Partnership Questions in English

(B) The ratio of their investments is calculated based on the product of capital and time period for each partner.
Satheesh invested for $12$ months,Sunil for $9$ months $(12 - 3 = 9)$,and Subhash for $3$ months $(9 - 6 = 3)$.
Ratio of investments = $(60000 \times 12) : (75000 \times 9) : (105000 \times 3)$
$= 720000 : 675000 : 315000$
Dividing by $45000$,we get the ratio = $16 : 15 : 7$.
Sum of ratio parts = $16 + 15 + 7 = 38$.
Total profit = ₹ $19000$.
Share of Satheesh = $(16 / 38) \times 19000 = ₹ 8000$.
Share of Sunil = $(15 / 38) \times 19000 = ₹ 7500$.
Share of Subhash = $(7 / 38) \times 19000 = ₹ 3500$.

(C) Let the investments of $A$ and $B$ be $3x$ and $5x$ respectively.
Since $C$ joined with an amount equal to $B$,the investment of $C$ is $5x$.
$A$ and $B$ invested for $12$ months,while $C$ invested for $6$ months.
The ratio of profit distribution is equal to the ratio of (Investment $\times$ Time).
Ratio $= (3x \times 12) : (5x \times 12) : (5x \times 6)$
$= 36x : 60x : 30x$
Dividing by $6x$,we get the ratio as $6 : 10 : 5$.

(D) Given: Investment of Nikita $(C_1)$ = ₹ $50,000$,Investment of Nishita $(C_2)$ = ₹ $40,000$,Total Profit $(P)$ = ₹ $22,500$.
The ratio of their capitals is $C_1 : C_2 = 50,000 : 40,000 = 5 : 4$.
Total ratio parts = $5 + 4 = 9$.
Nikita's share of profit = $\frac{5}{9} \times 22,500$.
Nikita's share = $5 \times 2,500 = ₹ 12,500$.

(A) The ratio of the investments made by Niki,Nisha,and Anu is $75,000 : 60,000 : 40,000$.
Simplifying this ratio by dividing by $5,000$,we get $15 : 12 : 8$.
Since the time period for all partners is the same ($3$ years),the profit-sharing ratio is equal to the investment ratio,which is $15 : 12 : 8$.
The sum of the ratio terms is $15 + 12 + 8 = 35$.
The total profit earned is ₹ $26,250$.
Anu's share in the profit is calculated as $\frac{8}{35} \times 26,250$.
$\frac{26,250}{35} = 750$.
Therefore,Anu's share $= 8 \times 750 = ₹ 6,000$.

(C) 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 shares = $(16000 \times 9) : (12000 \times 6) : (8000 \times 12)$
$= 144000 : 72000 : 96000$
Dividing by $24000$,we get the ratio as $6 : 3 : 4$.
Total parts = $6 + 3 + 4 = 13$.
Suresh's share = $\frac{3}{13} \times 26000 = 3 \times 2000 = ₹ 6000$.

(A) Step $1$: Calculate the equivalent capital for each partner for $1$ year ($12$ months).
Sita's investment: She invested ₹ $5,000$ for $3$ months and then added ₹ $2,000$,making it ₹ $7,000$ for the remaining $9$ months.
Sita's equivalent capital $= (5000 \times 3) + (7000 \times 9) = 15000 + 63000 = 78000$.
Step $2$: Gita's investment: She invested ₹ $4,000$ for $1$ month. Then she withdrew $\frac{1}{4}$ of $4000$ (which is $1000$),leaving $3000$ for the remaining $11$ months.
Gita's equivalent capital $= (4000 \times 1) + (3000 \times 11) = 4000 + 33000 = 37000$.
Step $3$: Rita's investment: Rita joined after $1$ month and invested ₹ $7,000$ for the remaining $11$ months.
Rita's equivalent capital $= 7000 \times 11 = 77000$.
Step $4$: Calculate the ratio of their shares: $78000 : 37000 : 77000 = 78 : 37 : 77$.
Total ratio sum $= 78 + 37 + 77 = 192$.
Step $5$: Rita's share $= \frac{77}{192} \times 1218 = \frac{93786}{192} \approx 488.47$.

(C) Let the investment of $B$ be $x$.
$A$ invested ₹ $3500$ for $12$ months.
$B$ invested $x$ for $(12 - 5) = 7$ months.
The ratio of profits is equal to the ratio of the product of investment and time.
$\frac{\text{Profit of } A}{\text{Profit of } B} = \frac{3500 \times 12}{x \times 7}$
Given the ratio is $2:3$,so:
$\frac{2}{3} = \frac{42000}{7x}$
$14x = 126000$
$x = \frac{126000}{14} = 9000$
Therefore,$B$ contributed ₹ $9000$.

(B) Let the capital of Gupta be $5k$ and Bansal be $6k$.
Let Bansal's capital be invested for $x$ months.
The profit ratio is given by the product of capital and time.
Ratio of profits = $\frac{5k \times 8}{6k \times x} = \frac{5}{9}$.
Simplifying the equation: $\frac{40}{6x} = \frac{5}{9}$.
Cross-multiplying gives $5 \times 6x = 40 \times 9$.
$30x = 360$.
$x = \frac{360}{30} = 12$ months.
Therefore,Bansal's capital was invested for $12$ months.

(B) Let $x$ be the number of months Brij remained in the business.
Arvind invested ₹ $550$ for $12$ months.
Brij invested ₹ $330$ for $x$ months.
The ratio of profits is equal to the ratio of the product of capital and time.
$\frac{\text{Arvind's profit}}{\text{Brij's profit}} = \frac{550 \times 12}{330 \times x}$
Given ratio is $\frac{10}{3}$.
$\frac{10}{3} = \frac{6600}{330x}$
$3300x = 19800$
$x = \frac{19800}{3300} = 6$ months.
Since Brij was in the business for $6$ months out of $12$ months,he joined after $12 - 6 = 6$ months.

(B) Let $B$ remain in the business for $x$ months.
Investment of $A = ₹ 3750$,Time = $12$ months.
Investment of $B = ₹ 5000$,Time = $x$ months.
Since the profits are divided equally,the ratio of their shares is $1:1$.
Profit ratio = (Investment of $A \times$ Time of $A$) : (Investment of $B \times$ Time of $B$).
$1/1 = (3750 \times 12) / (5000 \times x)$.
$5000x = 45000$.
$x = 45000 / 5000 = 9$ months.
$B$ remained in the business for $9$ months,which means $B$ joined after $12 - 9 = 3$ months.
Therefore,the correct option is $B$ (After $3$ months).

(B) Let Brijesh's capital be invested for $x$ months.
The ratio of the capitals of Anju and Brijesh is $5:9$.
Let the capitals of Anju and Brijesh be $5y$ and $9y$ respectively.
Anju's investment product is $C_1 \times t_1 = 5y \times 8 = 40y$.
Brijesh's investment product is $C_2 \times t_2 = 9y \times x = 9yx$.
The ratio of their profits is equal to the ratio of their investment products:
$\frac{\text{Anju's profit}}{\text{Brijesh's profit}} = \frac{C_1 \times t_1}{C_2 \times t_2}$
Given the profit ratio is $4:9$,we have:
$\frac{4}{9} = \frac{40y}{9yx}$
$\frac{4}{9} = \frac{40}{9x}$
$4x = 40$
$x = 10$ months.
Thus,Brijesh's capital was invested for $10$ months.

(A) The profit earned by partners is proportional to the product of their invested capital and the time period for which the capital is invested.
Let the capitals of $A$, $B$, and $C$ be $3x$, $5x$, and $9x$ respectively.
Let the time periods for which $A$, $B$, and $C$ invested their capitals be $2y$, $3y$, and $y$ respectively.
The ratio of their profits is given by $(Capital_A \times Time_A) : (Capital_B \times Time_B) : (Capital_C \times Time_C)$.
Profit ratio $= (3x \times 2y) : (5x \times 3y) : (9x \times y)$.
Profit ratio $= 6xy : 15xy : 9xy$.
Dividing by $3xy$, we get the ratio as $2 : 5 : 3$.

(A) We know that the profit share $P$ is given by the product of capital $C$ and time $T$,i.e.,$P = C \times T$.
Therefore,the ratio of time $T$ is given by $T = P / C$.
Given,the ratio of capitals $C_1 : C_2 : C_3 = 1 : 2 : 3$.
Given,the ratio of profits $P_1 : P_2 : P_3 = 1 : 2 : 3$.
Thus,the ratio of time is $T_1 : T_2 : T_3 = (P_1 / C_1) : (P_2 / C_2) : (P_3 / C_3)$.
Substituting the values,we get $T_1 : T_2 : T_3 = (1 / 1) : (2 / 2) : (3 / 3) = 1 : 1 : 1$.
Hence,Sumit,Punit,and Ramit invested their capitals for an equal period of time.

(D) We know that $\text{Profit} = \text{Capital} \times \text{Time}$, therefore $\text{Capital} = \frac{\text{Profit}}{\text{Time}}$.
Given, the ratio of profits $(P_A : P_B : P_C) = 4 : 5 : 6$.
The ratio of time periods $(t_A : t_B : t_C) = 2 : 3 : 6$.
The ratio of capitals $(C_A : C_B : C_C) = \frac{P_A}{t_A} : \frac{P_B}{t_B} : \frac{P_C}{t_C}$.
Substituting the values, we get $\frac{4}{2} : \frac{5}{3} : \frac{6}{6} = 2 : \frac{5}{3} : 1$.
To simplify, multiply each term by $3$: $(2 \times 3) : (\frac{5}{3} \times 3) : (1 \times 3) = 6 : 5 : 3$.
Since $6:5:3$ is not among the options, the correct answer is $D$ (None of these).

(C) The rent is distributed in the ratio of (number of oxen $\times$ time period).
Ratio of shares of $A, B, C = (12 \times 6) : (8 \times 7) : (6 \times 8)$
$= 72 : 56 : 48$
Dividing by the common factor $8$,we get the ratio as $9 : 7 : 6$.
Sum of the ratio terms $= 9 + 7 + 6 = 22$.
Rent paid by $A = \frac{9}{22} \times 396$
$= 9 \times 18 = 162$.
Thus,the rent paid by $A$ is ₹ $162$.

(D) The total capital is considered as $1$ unit.
$D$'s share of capital $= 1 - (\frac{1}{3} + \frac{1}{4} + \frac{1}{5})$.
To add the fractions,find the least common multiple $(LCM)$ of $3, 4, 5$,which is $60$.
$D$'s share $= 1 - (\frac{20}{60} + \frac{15}{60} + \frac{12}{60}) = 1 - \frac{47}{60} = \frac{13}{60}$.
Since the profit is distributed in proportion to the capital contributed,$D$'s share of the profit is $\frac{13}{60}$ of the total profit.
$D$'s share $= \frac{13}{60} \times 6000 = 13 \times 100 = ₹ 1300$.

(D) The profit sharing ratio is determined by the product of the capital invested and the time period for which it was invested.
$1$. $A$ invests ₹ $1500$ for $12$ months: $1500 \times 12 = 18000$.
$2$. $B$ invests ₹ $2000$ for $9$ months: $2000 \times 9 = 18000$.
$3$. $C$ joins after $4$ months and stays for the remaining $8$ months $(12 - 4 = 8)$. $C$ invests ₹ $2250$ for $8$ months: $2250 \times 8 = 18000$.
Therefore,the ratio of their profits is $18000 : 18000 : 18000$,which simplifies to $1: 1: 1$.

(C) The ratio of profit is given by the product of investment and time period.
Let the investments of $A$ and $B$ be $5x$ and $7x$ respectively.
Let the time period for $B$'s investment be $y$ months.
Given that the time period for $A$'s investment is $7$ months.
The ratio of their profits is $\frac{P_A}{P_B} = \frac{I_A \times T_A}{I_B \times T_B}$.
Substituting the given values: $\frac{1}{2} = \frac{5x \times 7}{7x \times y}$.
Simplifying the equation: $\frac{1}{2} = \frac{5}{y}$.
Therefore,$y = 5 \times 2 = 10$ months.

(B) Total profit in the business $= ₹ 1040$.
$A$ is an active partner and receives $25 \%$ of the total profit separately.
Separate profit for $A = 25 \% \text{ of } ₹ 1040 = \frac{1040 \times 25}{100} = ₹ 260$.
Remaining profit $= ₹ 1040 - ₹ 260 = ₹ 780$.
The remaining profit is divided in the ratio of their capital investments.
Ratio of capitals of $A$ and $B = 2100 : 3100 = 21 : 31$.
Sum of ratio terms $= 21 + 31 = 52$.
$A$'s share from remaining profit $= \frac{21}{52} \times 780 = ₹ 315$.
$B$'s share from remaining profit $= \frac{31}{52} \times 780 = ₹ 465$.
Total profit for $A = ₹ 315 + ₹ 260 = ₹ 575$.
Total profit for $B = ₹ 465$.
Thus,the profit shares for $A$ and $B$ are ₹ $575$ and ₹ $465$ respectively.

(A) Let the total profit in the business be $₹ x$.
$60 \%$ of the total profit $= 0.6x = \frac{3x}{5}$.
This amount is divided equally,so each partner receives $\frac{1}{2} \times \frac{3x}{5} = \frac{3x}{10}$.
The remaining profit is $x - \frac{3x}{5} = \frac{2x}{5}$.
This remaining profit is divided in the ratio of their capitals,which is $12500 : 8500 = 25 : 17$.
The share of the first partner from the remaining profit is $\frac{25}{25+17} \times \frac{2x}{5} = \frac{25}{42} \times \frac{2x}{5} = \frac{5x}{21}$.
The share of the second partner from the remaining profit is $\frac{17}{42} \times \frac{2x}{5} = \frac{17x}{105}$.
Total profit for the first partner $= \frac{3x}{10} + \frac{5x}{21}$.
Total profit for the second partner $= \frac{3x}{10} + \frac{17x}{105}$.
Given that the difference is $₹ 300$:
$(\frac{3x}{10} + \frac{5x}{21}) - (\frac{3x}{10} + \frac{17x}{105}) = 300$.
$\frac{5x}{21} - \frac{17x}{105} = 300$.
$\frac{25x - 17x}{105} = 300 \implies \frac{8x}{105} = 300$.
$x = \frac{300 \times 105}{8} = 37.5 \times 105 = 3937.50$.
Thus,the total profit is $₹ 3937.50$.

(D) The given ratio of shares is $\frac{7}{2}: \frac{4}{3}: \frac{6}{5}.$ To simplify,multiply by the $LCM$ of $2, 3, 5$ which is $30.$
Ratio $= (\frac{7}{2} \times 30) : (\frac{4}{3} \times 30) : (\frac{6}{5} \times 30) = 105 : 40 : 36.$
Let the initial investments be $105x, 40x,$ and $36x.$
$A$ increases his share by $50\%$ after $4 \text{ months}.$ New share of $A = 105 + (50\% \text{ of } 105) = 105 + 52.5 = 157.5.$
Ratio of equivalent capitals for $12 \text{ months}$:
$A = (105 \times 4) + (157.5 \times 8) = 420 + 1260 = 1680.$
$B = 40 \times 12 = 480.$
$C = 36 \times 12 = 432.$
Ratio $= 1680 : 480 : 432.$ Dividing by $48$,we get $35 : 10 : 9.$
Total parts $= 35 + 10 + 9 = 54.$
$B$'s share $= \frac{10}{54} \times 21600 = 10 \times 400 = ₹ 4000.$

(C) Let the total profit be $₹ P$.
After donating $5\%$ to charity,the remaining profit is $P - 0.05P = 0.95P$.
$A$'s share is $\frac{3}{3+2} = \frac{3}{5}$ of the remaining profit.
According to the problem,$A$'s share is $₹ 855$.
So,$\frac{3}{5} \times 0.95P = 855$.
$0.57P = 855$.
$P = \frac{855}{0.57} = \frac{85500}{57} = 1500$.
Therefore,the total profit is $₹ 1500$.

(D) $1$. $A$'s share for managing the business $= 12 \frac{1}{2} \% \text{ of } ₹ 880 = \frac{25}{200} \times 880 = ₹ 110$.
$2$. Remaining profit $= ₹ 880 - ₹ 110 = ₹ 770$.
$3$. The ratio of investments of $A$ and $B = 5000 : 6000 = 5 : 6$.
$4$. $A$'s share from the remaining profit $= \frac{5}{5+6} \times 770 = \frac{5}{11} \times 770 = 5 \times 70 = ₹ 350$.
$5$. $A$'s total profit $= ₹ 350 + ₹ 110 = ₹ 460$.

(A) The profit ratio is determined by the product of the investment and the time period for which it was invested.
Let $B$'s investment be $x$ and $C$'s investment be $y$.
$A$ invested ₹ $1200$ for $12$ months.
$B$ invested $x$ for $(12 - 3) = 9$ months.
$C$ invested $y$ for $(12 - 6) = 6$ months.
The ratio of profits is $(1200 \times 12) : (x \times 9) : (y \times 6) = 2 : 3 : 5$.
Comparing $A$ and $B$'s profit ratio:
$\frac{1200 \times 12}{x \times 9} = \frac{2}{3}$
$\frac{14400}{9x} = \frac{2}{3}$
$18x = 14400 \times 3$
$18x = 43200$
$x = \frac{43200}{18} = 2400$.
Thus,$B$'s investment is ₹ $2400$.

(D) The ratio of profit sharing is equal to the ratio of the product of capital and time for each partner.
Investment of $A = (3512420 \times 2) + (3512420 - 1100000) \times 1 = 7024840 + 2412420 = 9437260$.
Investment of $B = 4222180 \times 3 = 12666540$.
Investment of $C = (4065400 \times 2) + (4065400 + 800000) \times 1 = 8130800 + 4865400 = 12996200$.
Total investment ratio $= 9437260 : 12666540 : 12996200 = 943726 : 1266654 : 1299620$.
Sum of ratios $= 943726 + 1266654 + 1299620 = 3510000$.
Share of $C = \frac{1299620}{3510000} \times 1053000 = \frac{129962}{351} \times 1053 = 129962 \times 3 = 389886$.
Thus,the share of $C$ is ₹ $389886$.

(A) Let the total yearly profit be ₹ $x$.
Salary received by $B$ per year $= 120 \times 12 = ₹ 1440$.
Net profit to be distributed equally $= x - 1440$.
Share of $A$ and $B$ each $= \frac{x - 1440}{2}$.
Interest paid by $B$ to $A$ on half the capital $= \frac{10}{100} \times \frac{45000}{2} = \frac{10}{100} \times 22500 = ₹ 2250$.
$A$'s total income $= \text{Share} + \text{Interest} = \frac{x - 1440}{2} + 2250 = \frac{x - 1440 + 4500}{2} = \frac{x + 3060}{2}$.
$B$'s total income $= \text{Share} + \text{Salary} - \text{Interest} = \frac{x - 1440}{2} + 1440 - 2250 = \frac{x - 1440 - 1620}{2} = \frac{x - 3060}{2}$.
Given that $B$'s income $= \frac{1}{2} \times A$'s income:
$\frac{x - 3060}{2} = \frac{1}{2} \times \frac{x + 3060}{2}$.
$2(x - 3060) = x + 3060$.
$2x - 6120 = x + 3060$.
$x = 9180$.
Thus,the total yearly profit is ₹ $9180$.

(C) The ratio of capital invested by $A, B,$ and $C$ is $12000: 15000: 18000 = 4: 5: 6$.
Let the total annual profit be $P$. $A$ receives $15 \%$ of $P$ as a working partner. The remaining profit $(85 \% \text{ of } P)$ is distributed among $A, B,$ and $C$ in the ratio of their investments $(4: 5: 6)$.
Since $B$ received ₹ $8500$ and the ratio of $B$ to $C$ is $5: 6$, we verify: $\frac{8500}{5} = 1700$. Thus, $C$'s share from the remaining profit is $6 \times 1700 = ₹ 10200$, which matches the given data.
$A$'s share from the remaining profit is $4 \times 1700 = ₹ 6800$.
Let $R$ be the remaining profit: $R = 6800 + 8500 + 10200 = ₹ 25500$.
Since $R$ represents $85 \%$ of the total profit $P$, we have $0.85P = 25500$, so $P = \frac{25500}{0.85} = ₹ 30000$.
$A$'s salary (working partner share) is $15 \%$ of $P = 0.15 \times 30000 = ₹ 4500$.
Total amount received by $A = (\text{Share from profit}) + (\text{Salary}) = 6800 + 4500 = ₹ 11300$.

(C) The investment ratio of $A : B : C$ is calculated based on the capital invested for the duration of $12$ months.
$A$'s investment: $(16000 \times 3) + (11000 \times 9) = 48000 + 99000 = 147000$.
$B$'s investment: $(12000 \times 3) + (17000 \times 9) = 36000 + 153000 = 189000$.
$C$'s investment: $21000 \times 6 = 126000$ (since $C$ joined after $6$ months total).
Ratio $A : B : C = 147000 : 189000 : 126000 = 49 : 63 : 42$.
Simplifying by dividing by $7$,we get $7 : 9 : 6$.
Total parts = $7 + 9 + 6 = 22$.
Share of $B$ exceeds $C$ by $(9 - 6) = 3$ parts.
Difference = $\frac{3}{22} \times 26400 = 3 \times 1200 = ₹ 3600$.

(B) To find the share of profit,we calculate the ratio of the equivalent capital invested by each partner for $1$ year (or $12$ months).
$A$'s total investment: $30000 \times 12 = 360000$.
$B$'s total investment: $(24000 \times 4) + (18000 \times 8) = 96000 + 144000 = 240000$.
$C$'s total investment: $(42000 \times 4) + (32000 \times 8) = 168000 + 256000 = 424000$.
Ratio of investments $= 360000 : 240000 : 424000 = 360 : 240 : 424$.
Dividing by $8$,we get the ratio $= 45 : 30 : 53$.
Sum of ratio parts $= 45 + 30 + 53 = 128$.
$B$'s share $= \frac{30}{128} \times 11960 = 2803.125$.
Rounding to the nearest integer,$B$'s share is ₹ $2803$.

(B) Let the score of Ajay be $x$.
Rahul's score $= x - 15$.
Since Rahul's score is $10$ more than Manish's,Manish's score $= (x - 15) - 10 = x - 25$.
Given that Ajay scored $30$ marks more than the average of Rahul,Manish,and Suresh $(63)$:
$x = 63 + 30 = 93$.
Thus,Ajay's score $= 93$.
Rahul's score $= 93 - 15 = 78$.
Manish's score $= 93 - 25 = 68$.
The total score of Rahul,Manish,and Suresh $= 3 \times 63 = 189$.
Suresh's score $= 189 - (78 + 68) = 189 - 146 = 43$.
The sum of Manish's and Suresh's scores $= 68 + 43 = 111$.

(C) Let the common multiplier for the ratio be $x$.
Then,the amounts received by $P$,$Q$,and $R$ are $3x$,$5x$,and $7x$,respectively.
According to the problem,the amount received by $R$ is ₹ $4,000$ more than the amount received by $Q$.
So,$7x - 5x = 4000$.
$2x = 4000$,which gives $x = 2000$.
The total amount received by $P$ and $Q$ together is $3x + 5x = 8x$.
Substituting the value of $x$,we get $8 \times 2000 = ₹ 16000$.