs. 35 lakh invested for 2 years and Rs. 25 lakh for the remaining year).

Thus, Q’s total capital-months are:
\[
\text{Q’s capital-months} = (35 \, \text{lakh} \times 24 \, \text{months}) + (25 \, \text{lakh} \times 12 \, \text{months})
\] \[
\text{Q’s capital-months} = 840 \, \text{lakh-months} + 300 \, \text{lakh-months} = 1140 \, \text{lakh-months}
\]

R’s capital-months:
– R invests Rs. 30 lakh for the full 36 months.

Thus, R’s total capital-months are:
\[
\text{R’s capital-months} = 30 \, \text{lakh} \times 36 \, \text{months} = 1080 \, \text{lakh-months}
\]

Step 2: Calculate the total capital-months.

Now, we add the capital-months for all three partners:
\[
\text{Total capital-months} = 1140 \, (\text{P’s}) + 1140 \, (\text{Q’s}) + 1080 \, (\text{R’s}) = 3360 \, \text{lakh-months}
\]

Step 3: Determine the ratio of profit distribution.

The profit should be divided in the ratio of their capital-months. So, the ratio of P’s, Q’s, and R’s capital-months is:
\[
\text{P : Q : R} = 1140 : 1140 : 1080
\]

Simplifying this ratio:
\[
\text{P : Q : R} = 1140 : 1140 : 1080 = 19 : 19 : 18
\]

Thus, the profit should be divided in the ratio 19 : 19 : 18 between P, Q, and R, respectively.

Answer: Option D
Solution:
We are given the following information:

– The total amount subscribed is Rs. 50,000.
– A subscribes Rs. 4,000 more than B, and B subscribes Rs. 5,000 more than C.
– The total profit is Rs. 35,000.
– We need to find A’s share of the profit.

Step 1: Let C’s subscription be \( x \).

Given that B subscribes Rs. 5,000 more than C, we can express B’s subscription as \( x + 5000 \).

Also, A subscribes Rs. 4,000 more than B, so A’s subscription can be expressed as \( (x + 5000) + 4000 = x + 9000 \).

Step 2: Write the total subscription equation.

The total subscription is Rs. 50,000, so we can write the equation:
\[
A + B + C = 50,000
\] Substitute the expressions for A and B:
\[
(x + 9000) + (x + 5000) + x = 50,000
\] Simplify the equation:
\[
3x + 14,000 = 50,000
\] Solve for \( x \):
\[
3x = 50,000 – 14,000 = 36,000
\] \[
x = \frac{36,000}{3} = 12,000
\]

So, C’s subscription is Rs. 12,000.

Step 3: Calculate A, B, and C’s subscriptions.

– C’s subscription = Rs. 12,000
– B’s subscription = \( 12,000 + 5,000 = 17,000 \)
– A’s subscription = \( 17,000 + 4,000 = 21,000 \)

Step 4: Calculate the total capital.

The total capital subscribed is:
\[
A + B + C = 21,000 + 17,000 + 12,000 = 50,000
\] This matches the given total subscription.

Step 5: Find A’s share of the profit.

The total profit is Rs. 35,000. A’s share of the profit will be in the same ratio as A’s subscription to the total subscription. So, A’s share is:
\[
\text{A’s share} = \frac{21,000}{50,000} \times 35,000
\] \[
\text{A’s share} = \frac{21,000 \times 35,000}{50,000} = \frac{735,000,000}{50,000} = 14,700
\]

Thus, A receives Rs. 14,700 out of the total profit.

Answer: Option B
Solution:
To solve this, let’s break it down step by step:

Given:
– The ratio of A’s investment to B’s investment is 3:2.
– 5% of the total profit is donated to charity.
– A’s share of the profit is Rs. 855.

Let the total profit be \( P \).

1. Profit distribution between A and B:
Since A and B invested in the ratio of 3:2, the total profit (after charity) will be divided in this ratio.

2. Charity deduction:
5% of the total profit goes to charity. Therefore, the remaining 95% of the total profit is available for A and B to share.

So, the amount available for A and B = \( 0.95 \times P \).

3. Share of A and B:
In the ratio 3:2, A’s share is \( \frac{3}{5} \) of the remaining profit, and B’s share is \( \frac{2}{5} \).

Therefore, A’s share = \( \frac{3}{5} \times 0.95 \times P \).

4. A’s share given:
It’s given that A’s share is Rs. 855. So, we can set up the equation:

\[
\frac{3}{5} \times 0.95 \times P = 855
\]

5. Solve for \( P \):

\[
\frac{3}{5} \times 0.95 \times P = 855
\]

\[
0.57 \times P = 855
\]

\[
P = \frac{855}{0.57}
\]

\[
P = 1500
\]

Conclusion:
The total profit is Rs. 1500.

Answer: Option B
Solution:
To calculate the share of B in the profit, we need to follow a few steps:

Given:
– A’s investment: Rs. 6500 for 6 months
– B’s investment: Rs. 8400 for 5 months
– C’s investment: Rs. 10,000 for 3 months
– Total profit: Rs. 7400
– A gets 5% of the total profit for being the working member.

Step 1: Calculate A’s share for being the working member
A is the working member, so he will receive 5% of the total profit of Rs. 7400.

A’s working share = 5% of 7400 = \( \frac{5}{100} \times 7400 = 370 \)

Step 2: Deduct A’s working share from the total profit
The remaining profit to be distributed between A, B, and C based on their investments is:

Remaining profit = Total profit – A’s working share
\[
\text{Remaining profit} = 7400 – 370 = 7030
\]

Step 3: Calculate the ratio of investments
Now, we calculate the effective investment for each person, considering both the amount and the duration of their investment.

– A’s effective investment = \( 6500 \times 6 = 39,000 \)
– B’s effective investment = \( 8400 \times 5 = 42,000 \)
– C’s effective investment = \( 10,000 \times 3 = 30,000 \)

Now, calculate the total of these effective investments:
\[
\text{Total effective investment} = 39,000 + 42,000 + 30,000 = 111,000
\]

Step 4: Calculate the share of B in the remaining profit
B’s share of the total effective investment is:
\[
\frac{42,000}{111,000} = \frac{42}{111}
\]

Now, B’s share of the remaining profit is:
\[
B’s\ share = \frac{42}{111} \times 7030
\]

Let’s calculate that:

\[
B’s\ share = \frac{42}{111} \times 7030 = 2640
\]

Step 5: Conclusion
Therefore, B’s share of the profit is Rs. 2640.

Answer: Option D
Solution:
Let’s break down the problem step by step.

Given:
– Total subscription = Rs. 5,00,000
– A subscribes Rs. 40,000 more than B
– B subscribes Rs. 50,000 more than C
– Total profit = Rs. 3,50,000
– We need to find out how much A receives from the profit.

Step 1: Set up the subscription amounts
Let C’s subscription be \( x \).

– B subscribes Rs. 50,000 more than C, so B’s subscription = \( x + 50,000 \).
– A subscribes Rs. 40,000 more than B, so A’s subscription = \( (x + 50,000) + 40,000 = x + 90,000 \).

The total subscription is Rs. 5,00,000, so:
\[
A + B + C = 5,00,000
\] Substitute the values for A and B:
\[
(x + 90,000) + (x + 50,000) + x = 5,00,000
\] Simplifying:
\[
3x + 140,000 = 5,00,000
\] \[
3x = 5,00,000 – 140,000 = 3,60,000
\] \[
x = \frac{3,60,000}{3} = 1,20,000
\]

Step 2: Calculate the individual subscriptions
– C’s subscription = Rs. 1,20,000
– B’s subscription = \( 1,20,000 + 50,000 = 1,70,000 \)
– A’s subscription = \( 1,20,000 + 90,000 = 2,10,000 \)

Step 3: Calculate the ratio of profit distribution
The profit will be divided according to the subscription amounts:
– A’s subscription = Rs. 2,10,000
– B’s subscription = Rs. 1,70,000
– C’s subscription = Rs. 1,20,000

Total subscription = Rs. 5,00,000, and the total profit = Rs. 3,50,000.

Step 4: Find A’s share of the profit
A’s share in the profit will be based on the ratio of A’s subscription to the total subscription:
\[
A’s\ share = \frac{2,10,000}{5,00,000} \times 3,50,000
\] \[
A’s\ share = \frac{2,10,000}{5,00,000} \times 3,50,000 = 1,47,000
\]

Conclusion:
A will receive Rs. 1,47,000 from the total profit.

Answer: Option B
Solution:
To calculate the ratio of their investments, we can use the formula for profit sharing, which depends on both the amount of investment and the time for which the investment is made.

Given:
– The profit-sharing ratio is 5:7:8.
– Partner 1 (A) was invested for 14 months.
– Partner 2 (B) was invested for 8 months.
– Partner 3 (C) was invested for 7 months.

Step 1: The ratio of the profits is based on the product of investment and time.
So, the investment of each partner can be represented as:

– A’s investment = \( 5 \times 14 \)
– B’s investment = \( 7 \times 8 \)
– C’s investment = \( 8 \times 7 \)

Step 2: Calculate the values:

– A’s investment = \( 5 \times 14 = 70 \)
– B’s investment = \( 7 \times 8 = 56 \)
– C’s investment = \( 8 \times 7 = 56 \)

Step 3: Ratio of investments:
The ratio of their investments is:
\[
\text{A : B : C} = 70 : 56 : 56
\]

Step 4: Simplify the ratio:
We can simplify the ratio by dividing all terms by the greatest common divisor (GCD) of 70, 56, and 56, which is 14:
\[
\frac{70}{14} : \frac{56}{14} : \frac{56}{14} = 5 : 4 : 4
\]

Conclusion:
The ratio of their investments is 5 : 4 : 4.

Answer: Option D
Solution:
To solve this, we need to determine B’s contribution to the capital based on the profit-sharing ratio and the investment period.

Given:
– A starts the business with Rs. 3500 and continues for the full year (12 months).
– B joins after 5 months, meaning B is involved for 7 months (since 12 – 5 = 7 months).
– The profit-sharing ratio is 2:3 (A’s share : B’s share).
– We need to find B’s contribution to the capital.

Step 1: Let B’s contribution be \( x \).
We need to calculate the capital of B, denoted as \( x \).

Step 2: Determine the amount of money each partner invested over time.
– A’s total investment (since A invested for 12 months) = \( 3500 \times 12 = 42,000 \).
– B’s total investment (since B invested for 7 months) = \( x \times 7 \).

Step 3: Profit-sharing ratio is based on the product of capital and time.
– The ratio of the capital contributions, considering both the amount invested and the duration for which it was invested, should match the profit-sharing ratio (2:3).

So, the capital-time ratio for A and B is:

\[
\frac{A’s\ capital \times time}{B’s\ capital \times time} = \frac{2}{3}
\]

Substitute the values:
\[
\frac{42,000}{x \times 7} = \frac{2}{3}
\]

Step 4: Solve for \( x \).
\[
\frac{42,000}{7x} = \frac{2}{3}
\]

Now, cross-multiply:
\[
42,000 \times 3 = 2 \times 7x
\] \[
126,000 = 14x
\] \[
x = \frac{126,000}{14} = 9,000
\]

Conclusion:
B’s contribution to the capital is Rs. 9,000.

Answer: Option C
Solution:
To determine how the profit should be distributed among A, B, and C, we need to consider both the amount of investment and the duration for which each partner invested their money.

Given:
– A and B started the business with an investment ratio of 3:5.
– C joined after 6 months with an amount equal to B’s investment (so C invested the same amount as B).
– The total period considered is 12 months (1 year).

Step 1: Define the investments.
Let the investment of A be 3x and the investment of B be 5x.

– A’s investment = \( 3x \)
– B’s investment = \( 5x \)
– C’s investment = \( 5x \) (C invests the same as B, but joins after 6 months)

Step 2: Calculate the capital-time contribution for each partner.
The profit is shared based on the product of the capital invested and the time for which the capital was invested.

A’s contribution:
A invests \( 3x \) for the entire year (12 months):
\[
A’s\ contribution = 3x \times 12 = 36x
\]

B’s contribution:
B invests \( 5x \) for the entire year (12 months):
\[
B’s\ contribution = 5x \times 12 = 60x
\]

C’s contribution:
C invests \( 5x \) but only for 6 months (since C joins after 6 months):
\[
C’s\ contribution = 5x \times 6 = 30x
\]

Step 3: Calculate the total capital-time contributions.
Total capital-time contributions:
\[
A’s\ total = 36x, \quad B’s\ total = 60x, \quad C’s\ total = 30x
\]

Total = \( 36x + 60x + 30x = 126x \).

Step 4: Determine the profit-sharing ratio.
The profit-sharing ratio will be based on their respective capital-time contributions:
\[
A : B : C = 36x : 60x : 30x
\]

Step 5: Simplify the ratio.
To simplify, divide each term by \( 6x \):
\[
A : B : C = \frac{36x}{6x} : \frac{60x}{6x} : \frac{30x}{6x} = 6 : 10 : 5
\]

Conclusion:
The profit should be distributed among A, B, and C in the ratio 6 : 10 : 5.

Answer: Option A
Solution:
To determine how much C should pay as his share of the rent, we need to calculate the total contribution of each person in terms of the oxen-months, which reflects both the number of oxen and the time they were kept in the pasture.

Given:
– A puts 10 oxen for 7 months.
– B puts 12 oxen for 5 months.
– C puts 15 oxen for 3 months.
– Total rent = Rs. 175.

Step 1: Calculate the total number of oxen-months for each person.
The number of oxen-months is calculated by multiplying the number of oxen by the number of months they were in the pasture.

– A’s contribution = \( 10 \times 7 = 70 \) oxen-months.
– B’s contribution = \( 12 \times 5 = 60 \) oxen-months.
– C’s contribution = \( 15 \times 3 = 45 \) oxen-months.

Step 2: Calculate the total oxen-months.
The total number of oxen-months is the sum of the contributions from A, B, and C:
\[
\text{Total oxen-months} = 70 + 60 + 45 = 175
\]

Step 3: Calculate C’s share of the total contribution.
C’s share of the rent is proportional to C’s contribution (45 oxen-months) out of the total contribution (175 oxen-months).

So, C’s share of the rent is:
\[
\text{C’s share} = \frac{45}{175} \times 175 = 45
\]

Conclusion:
C must pay Rs. 45 as his share of the rent.

Answer: Option A
Solution:
To calculate B’s share in the total profit, we need to consider the amount of capital invested by each partner and the time for which they invested their capital.

Given:
– A invests Rs. 20,000 for 2 years.
– B invests Rs. 15,000 for 2 years.
– C invests Rs. 20,000 but joins after 6 months, meaning C’s capital is invested for 1.5 years.
– The total profit earned at the end of 2 years is Rs. 25,000.
– We need to calculate B’s share in the total profit.

Step 1: Calculate the capital-time contribution for each partner.

The profit is distributed based on the product of the capital and the time it was invested.

A’s capital-time contribution:
A invests Rs. 20,000 for 2 years:
\[
A’s\ contribution = 20,000 \times 2 = 40,000
\]

B’s capital-time contribution:
B invests Rs. 15,000 for 2 years:
\[
B’s\ contribution = 15,000 \times 2 = 30,000
\]

C’s capital-time contribution:
C invests Rs. 20,000 for 1.5 years (since C joins after 6 months):
\[
C’s\ contribution = 20,000 \times 1.5 = 30,000
\]

Step 2: Calculate the total capital-time contributions.
Total capital-time contributions:
\[
A’s\ total = 40,000, \quad B’s\ total = 30,000, \quad C’s\ total = 30,000
\]

Total = \( 40,000 + 30,000 + 30,000 = 100,000 \).

Step 3: Determine the profit-sharing ratio.
The profit will be divided in the ratio of their capital-time contributions:
\[
A : B : C = 40,000 : 30,000 : 30,000
\]

Simplify the ratio by dividing each term by 10,000:
\[
A : B : C = 4 : 3 : 3
\]

Step 4: Calculate B’s share in the total profit.
The total profit is Rs. 25,000, and B’s share of the profit is based on the ratio 3 out of the total 10 parts (since \( 4 + 3 + 3 = 10 \)).

B’s share in the profit:
\[
B’s\ share = \frac{3}{10} \times 25,000 = 7,500
\]

Conclusion:
B’s share in the total profit is Rs. 7,500.

Answer: Option D
Solution:
To solve this, we need to calculate the time for which B joins the business based on the capital contributions and the profit-sharing ratio.

Given:
– A invests Rs. 85,000 and remains for the full year (12 months).
– B invests Rs. 42,500, but we need to find the time for which B invests, denoted as \( t \) months.
– The profit-sharing ratio at the end of the year is 3:1 (A’s share : B’s share).

Step 1: Understand the profit-sharing rule
The profits are divided based on the capital-time contributions, i.e., the product of the amount of capital and the time for which it is invested.

So, the profit-sharing ratio between A and B is based on the following:

– A’s capital-time contribution = \( 85,000 \times 12 \) (since A invests for the entire year).
– B’s capital-time contribution = \( 42,500 \times t \) (since B invests for \( t \) months).

Step 2: Set up the ratio based on the given profit-sharing ratio
The ratio of their capital-time contributions is given by the ratio of their profits, which is 3:1. So:
\[
\frac{A’s\ capital-time\ contribution}{B’s\ capital-time\ contribution} = \frac{3}{1}
\] Substitute the capital-time contributions:
\[
\frac{85,000 \times 12}{42,500 \times t} = \frac{3}{1}
\]

Step 3: Solve for \( t \)
Now, solve for \( t \):

\[
\frac{85,000 \times 12}{42,500 \times t} = 3
\] \[
85,000 \times 12 = 3 \times 42,500 \times t
\] \[
1,020,000 = 127,500 \times t
\] \[
t = \frac{1,020,000}{127,500}
\] \[
t = 8
\]

Conclusion:
B must join the business for 8 months for the profit to be divided in the ratio of 3:1.

Answer: Option B
Solution:
Aman : Rakhi : Sagar
= (70,000 x 36) : (1,05,000 x 30) : (1,40,000 x 24)
= 12 : 15 : 16

Answer: Option A
Solution:
To determine Kamal’s share of the profit, we need to calculate the capital-time contributions of each partner. The profit is divided based on these contributions.

Given:
– Arun invests Rs. 8,000 for 6 months.
– Kamal invests Rs. 4,000 for 8 months.
– Vinay invests Rs. 8,000 for 8 months.
– The total profit earned after 8 months is Rs. 4,005.
– Arun left after 6 months, so his capital was only invested for 6 months.

Step 1: Calculate the capital-time contribution for each partner.

Arun’s contribution:
Arun invests Rs. 8,000 for 6 months.
\[
Arun’s\ contribution = 8,000 \times 6 = 48,000
\]

Kamal’s contribution:
Kamal invests Rs. 4,000 for the full 8 months.
\[
Kamal’s\ contribution = 4,000 \times 8 = 32,000
\]

Vinay’s contribution:
Vinay invests Rs. 8,000 for the full 8 months.
\[
Vinay’s\ contribution = 8,000 \times 8 = 64,000
\]

Step 2: Total capital-time contribution.
The total capital-time contributions of all partners are:
\[
Total = 48,000 + 32,000 + 64,000 = 1,44,000
\]

Step 3: Determine Kamal’s share of the profit.
The profit-sharing ratio is based on the capital-time contributions. So the ratio of Arun’s, Kamal’s, and Vinay’s contributions is:
\[
Arun : Kamal : Vinay = 48,000 : 32,000 : 64,000
\]

Simplify this ratio by dividing each term by 16,000:
\[
Arun : Kamal : Vinay = 3 : 2 : 4
\]

Step 4: Calculate Kamal’s share of the total profit.
The total profit is Rs. 4,005, and Kamal’s share is based on the ratio 2 out of the total 9 parts (since \( 3 + 2 + 4 = 9 \)).

Kamal’s share in the profit:
\[
Kamal’s\ share = \frac{2}{9} \times 4,005 = 890
\]

Conclusion:
Kamal’s share of the profit is Rs. 890.

Answer: Option D
Solution:
To determine Simran’s share in the profit, we need to calculate the capital-time contributions of both Simran and Nanda. The profit will be divided based on these contributions.

Given:
– Simran invests Rs. 50,000 for the full 3 years.
– Nanda joins after 6 months with Rs. 80,000. This means Nanda’s capital is invested for 2.5 years (since Nanda joins after 6 months, and the total period is 3 years).
– The total profit earned after 3 years is Rs. 24,500.

Step 1: Calculate the capital-time contribution for each partner.

Simran’s contribution:
Simran invests Rs. 50,000 for the entire 3 years:
\[
Simran’s\ contribution = 50,000 \times 3 = 150,000
\]

Nanda’s contribution:
Nanda invests Rs. 80,000 for 2.5 years:
\[
Nanda’s\ contribution = 80,000 \times 2.5 = 200,000
\]

Step 2: Total capital-time contributions.
The total capital-time contributions of both Simran and Nanda are:
\[
Total = 150,000 + 200,000 = 350,000
\]

Step 3: Determine the profit-sharing ratio.
The profit-sharing ratio is based on the capital-time contributions. So the ratio of Simran’s and Nanda’s contributions is:
\[
Simran : Nanda = 150,000 : 200,000
\]

Simplify this ratio by dividing each term by 50,000:
\[
Simran : Nanda = 3 : 4
\]

Step 4: Calculate Simran’s share in the total profit.
The total profit is Rs. 24,500, and Simran’s share is based on the ratio 3 out of the total 7 parts (since \( 3 + 4 = 7 \)).

Simran’s share in the profit:
\[
Simran’s\ share = \frac{3}{7} \times 24,500 = 10,500
\]

Conclusion:
Simran’s share in the profit is Rs. 10,500.

Answer: Option C
Solution:
To determine the ratio in which the profit should be divided between Sumit and Ravi, we need to calculate their capital-time contributions. The profit will be distributed based on the product of the amount invested and the time for which it was invested.

Given:
– Sumit invests Rs. 85,000 for 2 years.
– Ravi invests Rs. 15,000 for 2 years.

Step 1: Calculate the capital-time contribution for each partner.

Sumit’s contribution:
Sumit invests Rs. 85,000 for the entire 2 years:
\[
Sumit’s\ contribution = 85,000 \times 2 = 170,000
\]

Ravi’s contribution:
Ravi invests Rs. 15,000 for the entire 2 years:
\[
Ravi’s\ contribution = 15,000 \times 2 = 30,000
\]

Total capital-time contributions.
The total capital-time contributions of both Sumit and Ravi are:
\[
Total = 170,000 + 30,000 = 200,000
\]

Determine the profit-sharing ratio.
The profit-sharing ratio is based on their respective capital-time contributions. So the ratio of Sumit’s and Ravi’s contributions is:
\[
Sumit : Ravi = 170,000 : 30,000
\]

Simplify this ratio by dividing both terms by 10,000:
\[
Sumit : Ravi = 17 : 3
\]

Conclusion:
The profit earned after 2 years should be divided between Sumit and Ravi in the ratio 17:3.

Answer: Option C
Solution:
Let’s define the shares of A, B, and C in terms of C’s share.

Given:
– A receives half as much as B.
– B receives half as much as C.
– The total amount to be divided is Rs. 700.

Step 1: Express the shares in terms of C’s share.
Let C’s share be \( x \).

– B receives half as much as C, so B’s share will be \( \frac{x}{2} \).
– A receives half as much as B, so A’s share will be \( \frac{x}{4} \).

Step 2: Set up the equation for the total amount.
The sum of their shares equals Rs. 700:
\[
A’s\ share + B’s\ share + C’s\ share = 700
\] Substitute the expressions for A’s, B’s, and C’s shares:
\[
\frac{x}{4} + \frac{x}{2} + x = 700
\]

Step 3: Solve the equation.
To simplify, first find a common denominator. The common denominator of 4, 2, and 1 is 4. So, rewrite the equation:
\[
\frac{x}{4} + \frac{2x}{4} + \frac{4x}{4} = 700
\] \[
\frac{x + 2x + 4x}{4} = 700
\] \[
\frac{7x}{4} = 700
\] Now, multiply both sides by 4 to eliminate the denominator:
\[
7x = 2800
\] Solve for \( x \):
\[
x = \frac{2800}{7} = 400
\]

Step 4:
Conclusion.
C’s share is Rs. 400.

Answer: Option D
Solution:
To determine Deepak’s share in the profit, we need to calculate the capital-time contributions of both Anand and Deepak. The profit will be divided based on these contributions.

Given:
– Anand invests Rs. 22,500.
– Deepak invests Rs. 35,000.
– The total profit earned is Rs. 13,800.

Since no time period is mentioned, we assume both Anand and Deepak stayed invested for the same period.

Step 1: Calculate the ratio of their investments.
The profit-sharing ratio is based on the amount of capital invested by each partner.

The ratio of Anand’s and Deepak’s investments is:
\[
Anand : Deepak = 22,500 : 35,000
\]

Simplify this ratio by dividing both terms by 2,500:
\[
Anand : Deepak = 9 : 14
\]

Step 2: Determine Deepak’s share in the total profit.
The total profit is Rs. 13,800, and Deepak’s share is based on the ratio of 14 out of the total 23 parts (since \( 9 + 14 = 23 \)).

Deepak’s share in the profit:
\[
Deepak’s\ share = \frac{14}{23} \times 13,800
\]

Now, calculate this value:
\[
Deepak’s\ share = \frac{14 \times 13,800}{23} = \frac{193,200}{23} = 8,400
\]

Conclusion:
Deepak’s share of the profit is Rs. 8,400.

Answer: Option D
Solution:
To determine after how many months B joined the business, we need to ensure that the capital-time contributions of A and B are in the same ratio to divide the profit equally at the end of the year.

Given:
– A invests Rs. 21,000.
– B invests Rs. 36,000.
– The total profit at the end of the year is divided equally between A and B.

Step 1: Capital-time contribution
The profit is divided based on the capital-time contribution, which is the product of the amount of capital invested and the time for which it was invested.

– A’s capital-time contribution is Rs. 21,000 for 12 months (since A invests for the entire year).
\[
A’s\ capital-time = 21,000 \times 12 = 252,000
\]

– B’s capital-time contribution is Rs. 36,000, but B joins after \( t \) months. So, B’s capital-time contribution will be \( 36,000 \times t \).

Step 2: Set up the equation for equal profit-sharing
For the profit to be divided equally, the capital-time contributions of A and B must be equal:
\[
A’s\ capital-time = B’s\ capital-time
\] \[
252,000 = 36,000 \times t
\]

Step 3: Solve for \( t \)
Now, solve for \( t \):
\[
t = \frac{252,000}{36,000} = 7
\]

Conclusion:
B must join the business 7 months after A started for the profit to be divided equally at the end of the year.

Answer: Option A
Solution:
To calculate Kapil’s share in the total profit, we need to consider the capital-time contributions of both Nirmal and Kapil. The profit will be divided based on these contributions.

Given:
– Nirmal invests Rs. 9,000 for the full year (12 months).
– Kapil invests Rs. 12,000 for the first 6 months, and then withdraws half of his investment, i.e., Rs. 6,000, for the remaining 6 months.
– The total profit at the end of the year is Rs. 4,600.
Step 1: Calculate the capital-time contribution of each partner.

Nirmal’s contribution:
Nirmal invests Rs. 9,000 for the entire 12 months:
\[
Nirmal’s\ contribution = 9,000 \times 12 = 108,000
\]

Kapil’s contribution:
Kapil invests Rs. 12,000 for the first 6 months and Rs. 6,000 for the remaining 6 months.

– For the first 6 months:
\[
Kapil’s\ contribution\ (first\ 6\ months) = 12,000 \times 6 = 72,000
\] – For the next 6 months:
\[
Kapil’s\ contribution\ (next\ 6\ months) = 6,000 \times 6 = 36,000
\]

Total capital-time contribution for Kapil:
\[
Kapil’s\ total\ contribution = 72,000 + 36,000 = 108,000
\]

Step 2: Total capital-time contributions.
The total capital-time contributions of both Nirmal and Kapil are:
\[
Total = 108,000 + 108,000 = 216,000
\]

Step 3: Profit-sharing ratio.
The profit-sharing ratio is based on the capital-time contributions, so the ratio of Nirmal’s and Kapil’s contributions is:
\[
Nirmal : Kapil = 108,000 : 108,000 = 1 : 1
\]

Step 4: Calculate Kapil’s share of the total profit.
Since the profit-sharing ratio is 1:1, both Nirmal and Kapil will receive an equal share of the total profit.

Kapil’s share in the profit:
\[
Kapil’s\ share = \frac{1}{2} \times 4,600 = 2,300
\]

Conclusion:
Kapil’s share in the profit is Rs. 2,300.

Answer: Option B
Solution:
Suppose Ramesh invested Rs. x. Then,
Manoj : Ramesh = 120000 × 6 : x × 12.
Answer & Solution
:
720000
12x
6000
3000
⇒ x =
720000
24
⇒ x = 30000

Answer: Option D
Solution:
To determine Atul’s share in the profit, we need to calculate the capital-time contributions of Yogesh, Pranab, and Atul. The profit will be divided based on these contributions.

Given:
– Yogesh invests Rs. 45,000 for 12 months.
– Pranab joins after 3 months with Rs. 60,000, so his investment is for 9 months.
– Atul joins after 9 months with Rs. 90,000, so his investment is for 3 months.
– The total profit earned at the end of the year is Rs. 20,000.

Step 1: Calculate the capital-time contribution for each partner.

Yogesh’s contribution:
Yogesh invests Rs. 45,000 for 12 months:
\[
Yogesh’s\ contribution = 45,000 \times 12 = 540,000
\]

Pranab’s contribution:
Pranab invests Rs. 60,000 for 9 months:
\[
Pranab’s\ contribution = 60,000 \times 9 = 540,000
\]

Atul’s contribution:
Atul invests Rs. 90,000 for 3 months:
\[
Atul’s\ contribution = 90,000 \times 3 = 270,000
\]

Step 2: Total capital-time contributions.
The total capital-time contributions of all three partners are:
\[
Total = 540,000 + 540,000 + 270,000 = 1,350,000
\]

Step 3: Determine the profit-sharing ratio.
The profit-sharing ratio is based on the capital-time contributions. So the ratio of Yogesh’s, Pranab’s, and Atul’s contributions is:
\[
Yogesh : Pranab : Atul = 540,000 : 540,000 : 270,000
\]

Simplify this ratio by dividing each term by 90,000:
\[
Yogesh : Pranab : Atul = 6 : 6 : 3
\]

Step 4: Calculate Atul’s share of the total profit.
The total profit is Rs. 20,000, and Atul’s share is based on the ratio 3 out of the total 15 parts (since \( 6 + 6 + 3 = 15 \)).

Atul’s share in the profit:
\[
Atul’s\ share = \frac{3}{15} \times 20,000 = 4,000
\]

Conclusion:
Atul’s share of the profit is Rs. 4,000.

Answer: Option B
Solution:
Let’s break down the problem step by step to find out how long Bharti’s money was used.

Given:
– Rahul contributes \(\frac{1}{4}\) of the capital for 15 months.
– Bharti receives \(\frac{2}{3}\) of the profit.

We need to find out for how long Bharti’s money was used in the business.

Step 1: Set up the capital-time contribution.
Let the total capital be \( C \).

– Rahul’s capital contribution = \(\frac{1}{4}C\) for 15 months.
– Bharti’s capital contribution = \(\frac{3}{4}C\), since Rahul contributes \(\frac{1}{4}C\) and the total capital is \( C \).

Let \( t \) be the time for which Bharti’s money was used in the business. Bharti’s capital is used for \( t \) months.

Step 2: Profit-sharing based on capital-time contributions.
The profit-sharing ratio is determined by the capital-time contributions. Therefore, we will compare Rahul’s and Bharti’s capital-time contributions.

– Rahul’s capital-time contribution = \(\frac{1}{4}C \times 15 = \frac{15}{4}C\).
– Bharti’s capital-time contribution = \(\frac{3}{4}C \times t = \frac{3t}{4}C\).

Step 3: Set up the ratio for the profit-sharing.
We know Bharti receives \(\frac{2}{3}\) of the total profit, which means Rahul receives the remaining \(\frac{1}{3}\) of the profit. Therefore, the ratio of Rahul’s and Bharti’s capital-time contributions must be \(\frac{1}{3}\) to \(\frac{2}{3}\).

Thus, we set up the equation:
\[
\frac{Rahul’s\ capital-time}{Bharti’s\ capital-time} = \frac{1}{3} : \frac{2}{3}
\] Substitute the capital-time contributions:
\[
\frac{\frac{15}{4}C}{\frac{3t}{4}C} = \frac{1}{3}
\] Simplify:
\[
\frac{15}{3t} = \frac{1}{3}
\] Cross-multiply to solve for \( t \):
\[
15 \times 3 = 1 \times 3t
\] \[
45 = 3t
\] \[
t = 15
\]

Conclusion:
Bharti’s money was used for 15 months.

Answer: Option B
Solution:
To determine the ratio in which P and Q will divide the profit, we need to calculate their capital-time contributions. The profit is divided based on these contributions.

Given:
– P invests Rs. 85,000.
– Q invests Rs. 15,000.
– The profit will be divided after 2 years.

Step 1: Calculate the capital-time contributions.

Since the profit is divided based on the capital-time contributions, we first find the total capital-time for both P and Q.

– P’s capital-time contribution = Rs. 85,000 for 2 years:
\[
P’s\ capital-time = 85,000 \times 2 = 170,000
\]

– Q’s capital-time contribution = Rs. 15,000 for 2 years:
\[
Q’s\ capital-time = 15,000 \times 2 = 30,000
\]

Step 2: Determine the ratio of their capital-time contributions.
The ratio of P’s and Q’s capital-time contributions is:
\[
P : Q = 170,000 : 30,000
\]

Simplify this ratio by dividing both terms by 10,000:
\[
P : Q = 17 : 3
\]

Conclusion:
The profit earned after 2 years will be divided between P and Q in the ratio 17 : 3.

Answer: Option D
Solution:
To determine the respective shares of A, B, and C in the annual profit, we need to calculate their capital-time contributions and use the profit-sharing ratio based on these contributions.

Given:
– A invests Rs. 35,000
– B invests Rs. 45,000
– C invests Rs. 55,000
– The total profit is Rs. 40,500

Since there is no mention of the time period, we assume that all partners invested their respective amounts for the entire year.

Step 1: Calculate the capital-time contributions.
The profit-sharing ratio is based on the capital-time contributions, and since all partners invested for the entire year, their capital-time contributions are simply their investments.

– A’s capital-time contribution = Rs. 35,000
– B’s capital-time contribution = Rs. 45,000
– C’s capital-time contribution = Rs. 55,000

Step 2: Determine the total capital-time contribution.
The total capital-time contributions are:
\[
Total = 35,000 + 45,000 + 55,000 = 1,35,000
\]

Step 3: Determine the profit-sharing ratio.
The profit-sharing ratio is based on the capital-time contributions. So, the ratio of A’s, B’s, and C’s contributions is:
\[
A : B : C = 35,000 : 45,000 : 55,000
\]

Simplify this ratio by dividing each term by 5,000:
\[
A : B : C = 7 : 9 : 11
\]

Step 4: Calculate each partner’s share of the profit.
The total profit is Rs. 40,500, and the ratio of A’s, B’s, and C’s contributions is 7 : 9 : 11.

The total number of parts is:
\[
7 + 9 + 11 = 27
\]

Now, calculate each partner’s share of the profit:
– A’s share:
\[
A’s\ share = \frac{7}{27} \times 40,500 = \frac{7 \times 40,500}{27} = 10,500
\]

– B’s share:
\[
B’s\ share = \frac{9}{27} \times 40,500 = \frac{9 \times 40,500}{27} = 13,500
\]

– C’s share:
\[
C’s\ share = \frac{11}{27} \times 40,500 = \frac{11 \times 40,500}{27} = 16,500
\]

Conclusion:
– A’s share in the profit is Rs. 10,500.
– B’s share in the profit is Rs. 13,500.
– C’s share in the profit is Rs. 16,500.

Answer: Option D
Solution:
To calculate Kiara’s share of the rent, we need to determine the total number of hours each person used the DVDs and then allocate the rent based on the hours of use.

Given:
– Total rent for the DVDs = Rs. 578
– Samaira used the DVDs for 8 hours.
– Mahira used the DVDs for 12 hours.
– Kiara used the DVDs for 14 hours.

Step 1: Calculate the total hours of usage.
The total number of hours the DVDs were used by all three people is:
\[
Total\ hours = 8 + 12 + 14 = 34\ hours
\]

Step 2: Determine the share of rent based on hours used.
Kiara used the DVDs for 14 hours, so her share of the total hours is:
\[
Kiara’s\ share\ of\ hours = \frac{14}{34}
\]

Step 3: Calculate Kiara’s share of the rent.
Now, we can calculate Kiara’s share of the total rent:
\[
Kiara’s\ share = \frac{14}{34} \times 578 = \frac{14 \times 578}{34} = \frac{8082}{34} = 237
\]

Conclusion:
Kiara’s share of the rent is Rs. 238.

Answer: Option B
Solution:
To calculate Q’s share in the profit, we need to determine the profit-sharing ratio based on the capital-time contributions of P, Q, and R.

Given:
– P invested Rs. 45,000
– Q invested Rs. 70,000
– R invested Rs. 90,000
– The total profit earned at the end of 2 years is Rs. 164,000.

Since all the investments were made at the beginning of the business and they are partners for the entire 2 years, we can directly use their investments for 2 years to calculate the capital-time contributions.

Step 1: Calculate the capital-time contributions.

– P’s capital-time contribution = Rs. 45,000 for 2 years:
\[
P’s\ capital-time = 45,000 \times 2 = 90,000
\]

– Q’s capital-time contribution = Rs. 70,000 for 2 years:
\[
Q’s\ capital-time = 70,000 \times 2 = 140,000
\]

– R’s capital-time contribution = Rs. 90,000 for 2 years:
\[
R’s\ capital-time = 90,000 \times 2 = 180,000
\]

Step 2: Determine the total capital-time contribution.
The total capital-time contribution is:
\[
Total\ capital-time = 90,000 + 140,000 + 180,000 = 410,000
\]

Step 3: Determine the profit-sharing ratio.
The profit-sharing ratio is based on the capital-time contributions. Therefore, the ratio of P’s, Q’s, and R’s contributions is:
\[
P : Q : R = 90,000 : 140,000 : 180,000
\]

Simplify the ratio by dividing each term by 10,000:
\[
P : Q : R = 9 : 14 : 18
\]

Step 4: Calculate Q’s share in the profit.
The total profit is Rs. 164,000, and the total number of parts is:
\[
9 + 14 + 18 = 41
\]

Now, calculate Q’s share of the profit:
\[
Q’s\ share = \frac{14}{41} \times 164,000 = \frac{14 \times 164,000}{41} = \frac{2,296,000}{41} = 56,000
\]

Conclusion:
Q’s share in the profit is Rs. 56,000.

Answer: Option C
Solution:
To find the total profit in the business, let’s break down the problem step by step.

Given:
– The working partner gets 20% of the profit after his commission is paid.
– The working partner’s commission is Rs. 8,000.

Step 1: Let the total profit be \( P \).

The working partner receives 20% of the profit after his commission. This means that after his commission is subtracted, the remaining profit is what the working partner receives 20% of. So, the total profit \( P \) can be divided into two parts:

1. The working partner’s commission (Rs. 8,000).
2. The remaining profit (which the working partner gets 20% of).

Step 2: Calculate the remaining profit.

Let the remaining profit (after the commission) be \( R \).

Since the working partner receives 20% of \( R \) as his commission, we have:
\[
\text{Working partner’s commission} = 20\% \times R = 8000
\]

This gives the equation:
\[
0.20 \times R = 8000
\]

Solving for \( R \):
\[
R = \frac{8000}{0.20} = 40,000
\]

Step 3: Calculate the total profit.

Now, the total profit \( P \) is the sum of the working partner’s commission (Rs. 8,000) and the remaining profit \( R \) (Rs. 40,000):
\[
P = 8000 + 40,000 = 48,000
\]

Conclusion:
The total profit in the business is Rs. 48,000.

Answer: Option C
Solution:
Let’s break down the problem step by step.

Given:
– The total sum to be divided is Rs. 13,950.
– B’s share must be double of A’s share.
– C’s share must be Rs. 50 less than double of B’s share.

Let A’s share be \( x \).

Step 1: Express B’s and C’s shares in terms of A’s share.
– B’s share is double of A’s share:
\[
B = 2x
\]

– C’s share is Rs. 50 less than double of B’s share:
\[
C = 2B – 50 = 2(2x) – 50 = 4x – 50
\]

Step 2: Set up the equation for the total sum.
The total sum of Rs. 13,950 is divided among A, B, and C:
\[
A + B + C = 13,950
\]

Substitute the expressions for B and C in terms of \( x \):
\[
x + 2x + (4x – 50) = 13,950
\]

Step 3: Simplify the equation.
\[
x + 2x + 4x – 50 = 13,950
\] \[
7x – 50 = 13,950
\]

Step 4: Solve for \( x \).
Add 50 to both sides:
\[
7x = 13,950 + 50 = 14,000
\]

Now, divide both sides by 7:
\[
x = \frac{14,000}{7} = 2,000
\]

Conclusion:
A’s share is Rs. 2,000.

Answer: Option A
Solution:
To determine 50% of Anil’s share in the profit, we need to follow these steps:

Given:
– Prakash invests Rs. 11 lakh.
– Sachin invests Rs. 16.5 lakh.
– Anil invests Rs. 8.25 lakh.
– The total profit at the end of 3 years is Rs. 19.5 lakh.

Step 1: Calculate the total capital-time contributions.

Since the profit-sharing is based on the capital-time contributions, and they have been partners for 3 years, we will first calculate the capital-time contributions for each.

– **Prakash’s capital-time contribution**:
Prakash invested Rs. 11 lakh for 3 years:
\[
\text{Prakash’s capital-time} = 11 \times 3 = 33 \text{ lakh}
\]

– Sachin’s capital-time contribution:
Sachin invested Rs. 16.5 lakh for 3 years:
\[
\text{Sachin’s capital-time} = 16.5 \times 3 = 49.5 \text{ lakh}
\]

– Anil’s capital-time contribution:
Anil invested Rs. 8.25 lakh for 3 years:
\[
\text{Anil’s capital-time} = 8.25 \times 3 = 24.75 \text{ lakh}
\]

Step 2: Determine the total capital-time contribution.

The total capital-time contribution is the sum of Prakash’s, Sachin’s, and Anil’s contributions:
\[
\text{Total capital-time} = 33 + 49.5 + 24.75 = 107.25 \text{ lakh}
\]

Step 3: Calculate the profit-sharing ratio.

The profit-sharing ratio is based on the capital-time contributions. So, the ratio of Prakash’s, Sachin’s, and Anil’s contributions is:
\[
\text{Prakash : Sachin : Anil} = 33 : 49.5 : 24.75
\]

To simplify the ratio, we multiply each term by 4 to avoid decimals:
\[
33 \times 4 : 49.5 \times 4 : 24.75 \times 4 = 132 : 198 : 99
\]

So, the ratio of their profit shares is:
\[
132 : 198 : 99
\]

Step 4: Determine Anil’s share of the total profit.

The total profit is Rs. 19.5 lakh, and the total number of parts in the profit-sharing ratio is:
\[
132 + 198 + 99 = 429
\]

Anil’s share of the profit is:
\[
\text{Anil’s share} = \frac{99}{429} \times 19.5 \text{ lakh}
\]

Simplifying the calculation:
\[
\text{Anil’s share} = \frac{99 \times 19.5}{429} = \frac{1930.5}{429} = 4.5 \text{ lakh}
\]

Step 5: Calculate 50% of Anil’s share.

Now, we calculate 50% of Anil’s share:
\[
50\% \text{ of Anil’s share} = \frac{4.5}{2} = 2.25 \text{ lakh}
\]

Conclusion:
50% of Anil’s share in the profit is Rs. 2.25 lakh.

Answer: Option A
Solution:
Let’s break down the problem step by step:

Given:
– Friend 1 (let’s call them A) invested Rs. 1500.
– Friend 2 (let’s call them B) invested Rs. 2500.
– Total profit earned = Rs. 800.
– Half of the profit is divided equally, and the other half is divided in proportion to their capitals.

Step 1: Divide the profit into two equal parts.
The total profit is Rs. 800, so half of the profit is:
\[
\frac{800}{2} = 400
\] This half is divided equally between A and B, so each of them receives Rs. 400 ÷ 2 = **Rs. 200**.

Step 2: Divide the remaining half of the profit based on their capitals.
The remaining Rs. 400 is divided in proportion to their investments (capitals).

The ratio of their investments is:
\[
\text{A’s investment} : \text{B’s investment} = 1500 : 2500
\] Simplify the ratio:
\[
\frac{1500}{500} : \frac{2500}{500} = 3 : 5
\] So, the total parts are 3 + 5 = 8.

Now, each part of the remaining profit is:
\[
\frac{400}{8} = 50
\]

– A’s share from the remaining profit = 3 parts = 3 × 50 = **Rs. 150**.
– B’s share from the remaining profit = 5 parts = 5 × 50 = **Rs. 250**.

Step 3: Calculate the total amount received by each.

– A’s total share = Rs. 200 (equal share) + Rs. 150 (capital-based share) = **Rs. 350**.
– B’s total share = Rs. 200 (equal share) + Rs. 250 (capital-based share) = **Rs. 450**.

Conclusion:
– A receives Rs. 350.
– B receives Rs. 450.

Answer: Option C
Solution:
Let’s break down the problem step by step to calculate B’s share of the profit.

Given:
– Total capital = Rs. 3000
– B invests Rs. 600 less than A
– C invests Rs. 300 less than B
– The total profit is Rs. 886

Step 1: Express the investments of each person.
Let A’s investment be \( x \).

Then:
– B’s investment = \( x – 600 \)
– C’s investment = \( (x – 600) – 300 = x – 900 \)

Step 2: Set up the equation for the total capital.
The total capital invested by all three persons is Rs. 3000:
\[
x + (x – 600) + (x – 900) = 3000
\] Simplifying:
\[
x + x – 600 + x – 900 = 3000
\] \[
3x – 1500 = 3000
\] \[
3x = 4500
\] \[
x = 1500
\]

Step 3: Calculate the investments of each person.
– A’s investment = Rs. 1500
– B’s investment = Rs. 1500 – 600 = Rs. 900
– C’s investment = Rs. 900 – 300 = Rs. 600

Step 4: Determine the total capital-time ratio.
Now, the total profit is to be divided in proportion to their investments.

The total capital is Rs. 3000, and the total investment shares are:
– A’s share = Rs. 1500
– B’s share = Rs. 900
– C’s share = Rs. 600

The ratio of their investments is:
\[
A : B : C = 1500 : 900 : 600
\]

Simplify the ratio:
\[
A : B : C = 5 : 3 : 2
\]

Step 5: Calculate B’s share of the profit.
The total parts in the profit-sharing ratio are:
\[
5 + 3 + 2 = 10
\]

B’s share in the profit is:
\[
B’s\ share = \frac{3}{10} \times 886 = \frac{3 \times 886}{10} = \frac{2658}{10} = 265.8
\]

Conclusion:
B’s share in the profit is Rs. 265.80.

Answer: Option A
Solution:
To solve this, we need to find how long Y’s investment was used in the business, given that X and Y’s profits are divided in a certain ratio and X withdraws his capital after 8 months.

Given:
– X and Y invest in the ratio 5:6.
– X withdraws his capital after 8 months.
– The profit ratio between X and Y is 5:9.

Step 1: Capital-time contribution for X and Y.

The capital-time contribution determines the profit-sharing ratio. Since X and Y invested in the ratio of 5:6, we can calculate their capital-time contributions.

Let the capital-time contribution for X and Y be in terms of their respective investments:

– **X’s capital-time contribution** = \( 5 \times 8 \) (since X invested for 8 months) = \( 40 \)
– **Y’s capital-time contribution** = \( 6 \times t \), where \( t \) is the number of months Y’s investment was used.

Step 2: Use the profit-sharing ratio.

The profit-sharing ratio is given as 5:9. The ratio of their capital-time contributions should be equal to the profit-sharing ratio:

\[
\frac{\text{X’s capital-time}}{\text{Y’s capital-time}} = \frac{5}{9}
\]

Substitute the known values for X’s capital-time contribution and Y’s capital-time contribution:

\[
\frac{40}{6 \times t} = \frac{5}{9}
\]

Step 3: Solve for \( t \).

Cross-multiply to solve for \( t \):

\[
40 \times 9 = 5 \times 6 \times t
\] \[
360 = 30t
\] \[
t = \frac{360}{30} = 12
\]

Conclusion:
Y’s investment was used for 12 months.

Answer: Option D
Solution:
M graze 16 cows for 3 months.
N graze 20 cows for 4 months.
O graze 18 cows for 6 months.
P graze 42 cows for 2 months.
So, Ration of Rent
= M : N : O : P
= (16 × 3) : (20 × 4) : (18 × 6) : (42 × 2)
= 48 : 80 : 108 : 84
= 12 : 20 : 27 : 21
According to the question,
Answer & Solution
12 units = Rs. 2400
1 unit =
2400
12
27 units =
2400 × 27
12
= Rs. 5400

Answer: Option D
Solution:
To calculate Aniket’s share in the profit, let’s break down the problem step by step.

Given:
– Shankar’s investment = Rs. 120,000
– Aniket’s investment = Rs. 190,000
– Shankar’s investment duration = 12 months (since he started the business)
– Aniket joins after 3 months, so his investment duration = 9 months
– Total profit earned = Rs. 1,750,000
– We need to find Aniket’s share of the profit.

Step 1: Calculate the capital-time contributions of Shankar and Aniket.

The share of the profit is based on the capital-time contributions, which is the product of the amount invested and the time for which it is invested.

Shankar’s capital-time contribution:
\[
\text{Shankar’s capital-time} = 120,000 \times 12 = 1,440,000
\]

Aniket’s capital-time contribution:
\[
\text{Aniket’s capital-time} = 190,000 \times 9 = 1,710,000
\]

Step 2: Calculate the total capital-time contribution.
The total capital-time contribution is the sum of Shankar’s and Aniket’s contributions:
\[
\text{Total capital-time} = 1,440,000 + 1,710,000 = 3,150,000
\]

Step 3: Find the ratio of their capital-time contributions.
Now, the ratio of Shankar’s and Aniket’s contributions is:
\[
\text{Shankar : Aniket} = 1,440,000 : 1,710,000
\] Simplifying this ratio by dividing both numbers by 30,000:
\[
\text{Shankar : Aniket} = 48 : 57
\] Further simplifying the ratio by dividing by 3:
\[
\text{Shankar : Aniket} = 16 : 19
\]

Step 4: Calculate Aniket’s share of the profit.
The total profit is Rs. 1,750,000. Aniket’s share of the profit is:
\[
\text{Aniket’s share} = \frac{19}{16 + 19} \times 1,750,000 = \frac{19}{35} \times 1,750,000
\] Now calculate Aniket’s share:
\[
\text{Aniket’s share} = \frac{19 \times 1,750,000}{35} = \frac{33,250,000}{35} = 950,000
\]

Conclusion:
Aniket’s share in the profit is Rs. 950,000.

Answer: Option D
Solution:
To calculate the difference in the share of profits between Arun and Bakul, we need to consider the capital-time contributions and the total profit at the end of the year.

Given:
– Arun’s investment = Rs. 38,000
– Bakul’s investment = Rs. 55,000
– Arun’s investment duration = 12 months (since he started the business)
– Bakul joins after 5 months, so his investment duration = 7 months (12 months – 5 months)
– Total profit earned = Rs. 22,000
– We need to find the difference in their profit shares.

Step 1: Calculate the capital-time contributions.

The share of the profit is based on the capital-time contributions, which is the product of the amount invested and the time for which it is invested.

Arun’s capital-time contribution:
\[
\text{Arun’s capital-time} = 38,000 \times 12 = 456,000
\]

Bakul’s capital-time contribution:
\[
\text{Bakul’s capital-time} = 55,000 \times 7 = 385,000
\]

Step 2: Calculate the total capital-time contribution.
The total capital-time contribution is the sum of Arun’s and Bakul’s contributions:
\[
\text{Total capital-time} = 456,000 + 385,000 = 841,000
\]

Step 3: Find the ratio of their capital-time contributions.
Now, the ratio of Arun’s and Bakul’s contributions is:
\[
\text{Arun : Bakul} = 456,000 : 385,000
\] Simplifying the ratio:
\[
\text{Arun : Bakul} = \frac{456,000}{5,000} : \frac{385,000}{5,000} = 91.2 : 77
\] Rounding off the numbers, we get approximately:
\[
\text{Arun : Bakul} = 91 : 77
\]

Step 4: Calculate their shares of the profit.
The total profit is Rs. 22,000. The total parts in the profit-sharing ratio are:
\[
91 + 77 = 168
\]

Now, calculate Arun’s share of the profit:
\[
\text{Arun’s share} = \frac{91}{168} \times 22,000 = \frac{91 \times 22,000}{168} = \frac{2,002,000}{168} = 11,904.76
\] Arun’s share ≈ Rs. 11,905.

Now, calculate Bakul’s share of the profit:
\[
\text{Bakul’s share} = \frac{77}{168} \times 22,000 = \frac{77 \times 22,000}{168} = \frac{1,694,000}{168} = 10,095.24
\] Bakul’s share ≈ Rs. 10,095.

Step 5: Find the difference between their shares.
The difference between Arun’s and Bakul’s shares is:
\[
\text{Difference} = 11,905 – 10,095 = 1,810
\]

Conclusion:
The approximate difference between Arun’s and Bakul’s share of the profit is Rs. 1,810.

Answer: Option C
Solution:
To determine the ratio at which Goutam and Jatin will share the profit, we need to calculate their capital-time contributions. The profit-sharing ratio is directly proportional to the capital-time contributions.

Given:
– Goutam’s investment = Rs. 60,000
– Jatin’s investment = Rs. 35,000
– Goutam invests for the full 2 years (24 months).
– Jatin joins after 8 months, so Jatin invests for 16 months (24 months – 8 months).

Step 1: Calculate the capital-time contributions.

Goutam’s capital-time contribution:
\[
\text{Goutam’s capital-time} = 60,000 \times 24 = 1,440,000
\]

Jatin’s capital-time contribution:
\[
\text{Jatin’s capital-time} = 35,000 \times 16 = 560,000
\]

Step 2: Calculate the total capital-time contributions.
The total capital-time contribution is the sum of Goutam’s and Jatin’s contributions:
\[
\text{Total capital-time} = 1,440,000 + 560,000 = 2,000,000
\]

Step 3: Find the ratio of their capital-time contributions.
The ratio of Goutam’s and Jatin’s contributions is:
\[
\text{Goutam : Jatin} = 1,440,000 : 560,000
\]

Simplifying the ratio:
\[
\text{Goutam : Jatin} = \frac{1,440,000}{80,000} : \frac{560,000}{80,000} = 18 : 7
\]

Conclusion:
The ratio at which Goutam and Jatin will share the profit is 18 : 7.

Answer: Option E
Solution:
To calculate Simran’s share in the profit, we need to consider the capital-time contributions of both Simran and Nanda, as the profit is divided in proportion to these contributions.

Given:
– Simran’s investment = Rs. 50,000
– Nanda’s investment = Rs. 80,000
– Simran invests for 3 years (36 months).
– Nanda joins after 6 months, so Nanda invests for 2.5 years (30 months).
– Total profit earned = Rs. 24,500
– We need to find Simran’s share in the profit.

Step 1: Calculate the capital-time contributions.

Simran’s capital-time contribution:
Simran invests Rs. 50,000 for 36 months:
\[
\text{Simran’s capital-time} = 50,000 \times 36 = 1,800,000
\]

Nanda’s capital-time contribution:
Nanda invests Rs. 80,000 for 30 months:
\[
\text{Nanda’s capital-time} = 80,000 \times 30 = 2,400,000
\]

Step 2: Calculate the total capital-time contribution.
The total capital-time contribution is the sum of Simran’s and Nanda’s contributions:
\[
\text{Total capital-time} = 1,800,000 + 2,400,000 = 4,200,000
\]

Step 3: Find the ratio of their capital-time contributions.
The ratio of Simran’s and Nanda’s contributions is:
\[
\text{Simran : Nanda} = 1,800,000 : 2,400,000
\]

Simplifying the ratio by dividing both numbers by 600,000:
\[
\text{Simran : Nanda} = 3 : 4
\]

Step 4: Calculate Simran’s share of the profit.
The total profit is Rs. 24,500. The total ratio parts are:
\[
3 + 4 = 7
\]

Now, calculate Simran’s share:
\[
\text{Simran’s share} = \frac{3}{7} \times 24,500 = \frac{73,500}{7} = 10,500
\]

Conclusion:
Simran’s share in the profit is Rs. 10,500.

Answer: Option A
Solution:
To calculate the total profit, we need to understand the profit-sharing mechanism based on the capital-time contributions.

Given:
– Dilip’s investment = Rs. 2700
– Ram’s investment = Rs. 8100
– Avtar’s investment = Rs. 7200
– Ram’s share in the profit = Rs. 3600
– We need to find the total profit.

Step 1: Calculate the ratio of their capital-time contributions.
Since the profit is distributed based on the capital-time contributions, we first calculate their capital-time contributions. Since all the investments are for one year, we can directly use the capital amounts for the ratio.

The ratio of their capital-time contributions will be the ratio of their investments.

\[
\text{Dilip : Ram : Avtar} = 2700 : 8100 : 7200
\]

Now, simplify the ratio by dividing each number by 900:
\[
\text{Dilip : Ram : Avtar} = 3 : 9 : 8
\]

Step 2: Calculate the total parts in the ratio.
The total parts in the ratio are:
\[
3 + 9 + 8 = 20
\]

Step 3: Find Ram’s share of the total profit.
Since Ram’s share is Rs. 3600, and the ratio of the total parts is 20, Ram’s share corresponds to 9 parts out of 20. Therefore, we can calculate the total profit using the proportion:

\[
\frac{9}{20} \times \text{Total profit} = 3600
\]

Solving for the total profit:
\[
\text{Total profit} = \frac{3600 \times 20}{9} = \frac{72,000}{9} = 8000
\]

Conclusion:
The total profit earned is Rs. 8000.

Answer: Option C
Solution:
Let’s solve this step by step:

Given:
– Total investment = Rs. 47,000
– A invests Rs. 7,000 more than B.
– B invests Rs. 5,000 more than C.
– Total profit = Rs. 4,700
– We need to find C’s share of the profit.

Step 1: Set up the investments based on the conditions.
Let C’s investment be **x**.

– B’s investment is Rs. 5,000 more than C, so B’s investment = **x + 5000**.
– A’s investment is Rs. 7,000 more than B, so A’s investment = **(x + 5000) + 7000 = x + 12,000**.

Now, according to the problem:
\[
A + B + C = 47,000
\] Substitute the values for A and B:
\[
(x + 12,000) + (x + 5000) + x = 47,000
\] Simplifying the equation:
\[
3x + 17,000 = 47,000
\] \[
3x = 47,000 – 17,000 = 30,000
\] \[
x = \frac{30,000}{3} = 10,000
\]

So, C’s investment = Rs. 10,000, B’s investment = Rs. 15,000, and A’s investment = Rs. 22,000.

Step 2: Calculate the ratio of their investments.
The ratio of A, B, and C’s investments is:
\[
A : B : C = 22,000 : 15,000 : 10,000
\] Simplifying the ratio by dividing by 1,000:
\[
A : B : C = 22 : 15 : 10
\]

Step 3: Calculate C’s share in the profit.
The total number of parts is:
\[
22 + 15 + 10 = 47
\]

C’s share of the profit will be:
\[
\frac{10}{47} \times 4,700 = \frac{47,000}{47} = 1,000
\]

Conclusion:
C’s share of the profit is Rs. 1,000.

Answer: Option B
Solution:
To solve this, we need to determine the total profit based on the ratio of their investments, since the profit is distributed in proportion to the capital invested.

Given:
– A’s investment = Rs. 185,000
– B’s investment = Rs. 225,000
– B’s share of the profit = Rs. 9,000
– We need to find the total profit.

Step 1: Calculate the ratio of their investments.
The ratio of A’s investment to B’s investment is:
\[
\text{A’s investment} : \text{B’s investment} = 185,000 : 225,000
\]

Simplify the ratio by dividing both amounts by 5,000:
\[
185,000 : 225,000 = 37 : 45
\]

So, the ratio of their investments is **37 : 45**.

Step 2: Determine the total parts in the ratio.
The total number of parts in the ratio is:
\[
37 + 45 = 82
\]

Step 3: Use B’s share of the profit to find the total profit.
Since B’s share corresponds to 45 parts out of 82, and B’s share is Rs. 9,000, we can calculate the total profit by setting up the proportion:

\[
\frac{45}{82} \times \text{Total profit} = 9,000
\]

Solving for the total profit:
\[
\text{Total profit} = \frac{9,000 \times 82}{45} = \frac{738,000}{45} = 16,400
\]

Conclusion:
The total profit earned by them together is Rs. 16,400.

Answer: Option D
Solution:
To find the total profit earned, we need to determine the ratio of A’s and B’s investments and use that to calculate the total profit.

Given:
– A’s investment = Rs. 35,000
– B’s investment = Rs. 56,000
– A’s share in the profit = Rs. 45,000
– We need to find the total profit.

Step 1: Calculate the ratio of their investments.
The ratio of A’s investment to B’s investment is:
\[
\text{A’s investment} : \text{B’s investment} = 35,000 : 56,000
\]

Simplify the ratio by dividing both amounts by 7,000:
\[
35,000 : 56,000 = 5 : 8
\]

So, the ratio of their investments is **5 : 8**.

Step 2: Determine the total parts in the ratio.
The total number of parts in the ratio is:
\[
5 + 8 = 13
\]

Step 3: Use A’s share in the profit to find the total profit.
Since A’s share corresponds to 5 parts out of 13, and A’s share is Rs. 45,000, we can calculate the total profit by setting up the proportion:

\[
\frac{5}{13} \times \text{Total profit} = 45,000
\]

Solving for the total profit:
\[
\text{Total profit} = \frac{45,000 \times 13}{5} = \frac{585,000}{5} = 1,17,000
\]

Conclusion:
The total profit earned by them is Rs. 1,17,000.