Partnership Questions in English

(B) In a partnership business,the profit is distributed in the same ratio as the capital invested by the partners,assuming the time period of investment is the same for both.
Ratio of profits = Ratio of investments
Ratio of profits = $45000 : 30000$
Dividing both terms by $15000$,we get:
Ratio of profits = $3 : 2$

(A) Since the investments were made for the same period of time,the ratio of their profit shares is equal to the ratio of their investments.
Ratio of investments $= 60,000 : 50,000 = 6 : 5$.
Total ratio parts $= 6 + 5 = 11$.
Kishore's share $= \frac{5}{11} \times \text{Total Profit}$.
Kishore's share $= \frac{5}{11} \times 33,000 = 5 \times 3,000 = 15,000$.
Therefore,Kishore's share is $Rs. 15,000$.

(D) The ratio of profit shares is equal to the ratio of their investments.
Investment of Anil $= Rs. 3300$
Investment of Nikhil $= Rs. 5000$
Ratio of investments $= 3300 : 5000 = 33 : 50$
Total ratio parts $= 33 + 50 = 83$
Total profit $= Rs. 1660$
Anil's share $= \frac{33}{83} \times 1660$
Anil's share $= 33 \times 20 = Rs. 660$

(B) The profit distribution ratio is determined by the product of the investment amount and the time period for which the capital was invested.
Rahul invested $Rs. 45,000$ for $12$ months.
Sanjay invested $Rs. 30,000$ for $6$ months (since he joined $6$ months later).
Ratio of profits = (Rahul's investment $\times$ Rahul's time) : (Sanjay's investment $\times$ Sanjay's time)
Ratio $= (45000 \times 12) : (30000 \times 6)$
Ratio $= 540000 : 180000$
Ratio $= 54 : 18 = 3 : 1$.
Thus,the profit should be distributed in the ratio $3:1$.

(B) The profit sharing ratio is equal to the investment ratio,which is $6:5:8$.
Let the common ratio constant be $x$.
Then,the total profit is the sum of their individual shares: $6x + 5x + 8x = 83,600$.
$19x = 83,600$.
$x = 83,600 / 19 = 4,400$.
Saurabh's share corresponds to the ratio part $8$.
Therefore,Saurabh's share $= 8x = 8 \times 4,400 = 35,200$ $Rs.$

(D) The profit earned by partners is distributed in the ratio of the product of their capital invested and the time period for which it was invested.
Profit Ratio = (Capital Ratio) $\times$ (Time Ratio)
For Ram: $2 \times 6 = 12$
For Karan: $3 \times 4 = 12$
For Rohan: $4 \times 3 = 12$
Ratio of profits = $12 : 12 : 12$
Simplifying the ratio by dividing by $12$,we get $1 : 1 : 1$.

(D) In a partnership business,the ratio of profits is equal to the ratio of investments made by the partners,provided the time period of investment is the same for both.
Investment of Ravi $= Rs. 8000$
Investment of Kavi $= Rs. 72000$
Ratio of investments $= 8000 : 72000$
Dividing both sides by $8000$,we get:
Ratio $= 1 : 9$
Since the time period is the same (one year),the ratio of their profits is equal to the ratio of their investments.
Therefore,the ratio of their profits $= 1 : 9$.

(B) The ratio of investments made by Srikant and Vividh is $185000 : 225000 = 185 : 225 = 37 : 45$.
Let the total profit earned be $P$.
The share of profit is distributed in the ratio of their investments.
Vividh's share of profit $= \frac{45}{37 + 45} \times P = \frac{45}{82} \times P$.
Given that Vividh's share is $Rs. 9000$,we have:
$\frac{45}{82} \times P = 9000$.
Solving for $P$:
$P = \frac{9000 \times 82}{45} = 200 \times 82 = 16400$.
Therefore,the total profit earned by them is $Rs. 16400$.

(D) The ratio of investments made by Rajan and Sajan is $14200 : 15600$.
Simplifying this ratio: $142 : 156 = 71 : 78$.
The sum of the ratio parts is $71 + 78 = 149$.
The total profit is $Rs. 74500$.
Rajan's share $= \frac{71}{149} \times 74500$.
Since $74500 / 149 = 500$,we get $71 \times 500 = 35500$.
Therefore,Rajan's share is $Rs. 35500$.

(A) The profit ratio in a partnership is proportional to the product of the capital invested and the time period for which it was invested.
Equivalent amount invested by $A = 4000 \times 12 = 48000$.
Since $B$ joins after $3$ months,$B$ invests for $12 - 3 = 9$ months.
Equivalent amount invested by $B = 16000 \times 9 = 144000$.
The ratio of their profits $= 48000 : 144000$.
Simplifying the ratio: $48 : 144 = 1 : 3$.

(D) In a partnership business,the profit is distributed based on the product of the investment and the time period for which the investment is made.
Let the investments of Ram,Rohan,and Karn be $3x$,$2x$,and $5x$ respectively.
Let the profit earned by Rohan be $P$.
According to the problem,Ram earns $100\%$ more profit than Rohan,so Ram's profit $= P + 100\% \text{ of } P = 2P$.
Karn earns $40\%$ more profit than Rohan,so Karn's profit $= P + 40\% \text{ of } P = 1.4P$.
The total profit is the sum of individual profits: $2P + P + 1.4P = 4.4P$.
However,the problem does not provide the total profit amount earned by the business.
Since the total profit value is missing,it is impossible to calculate the specific numerical share of Rohan.
Therefore,the share of Rohan cannot be determined.

(C) To find the share of profit,we calculate the ratio of the product of capital and time for each partner.
Ram's investment: $Rs. 45000$ for $12$ months.
Sanjay's investment: $Rs. 60000$ for $9$ months (since he joined after $3$ months).
Aditya's investment: $Rs. 90000$ for $6$ months (since he joined after $6$ months).
Ratio of profits = $(45000 \times 12) : (60000 \times 9) : (90000 \times 6)$
$= 540000 : 540000 : 540000$
$= 1 : 1 : 1$
Total profit = $Rs. 16500$.
Ram's share = $\frac{1}{1+1+1} \times 16500 = \frac{1}{3} \times 16500 = Rs. 5500$.

(C) To find the share of profit,we calculate the product of the investment and the time period for each partner:
$1$. Ram's investment: $1200 \times 12 = 14400$
$2$. Anil's investment: $1500 \times 6 = 9000$
$3$. Aditya's investment: $900 \times 9 = 8100$ (Aditya joins after $3$ months,so he stays for $12 - 3 = 9$ months)
Ratio of profits = $14400 : 9000 : 8100$
Dividing by $900$,we get the ratio: $16 : 10 : 9$
Total ratio sum = $16 + 10 + 9 = 35$
Total profit = $Rs. 2450$
Aditya's share = $\frac{9}{35} \times 2450 = 9 \times 70 = Rs. 630$

(C) To find the share of profit,we calculate the equivalent capital for one month for each partner:
$P$'s investment: $(50 \times 4) + (25 \times 8) = 200 + 200 = 400$
$Q$'s investment: $(45 \times 5) + (22.5 \times 7) = 225 + 157.5 = 382.5$
$S$'s investment: $(70 \times 7) = 490$
Ratio of profits $(P : Q : S) = 400 : 382.5 : 490$
To simplify,multiply by $2$: $800 : 765 : 980$
Divide by $5$: $160 : 153 : 196$
Total ratio sum $= 160 + 153 + 196 = 509$
$S$'s share $= \frac{196}{509} \times 1272.5 = 196 \times 2.5 = Rs. 490$

(C) invests $Rs. 12,000$ for $4$ months and $(12,000 + 5,000) = Rs. 17,000$ for $8$ months.
$B$ invests $Rs. 16,000$ for $4$ months and $(16,000 - 6,000) = Rs. 10,000$ for $8$ months.
$C$ invests $Rs. 20,000$ for $6$ months (since $C$ joined $2$ months after $A$ and $B$ changed their investments,i.e.,$4+2=6$ months from the start).
Calculating the equivalent capital for one month:
$A = (12,000 \times 4) + (17,000 \times 8) = 48,000 + 136,000 = 184,000$
$B = (16,000 \times 4) + (10,000 \times 8) = 64,000 + 80,000 = 144,000$
$C = 20,000 \times 6 = 120,000$
Ratio of profits $= 184,000 : 144,000 : 120,000 = 184 : 144 : 120 = 23 : 18 : 15$.
Total ratio sum $= 23 + 18 + 15 = 56$.
$A$'s share $= (23 / 56) \times 30,100 = 23 \times 537.5 = Rs. 12,362.5$.

(A) We know that the profit is distributed in the ratio of the product of investment and time period.
Let the investments be $I_P = 8x, I_Q = 6x, I_R = 5x$.
Let the time periods be $T_P, T_Q, T_R$.
Profit ratio is given as $P_P : P_Q : P_R = 1 : 3 : 5$.
Since $\text{Profit} = \text{Investment} \times \text{Time}$,we have $\text{Time} = \frac{\text{Profit}}{\text{Investment}}$.
Therefore,the ratio of time periods is:
$T_P : T_Q : T_R = \frac{1}{8} : \frac{3}{6} : \frac{5}{5}$
$T_P : T_Q : T_R = \frac{1}{8} : \frac{1}{2} : 1$
To simplify,multiply by the least common multiple of the denominators,which is $8$:
$T_P : T_Q : T_R = (\frac{1}{8} \times 8) : (\frac{1}{2} \times 8) : (1 \times 8)$
$T_P : T_Q : T_R = 1 : 4 : 8$.
Thus,the correct option is $A$.

(D) Total time $= 36$ months.
Let Aditya's time of contribution $= t_A$ and Manish's time $= t_M$.
Since $t_A + t_M = 36$,we have $t_M = 36 - t_A$.
The ratio of their profit shares is equal to the ratio of their investments multiplied by time: $(300 \times t_A) : (500 \times t_M)$.
Total profit $= Rs. 1020$.
Aditya's share $= Rs. 495$,so Manish's share $= 1020 - 495 = Rs. 525$.
The ratio of their profits $= 495 : 525 = 33 : 35$.
Equating the ratios: $\frac{300 \times t_A}{500 \times (36 - t_A)} = \frac{33}{35}$.
Simplifying: $\frac{3 \times t_A}{5 \times (36 - t_A)} = \frac{33}{35} \Rightarrow \frac{t_A}{36 - t_A} = \frac{33}{35} \times \frac{5}{3} = \frac{11}{7}$.
$7 \times t_A = 11 \times (36 - t_A) \Rightarrow 7t_A = 396 - 11t_A$.
$18t_A = 396 \Rightarrow t_A = 22$ months.

(A) First,simplify the ratio of shares by multiplying by the $LCM$ of denominators $(2, 3, 5)$,which is $30$:
Aditya : Manish : Gaurav = $(\frac{7}{2} \times 30) : (\frac{4}{3} \times 30) : (\frac{6}{5} \times 30) = 105 : 40 : 36$.
To make calculations easier,multiply by $2$: $210 : 80 : 72$.
Aditya's initial share is $210$. After $4$ months,he increases it by $50\%$,so his new share is $210 + (0.5 \times 210) = 210 + 105 = 315$.
Now,calculate the effective investment for the year:
Aditya's investment = $(210 \times 4) + (315 \times 8) = 840 + 2520 = 3360$.
Manish's investment = $80 \times 12 = 960$.
Gaurav's investment = $72 \times 12 = 864$.
The ratio of profit is $3360 : 960 : 864$. Dividing by $48$,we get $70 : 20 : 18$,or $35 : 10 : 9$.
Total parts = $35 + 10 + 9 = 54$.
Manish's share = $\frac{10}{54} \times 43200 = 10 \times 800 = Rs. 8000$.

(A) In a partnership business,the profit is distributed in the ratio of the product of capital invested and the time period for which it is invested.
Let the capital invested by both Sita and Gita be $C$.
Sita invested her capital for $12$ months.
Let Gita invest her capital for $x$ months.
The ratio of their profits is given by:
$\frac{\text{Sita's Profit}}{\text{Gita's Profit}} = \frac{C \times 12}{C \times x} = \frac{12}{x}$
Given that the profit ratio is $3:2$,we have:
$\frac{3}{2} = \frac{12}{x}$
$3x = 24$
$x = \frac{24}{3} = 8$
Therefore,Gita invested her capital for $8$ months.

(B) The profit earned by partners is proportional to the product of their capital and the time period for which it is invested.
Let the time periods for $S, T,$ and $U$ be $x, y,$ and $z$ respectively.
The ratio of capitals is $3: 4: 6$.
The ratio of profits is given by $(\text{Capital} \times \text{Time}) = 3x : 4y : 6z$.
We are given that the profit ratio is $1 : 2 : 3$.
Therefore,$3x : 4y : 6z = 1 : 2 : 3$.
Comparing the first two terms: $\frac{3x}{4y} = \frac{1}{2} \implies 6x = 4y \implies \frac{x}{y} = \frac{4}{6} = \frac{2}{3}$.
Comparing the last two terms: $\frac{4y}{6z} = \frac{2}{3} \implies 12y = 12z \implies y = z$.
Thus,the ratio $x : y : z = 2 : 3 : 3$.

(B) Let the investment of $A$ be $x$.
$B$'s investment $= x + 50,000$.
$C$'s investment $= (x + 50,000) - 25,000 = x + 25,000$.
Total investment: $x + (x + 50,000) + (x + 25,000) = 3,00,000$.
$3x + 75,000 = 3,00,000$.
$3x = 2,25,000$.
$x = 75,000$.
Investment of $A = 75,000$.
Investment of $B = 75,000 + 50,000 = 1,25,000$.
Investment of $C = 75,000 + 25,000 = 1,00,000$.
Ratio of investments $A:B:C = 75,000 : 1,25,000 : 1,00,000 = 3 : 5 : 4$.
Total ratio parts $= 3 + 5 + 4 = 12$.
$C$'s share of profit $= \frac{4}{12} \times 14,400 = \frac{1}{3} \times 14,400 = 4,800$.
Thus,$C$'s share is ₹ $4,800$.

(D) Let the total profit earned by the business be $x$.
Since $30\%$ of the profit is reinvested,the remaining profit distributed between $A$ and $B$ is $70\%$ of $x$,which is $0.7x$.
The ratio of investment is $5:8$,so the profit is also shared in the ratio $5:8$.
$A$'s share $= \frac{5}{5+8} \times 0.7x = \frac{5}{13} \times 0.7x$.
Given that $A$'s share is ₹ $87,500$,we have:
$\frac{5}{13} \times 0.7x = 87,500$
$0.7x = 87,500 \times \frac{13}{5}$
$0.7x = 17,500 \times 13 = 227,500$
$x = \frac{227,500}{0.7} = 325,000$.
Thus,the total profit made by the business is ₹ $325,000$.

(C) Let the investments of $A$ and $B$ be $3x$ and $8x$ respectively.
$C$ invested $\frac{3}{4}$ of $B$'s investment,so $C$'s investment $= \frac{3}{4} \times 8x = 6x$.
$A$ and $B$ invested for $12$ months,while $C$ invested for $(12 - 4) = 8$ months.
The ratio of their profit shares is the ratio of (investment $\times$ time):
$A:B:C = (3x \times 12) : (8x \times 12) : (6x \times 8)$
$= 36x : 96x : 48x$
Dividing by $12x$,we get the ratio $3 : 8 : 4$.
Let the total profit be $P$. $C$'s share is $\frac{4}{3+8+4} \times P = \frac{4}{15} \times P$.
Given $C$'s share $= ₹ 24000$,so $\frac{4}{15} \times P = 24000$.
$P = 24000 \times \frac{15}{4} = 6000 \times 15 = ₹ 90000$.

(A) The ratio of investment of $A$ and $B$ is $4:5$. Let the investments be $4x$ and $5x$ respectively.
$A$ stays in the business for the full year ($12$ months),while $B$ stays for $10$ months.
The ratio of their profit shares is calculated as (Investment $\times$ Time).
Profit ratio of $A:B = (4x \times 12) : (5x \times 10) = 48x : 50x = 24 : 25$.
Total profit = ₹ $49,000$.
Sum of ratio parts = $24 + 25 = 49$.
Value of one part = $49000 / 49 = 1000$.
$A$'s share = $24 \times 1000 = ₹ 24,000$.
(Note: The provided options did not contain the correct answer $24000$. Based on the calculation,$A$'s share is ₹ $24,000$.)

(A) Let $x$ be the number of months Simran's capital was invested in the business.
Rohit's investment = $75000$ for $12$ months.
Simran's investment = $60000$ for $x$ months.
The ratio of their profits is given by the ratio of the product of their capital and time.
Ratio of profit = $(75000 \times 12) : (60000 \times x) = 3 : 1$.
$\frac{75000 \times 12}{60000 \times x} = \frac{3}{1}$.
$\frac{75 \times 12}{60 \times x} = 3$.
$\frac{5 \times 12}{4 \times x} = 3$.
$\frac{15}{x} = 3$.
$x = 5$ months.
Simran joined after $12 - 5 = 7$ months.

(A) Let the investments of $A, B,$ and $C$ be $I_A, I_B,$ and $I_C$ respectively.
According to the problem:
$3 I_A = 4 I_B \implies I_A = \frac{4}{3} I_B$
$I_B = 2 I_C \implies I_C = \frac{1}{2} I_B$
The ratio of their investments is $I_A : I_B : I_C = \frac{4}{3} I_B : I_B : \frac{1}{2} I_B$.
To simplify,multiply by $6$:
Ratio $= 8 : 6 : 3$.
Since the profit is distributed in the ratio of investments,the ratio of their shares is $8 : 6 : 3$.

(D) To find the amount invested in Scheme $B$,we need to determine its value.
From Statement $III$,we know the amount invested in Scheme $A$ is $₹ 45,000$.
From Statement $I$,the ratio of investment in $A$ and $B$ is $2:3$. If we know $A = 45,000$,then $\frac{45,000}{B} = \frac{2}{3}$,which gives $B = \frac{45,000 \times 3}{2} = ₹ 67,500$.
From Statement $II$,we know $A$ is $40\%$ of the total. Since $A = 45,000$,we can find the total amount $(T)$ as $0.40 \times T = 45,000$,so $T = 112,500$. Then $B = T - A = 112,500 - 45,000 = ₹ 67,500$.
Thus,we need Statement $III$ combined with either Statement $I$ or Statement $II$ to find the answer.
Therefore,the correct option is $D$.

(A) The profit in a partnership is divided in the ratio of the product of the capital invested and the time period for which it is invested.
Ratio of profit = (Capital of 1st partner $\times$ Time) : (Capital of 2nd partner $\times$ Time)
Ratio of profit = $(1200 \times 5) : (750 \times 4)$
Ratio of profit = $6000 : 3000 = 2 : 1$
Total parts = $2 + 1 = 3$
Share of 1st partner = $450 \times (2/3) = ₹ 300$
Share of 2nd partner = $450 \times (1/3) = ₹ 150$
Thus,the profit is divided in the ratio $2:1$.

(C) To find the profit-sharing ratio,we calculate the equivalent capital for one month for each partner:
$A$'s contribution: $₹ 1500$ for $12$ months = $1500 \times 12 = 18000$.
$B$'s contribution: $₹ 2000$ for $9$ months = $2000 \times 9 = 18000$.
$C$'s contribution: $C$ joins after $4$ months and stays for the remaining $8$ months. Contribution = $2250 \times 8 = 18000$.
The ratio of their shares is $18000 : 18000 : 18000$,which simplifies to $1 : 1 : 1$.
Therefore,the profit of $₹ 900$ is divided equally among them,with each receiving $₹ 300$.

(B) To find the profit distribution,we calculate the equivalent capital for one month for each partner:
$A$'s equivalent capital $= (50 \times 4) + (25 \times 8) = 200 + 200 = 400$.
$B$'s equivalent capital $= (45 \times 5) + (22.5 \times 7) = 225 + 157.5 = 382.5$.
$C$'s equivalent capital $= 70 \times 7 = 490$.
Ratio of shares $= 400 : 382.5 : 490$.
Multiply by $2$ to remove decimals: $800 : 765 : 980$.
Divide by $5$: $160 : 153 : 196$.
Sum of ratios $= 160 + 153 + 196 = 509$.
$A$'s share $= (160 / 509) \times 254 = 160 \times 0.5 = 80$.
$B$'s share $= (153 / 509) \times 254 = 153 \times 0.5 = 76.5 \approx 76$.
$C$'s share $= (196 / 509) \times 254 = 196 \times 0.5 = 98$.
Thus,the shares are $80, 76$ and $98$.

(C) Total investment = ₹ $114000$.
Ratio of profits = $337.50 : 1125 : 675$.
Dividing by $112.5$,we get the ratio $3 : 10 : 6$.
Sum of ratio parts = $3 + 10 + 6 = 19$.
Investment of first partner = $(3/19) \times 114000 = ₹ 18000$.
Investment of second partner = $(10/19) \times 114000 = ₹ 60000$.
Investment of third partner = $(6/19) \times 114000 = ₹ 36000$.
Total profit = $337.50 + 1125 + 675 = ₹ 2137.50$.
Percentage of profit = $(\text{Total Profit} / \text{Total Investment}) \times 100 = (2137.50 / 114000) \times 100 = 1.875 \%$.

(D) The ratio of profits is equal to the product of the ratio of capitals and the ratio of time periods.
Let the time period for which $B$'s capital was invested be $x$ months.
Given: Ratio of capitals $= 5: 6$.
Time period for $A = 8$ months.
Time period for $B = x$ months.
Ratio of profits $= (5 \times 8) : (6 \times x) = 40 : 6x$.
According to the problem,the ratio of profits is $5: 9$.
Therefore,$\frac{40}{6x} = \frac{5}{9}$.
Cross-multiplying gives: $40 \times 9 = 5 \times 6x$.
$360 = 30x$.
$x = \frac{360}{30} = 12$.
Thus,$B$'s capital was used for $12$ months.

(C) Let the total investment of $P, Q,$ and $R$ be $5y, 6y,$ and $6y$ respectively.
$P$ invested for $12$ months,$Q$ invested for $12$ months,and $R$ invested for $6$ months.
The ratio of their equivalent capitals for one month is:
$(5y \times 12) : (6y \times 12) : (6y \times 6) = 60y : 72y : 36y = 5 : 6 : 3$.
Total profit is $20\%$ of total investment. Let total investment be $I$.
$0.20 \times I = 98000 \Rightarrow I = 98000 \times 5 = 490000$.
Total ratio sum $= 5 + 6 + 3 = 14$.
$R$'s share of investment $= \frac{3}{14} \times 490000 = 3 \times 35000 = 210000$.
Thus,the amount invested by $R$ is $₹ 210000$.

(B) First,calculate the total investment equivalent for one year for each partner:
Shekhar's investment:
Year $1$: ₹ $25000$
Year $2$: ₹ $25000 + 10000 = 35000$
Year $3$: ₹ $35000 + 10000 = 45000$
Total equivalent investment for Shekhar = $25000 + 35000 + 45000 = 105000$ (units of one year).
Rajeev's investment:
Joined in $2000$,invested ₹ $35000$ for $2$ years.
Total equivalent investment for Rajeev = $35000 \times 2 = 70000$ (units of one year).
Jatin's investment:
Joined in $2001$,invested ₹ $35000$ for $1$ year.
Total equivalent investment for Jatin = $35000 \times 1 = 35000$ (units of one year).
Ratio of shares = $105000 : 70000 : 35000 = 3 : 2 : 1$.
Total profit = ₹ $150000$.
Rajeev's share = $\frac{2}{3+2+1} \times 150000 = \frac{2}{6} \times 150000 = \frac{1}{3} \times 150000 = ₹ 50000$.

(D) Let $B$'s contribution be ₹ $x$.
$A$ invested ₹ $3500$ for $12$ months.
$B$ invested ₹ $x$ for $(12 - 5) = 7$ months.
The ratio of profit is equal to the ratio of the product of capital and time.
$\therefore A:B = (3500 \times 12) : (x \times 7) = 2:3$
$\therefore \frac{3500 \times 12}{7x} = \frac{2}{3}$
$\therefore 7x \times 2 = 3500 \times 12 \times 3$
$\therefore 14x = 126000$
$\therefore x = \frac{126000}{14} = 9000$
Thus,$B$'s contribution is ₹ $9000$.

(B) The ratio of the investments of $A, B$ and $C$ is calculated based on the product of their capital and the time period for which it was invested:
Ratio $= (6500 \times 6) : (8400 \times 5) : (10000 \times 3)$
$= 39000 : 42000 : 30000 = 39 : 42 : 30 = 13 : 14 : 10$
Total profit $= ₹ 7400$.
$A$ receives $5 \%$ of the total profit as a working member:
$A$'s salary $= \frac{5}{100} \times 7400 = ₹ 370$.
Remaining profit to be distributed among $A, B$ and $C$ in the ratio of their investments $= 7400 - 370 = ₹ 7030$.
Sum of the ratio terms $= 13 + 14 + 10 = 37$.
$B$'s share in the profit $= \frac{14}{37} \times 7030 = 14 \times 190 = ₹ 2660$.

(B) The ratio of investments at the start is $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$. Multiplying by the $LCM$ of $2, 3, 4$ (which is $12$),we get the ratio $6 : 4 : 3$.
Let the initial investments be $6k, 4k,$ and $3k$.
$A$ invests $6k$ for $2$ months,then withdraws half $(3k)$ for the remaining $10$ months. Total investment by $A = (6k \times 2) + (3k \times 10) = 12k + 30k = 42k$.
$B$ invests $4k$ for the full $12$ months. Total investment by $B = 4k \times 12 = 48k$.
$C$ invests $3k$ for the full $12$ months. Total investment by $C = 3k \times 12 = 36k$.
The ratio of their effective investments is $42k : 48k : 36k = 7 : 8 : 6$.
The total profit is ₹ $378$. The sum of the ratio parts is $7 + 8 + 6 = 21$.
$B$'s share = $\frac{8}{21} \times 378 = 8 \times 18 = ₹ 144$.

(D) The ratio of initial investments is $\frac{7}{2}: \frac{4}{3}: \frac{6}{5}$.
To simplify,multiply by the $LCM$ of denominators $(30)$: $(7/2 \times 30) : (4/3 \times 30) : (6/5 \times 30) = 105 : 40 : 36$.
Let the initial investments be $105k, 40k$,and $36k$.
After $4$ months,$A$ increases his share by $50\%$. New investment of $A = 105k + 0.5 \times 105k = 157.5k$.
Total investment ratio for the year:
$A = (105k \times 4) + (157.5k \times 8) = 420k + 1260k = 1680k$.
$B = 40k \times 12 = 480k$.
$C = 36k \times 12 = 432k$.
Ratio $A:B:C = 1680:480:432$. Dividing by $48$,we get $35:10:9$.
Total parts $= 35 + 10 + 9 = 54$.
$B$'s share $= \frac{10}{54} \times 21600 = 10 \times 400 = ₹ 4000$.

(A) Let the initial capitals of $A$ and $B$ be $4k$ and $5k$ respectively.
$A$'s investment for the first $3$ months is $4k$. After $3$ months,$A$ withdraws $\frac{1}{4}$ of $4k = k$. So,the remaining capital is $3k$ for the next $7$ months.
Total equivalent investment of $A = (4k \times 3) + (3k \times 7) = 12k + 21k = 33k$.
$B$'s investment for the first $3$ months is $5k$. After $3$ months,$B$ withdraws $\frac{1}{5}$ of $5k = k$. So,the remaining capital is $4k$ for the next $7$ months.
Total equivalent investment of $B = (5k \times 3) + (4k \times 7) = 15k + 28k = 43k$.
The ratio of their profits is $33k : 43k = 33 : 43$.
Total profit = ₹ $760$.
$A$'s share = $\frac{33}{33 + 43} \times 760 = \frac{33}{76} \times 760 = 33 \times 10 = ₹ 330$.

(C) Given the ratio of investments between $A$ and $C$ is $2:1$.
Given the ratio of investments between $A$ and $B$ is $3:2$.
To find the combined ratio $A:B:C$,we equate the value of $A$ in both ratios.
$A:C = 2:1 = 6:3$
$A:B = 3:2 = 6:4$
Thus,the ratio of investments $A:B:C = 6:4:3$.
Total profit = ₹ $1,57,300$.
Sum of ratio parts = $6 + 4 + 3 = 13$.
Value of one part = $1,57,300 / 13 = 12,100$.
Share of $B = 4 \times 12,100 = ₹ 48,400$.

(A) The ratio of the profit shares is equal to the ratio of the product of their investments and the time period for which they invested.
Ratio of investments (Arun : Kamal : Vinay) = $(8000 \times 6) : (4000 \times 8) : (8000 \times 8)$
Dividing by $8000$,we get the ratio as $6 : 4 : 8$,which simplifies to $3 : 2 : 4$.
Total ratio sum = $3 + 2 + 4 = 9$.
Total gain = ₹ $4005$.
Share of Kamal = $\frac{2}{9} \times 4005 = 2 \times 445 = ₹ 890$.

(B) The ratio of profit is determined by the product of the capital invested and the time period for which it was invested.
Kamal's investment period = $12$ months.
Sameer's investment period = $12 - 5 = 7$ months.
Ratio of profit (Kamal : Sameer) = $(9000 \times 12) : (8000 \times 7)$.
$= (9 \times 12) : (8 \times 7) = 108 : 56$.
Dividing both sides by $4$,we get $27 : 14$.
Total ratio parts = $27 + 14 = 41$.
Sameer's share = $\frac{14}{41} \times 6970$.
$= 14 \times 170 = ₹ 2380$.

(A) The ratio of investments of $A, B$ and $C$ is $35000 : 45000 : 55000 = 35 : 45 : 55 = 7 : 9 : 11$.
Total ratio sum $= 7 + 9 + 11 = 27$.
Total profit $= ₹ 40500$.
Share of $A = \frac{7}{27} \times 40500 = 7 \times 1500 = ₹ 10500$.
Share of $B = \frac{9}{27} \times 40500 = 9 \times 1500 = ₹ 13500$.
Share of $C = \frac{11}{27} \times 40500 = 11 \times 1500 = ₹ 16500$.

(D) Simran's investment duration $= 3 \text{ years} = 36 \text{ months}$.
Nanda's investment duration $= 36 - 6 = 30 \text{ months}$.
Ratio of investments $= (50000 \times 36) : (80000 \times 30) = (5 \times 36) : (8 \times 30) = 180 : 240 = 3 : 4$.
Total profit $= ₹ 24500$.
Simran's share $= \frac{3}{3+4} \times 24500 = \frac{3}{7} \times 24500 = 3 \times 3500 = ₹ 10500$.

(B) The ratio of profit is equal to the ratio of the product of capital and time.
$A$'s investment: $(5000 \times 4) + (2500 \times 8) = 20000 + 20000 = 40000$.
$B$'s investment: $(4500 \times 6) + (3000 \times 6) = 27000 + 18000 = 45000$.
$C$'s investment: $7000 \times 6 = 42000$.
Ratio $A : B : C = 40000 : 45000 : 42000 = 40 : 45 : 42$.
Total ratio sum $= 40 + 45 + 42 = 127$.
Total profit $= ₹ 5080$.
Share of $A = (40 / 127) \times 5080 = ₹ 1600$.
Share of $B = (45 / 127) \times 5080 = ₹ 1800$.
Share of $C = (42 / 127) \times 5080 = ₹ 1680$.

(D) Let the investments of the three partners be $I_1, I_2,$ and $I_3$ respectively.
Profit is distributed in the ratio of (Investment $\times$ Time).
Given ratio of profit = $5: 7: 8$.
Time periods are $T_1 = 14$ months,$T_2 = 8$ months,and $T_3 = 7$ months.
So,$(I_1 \times 14) : (I_2 \times 8) : (I_3 \times 7) = 5 : 7 : 8$.
From this,we have:
$I_1 = \frac{5}{14}, I_2 = \frac{7}{8}, I_3 = \frac{8}{7}$ (in terms of profit units per month).
To find the ratio $I_1 : I_2 : I_3$,we calculate $\frac{5}{14} : \frac{7}{8} : \frac{8}{7}$.
Multiply by the $LCM$ of denominators $(14, 8, 7)$,which is $56$.
$I_1 : I_2 : I_3 = (\frac{5}{14} \times 56) : (\frac{7}{8} \times 56) : (\frac{8}{7} \times 56)$.
$I_1 : I_2 : I_3 = (5 \times 4) : (7 \times 7) : (8 \times 8)$.
$I_1 : I_2 : I_3 = 20 : 49 : 64$.

(C) Let the total capital be $x$ and the total profit be $P$.
$A$ contributes $\frac{1}{4}$ of the capital,so $B$ contributes $1 - \frac{1}{4} = \frac{3}{4}$ of the capital.
Let $B$'s money be invested for $y$ months.
The ratio of profit is equal to the ratio of (capital $\times$ time).
Profit ratio $A : B = \frac{1}{3} : \frac{2}{3} = 1 : 2$.
Therefore,$\frac{\frac{1}{4}x \times 15}{\frac{3}{4}x \times y} = \frac{1}{2}$.
$\frac{15}{3y} = \frac{1}{2}$.
$\frac{5}{y} = \frac{1}{2}$.
$y = 10$ months.

(D) Let $B$ be in the business for $x$ months.
Since $A$ invested for the full year,his time period is $12$ months.
The ratio of profit is equal to the ratio of the product of investment and time.
$\text{Profit ratio } A:B = (85000 \times 12) : (42500 \times x) = 3:1$.
Simplifying the ratio: $\frac{85000 \times 12}{42500 \times x} = \frac{3}{1}$.
Since $85000 = 2 \times 42500$,we have $\frac{2 \times 12}{x} = \frac{3}{1}$.
$\frac{24}{x} = 3$.
$x = \frac{24}{3} = 8$ months.
Thus,$B$ joined for $8$ months.

(A) The ratio of profit sharing is determined by the product of the investment and the time period for which the money was invested.
$A$ invested ₹ $20000$ for $24$ months.
$B$ invested ₹ $15000$ for $24$ months.
$C$ invested ₹ $20000$ for $(24 - 6) = 18$ months.
Ratio of investments $A : B : C = (20000 \times 24) : (15000 \times 24) : (20000 \times 18)$.
Dividing by $6000$,we get $A : B : C = (4 \times 20) : (3 \times 20) : (4 \times 15) = 80 : 60 : 60 = 4 : 3 : 3$.
Total ratio sum $= 4 + 3 + 3 = 10$.
Total profit $= ₹ 25000$.
$B$'s share $= \frac{3}{10} \times 25000 = ₹ 7500$.

(B) The profit distribution ratio is determined by the product of the investment amount and the time period for which the investment was held.
$1$. Aman's investment period = $3$ years = $36$ months.
Investment = ₹ $70,000$.
Product = $70,000 \times 36 = 25,20,000$.
$2$. Rakhi's investment period = $36 - 6 = 30$ months.
Investment = ₹ $1,05,000$.
Product = $1,05,000 \times 30 = 31,50,000$.
$3$. Sagar's investment period = $30 - 6 = 24$ months.
Investment = ₹ $1,40,000$.
Product = $1,40,000 \times 24 = 33,60,000$.
Ratio of profits = $25,20,000 : 31,50,000 : 33,60,000$.
Dividing by $70,000$:
$= 36 : 45 : 48$.
Dividing by $3$:
$= 12 : 15 : 16$.