Partnership — Part 2 MCQs with Explanations

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, $$\eqalign{ & {\text{12 units}} = {\text{Rs}}{\text{. 2400}} \cr & {\text{1 unit}} = \frac{{2400}}{{12}} \cr & 27{\text{ units}} = \frac{{2400 \times 27}}{{12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}5400 \cr} $$

$$\eqalign{ & {\text{Shankar}}:{\text{Aniket}} \cr & = \left( {120000 \times 12} \right):\left( {190000 \times 9} \right) \cr & = 1440000:1710000 \cr & = 16:19 \cr & \therefore {\text{Aniket's share}} \cr & = {\text{Rs}}{\text{.}}\left( {1750000 \times \frac{{19}}{{35}}} \right) \cr & = {\text{Rs. }}{\text{950000 }} \cr} $$

$$\eqalign{ & {\text{Arun}}:{\text{Bakul}} \cr & = \left( {38000 \times 12} \right):\left( {55000 \times 7} \right) \cr & = 456000:385000 \cr & = 456:385 \cr & \therefore {\text{Required difference}} \cr & = {\text{Rs}}{\text{.}}\,\left[ {22000 \times \left( {\frac{{456}}{{841}} - \frac{{385}}{{841}}} \right)} \right] \cr & = {\text{Rs}}{\text{.}}\left( {22000 \times \frac{{71}}{{841}}} \right) \cr & = {\text{Rs. }}{\text{1857}}{\text{.31}} \cr & = {\text{Rs. }}{\text{1857 }} \cr} $$

$$\eqalign{ & {\text{Gautam}}:{\text{Jatin}} \cr & = \left( {6000 \times 12} \right):\left( {35000 \times 8} \right) \cr & = 720000:280000 \cr & = 18:7 \cr} $$

$$\eqalign{ & {\text{Dilip}}:{\text{Ram}}:{\text{Avtar}} \cr & = 2700:8100:7200 \cr & = 3:9:8 \cr & {\text{Let the total profit be Rs}}.\,x \cr & {\text{Then, Ram's share}} \cr & = {\text{Rs}}{\text{.}}\,\left( {\frac{9}{{20}}x} \right) \cr & \therefore \frac{9}{{20}}x = 3600 \cr & \Rightarrow x = \left( {\frac{{3600 \times 20}}{9}} \right) \cr & \Rightarrow x = 8000 \cr & {\text{Hence,}} \cr & {\text{Total profit}} = {\text{Rs}}{\text{. 8000}} \cr} $$

Let C subscribes the business = Rs. x A : B : C Capital → (x + 12000) : (x + 5000) : x Note: Profit would be divide in the ratio of their capitals According to the question, $$\eqalign{ & \Rightarrow \left( {x + 12000} \right) + \left( {x + 5000} \right) + x = 47000 \cr & \Rightarrow 3x + 17000 = 47000 \cr & \Rightarrow 3x = 30000 \cr & \Rightarrow x = 10000 \cr & {\text{ A }}:{\text{ B }}:{\text{ C}} \cr & {\text{Capital}} \to 22000:15000:10000 \cr & {\text{Profit}} \to {\text{ }}22{\text{ }}:{\text{ }}15{\text{ }}:{\text{ }}10 \cr & \left( {22 + 15 + 10} \right){\text{ units}} = {\text{4700}} \cr & {\text{1 unit}} = \frac{{4700}}{{47}} = 100 \cr & {\text{Share of C}} = 10{\text{ units}} = 10 \times 100 = {\text{Rs}}.1000 \cr & \cr} $$

A B Capital → 1850000 : 225000 Profit → 37 : 45 ↓×200 : ↓×200 7400 9000 Total profit = (7400 + 9000) = Rs. 16400

A B Capital → 35000 : 56000 Profit → 5 : 8 ↓×9000 : ↓×9000 45000 72000 Total profit = (45000 + 72000) = Rs. 117000

Sonia : Vivek : Sagar = (60000 × 24) : (140000 × 18) : (120000 × 12) = 1440000 : 2520000 : 1440000 = 4 : 7 : 4 $$\eqalign{ & \therefore {\text{Vivek's share}} \cr & \Rightarrow {\text{Rs}}{\text{.}}\,\left( {450000 \times \frac{7}{{15}}} \right) \cr & \Rightarrow {\text{Rs}}{\text{. 210000}} \cr} $$

Let the initial capital of P and Q be Rs. 2x and Rs. 3x respectively Then, ratio of profits $$\eqalign{ & = \left( {2x \times 12} \right):\left( {3x \times 24} \right) \cr & = 24x:72x \cr & = 1:3 \cr & \therefore {\text{Q's share}} \cr & = {\text{Rs}}{\text{.}}\left( {26000 \times \frac{3}{4}} \right) \cr & = {\text{Rs. }}{\text{19500}} \cr} $$

Let the money invested by A, B and C be Rs. x, Rs. 2x and Rs. 3x respectively Then, A : B : C $$\eqalign{ & = \left( {x \times 12} \right):\left( {2x \times 6} \right):\left( {3x \times 4} \right) \cr & = 12x:12x:12x \cr & = 1:1:1 \cr & \therefore {\text{C's share}} \cr & = {\text{Rs}}{\text{.}}\left( {27000 \times \frac{1}{3}} \right) \cr & = {\text{Rs. }}{\text{9000}} \cr} $$

Let the capitals of Y was used for T months According to the question, $$\eqalign{ & \frac{{5 \times 8}}{{6 \times {\text{T}}}}{\text{ = }}\frac{5}{9} \cr & \Rightarrow {\text{T}} = {\text{12 months}} \cr} $$ Hence capital of Y was used for = 12 months

1 st Partner 2 nd Partner Capital → 125000 : 85000 25 : 17 8 According to the question, Note: 60% of profit should be divided equally between them $$\eqalign{ & {\text{8 units}} = {\text{Rs}}{\text{.}}\,{\text{300}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{.}}\,\frac{{{\text{300}}}}{8} \cr & {\text{42 units}} = {\text{Rs}}{\text{.}}\,\frac{{300 \times 42}}{8} \cr & 40\% {\text{ of profit}} \cr & = {\text{Rs}}{\text{.}}\,\frac{{300 \times 42}}{8} \cr & 100\% {\text{ profit}} \cr & = {\text{Rs}}.\,\frac{{300 \times 42 \times 100}}{{8 \times 40}} \cr & = {\text{Rs}}{\text{.}}\,{\text{3937}}{\text{.50}} \cr} $$

30% of the total profit is divided in the ratio 50000 : 70000 = 5 : 7 between the two partners. Suppose x = 30% of the total profit. Then $$\eqalign{ & x \times \frac{{7 - 5}}{{7 + 5}} = 90 \cr & x \times \frac{1}{6} = 90 \cr & x = 540 \cr & 30\% \,{\text{of}}\,{\text{the}}\,{\text{total}}\,{\text{profit}} = 540 \cr & {\text{Total Profit}} \cr & = \frac{{540 \times 100}}{{30}} \cr & = 1800 \cr} $$ Alternative From the 30% of the profit, let first partner gets x and second partner gets (x + 90) $$\eqalign{ & x:(x + 90) = 50000:70000 \cr & x:(x + 90) = 5:7 \cr & 7x = 5x + 450 \cr & 2x = 450 \cr & x = 225 \cr & \therefore 30\% \,{\text{of}}\,{\text{the}}\,{\text{profit}} \cr & = x + {\text{x}} + 90 \cr & = 225 + 225 + 90 \cr & = 540 \cr & {\text{Total Profit}} = \frac{{540 \times 100}}{{30}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1800 \cr} $$

A : B : C (40000 × 36) :(80000 × 12 + 40000 × 24) : (120000 × 24 + 40000 × 12) = 144 : 192 : 336 = 3 : 4 : 7

A : B : C = (25 lakhs × 1 + 35 lakhs × 2) : (35 lakhs × 2 + 25 lakhs × 1) : (30 lakhs × 3) = 95 lakhs : 95 lakhs : 90 lakhs = 19 : 19 : 18

$$\eqalign{ & {\text{Subhas}}:{\text{Aditya}} \cr & = \left( {25000 \times 36} \right):\left( {15000 \times 6 + 30000 \times 24} \right) \cr & = 900000:810000 \cr & = 10:9 \cr & \therefore {\text{Aditya's share}} \cr & \Rightarrow {\text{Rs}}{\text{.}}\left( {247000 \times \frac{9}{{19}}} \right) \cr & \Rightarrow {\text{Rs}}{\text{.117000}} \cr} $$

$$\eqalign{ & {\text{Here}}\,{{\text{p}}_1}:{{\text{p}}_2}:{{\text{p}}_3} = 5:3:12 \cr & {\text{and}}\,{{\text{x}}_1}:{{\text{x}}_2}:{{\text{x}}_3} = 5:6:8 \cr & {\text{According}}\,{\text{to}}\,{\text{the}}\,{\text{formula}} \cr & {\text{Required}}\,{\text{ratio}} \cr & = \frac{{{{\text{p}}_1}}}{{{{\text{x}}_1}}}:\frac{{{{\text{p}}_2}}}{{{{\text{x}}_2}}}:\frac{{{{\text{p}}_3}}}{{{{\text{x}}_3}}} \cr & = \frac{5}{5}:\frac{3}{6}:\frac{{12}}{8} \cr & = 1:\frac{1}{2}:\frac{3}{2} \cr & = 2:1:3 \cr & \cr & {\bf{Alternate:}} \cr & {\text{Profit}} = {\text{Time}} \times {\text{Capital invested}} \cr & {\text{Time}} = \frac{{{\text{Profit}}}}{{{\text{Capital invested}}}} \cr & {\text{Required ratio of time}} \cr & = \frac{5}{6}:\frac{3}{6}:\frac{{12}}{8} \cr & = 1:\frac{1}{2}:\frac{3}{2} \cr & = 2:1:3 \cr} $$

Given X : Z = 2 : 1 X : Y = 3 : 2 X : Z = 2 : 1 (multiply with 3) X : Y = 3 : 2 (multiply with 2) i.e X : Z = 6 : 3 X : Y = 6 : 4 ∴ X : Y : Z = 6 : 4 : 3 According to the question (6 + 4 + 3) units = Rs.157300 13 units = Rs.157300 1 unit = Rs.12100 4 units = Rs.12100 × 4 = Rs. 48400 ∴ Share of Y = Rs. 48400

A : B : C (5000 × 4 + 2500 × 8) : (4500 × 6 + 3000 × 6) : (7000 × 6) = 40000 : 45000 : 42000 = 40 : 45 : 42 $$\eqalign{ & \therefore {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left( {5080 \times \frac{{40}}{{127}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1600}} \cr & \therefore {\text{B's share}} \cr & = {\text{Rs}}{\text{.}}\left( {5080 \times \frac{{45}}{{127}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1800}} \cr & \therefore {\text{C's share}} \cr & = {\text{Rs}}{\text{.}}\left( {5080 \times \frac{{42}}{{127}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1680}} \cr} $$

$$\eqalign{ & {\text{Let C}} = x \cr & {\text{Then, B}} = x + 5000{\text{ and}} \cr & {\text{A}} = x + 5000 + 4000 = x + 9000 \cr & {\text{So, }} \cr & \Rightarrow x + x + 5000 + x + 9000 = 50000 \cr & \Rightarrow 3x = 36000 \cr & \Rightarrow x = 12000 \cr & \therefore {\text{A}}:{\text{B}}:{\text{C}} \cr & = 21000:17000:12000 \cr & = 21:17:12 \cr & \therefore {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left( {35000 \times \frac{{21}}{{50}}} \right) \cr & = {\text{Rs.}}{\text{14700}} \cr} $$

Ratio of investments of A, B and C = Ratio of their profits $$\eqalign{ & = 337.50:1125:637.50 \cr & = 9:30:17 \cr & \therefore {\text{A's investment}} \cr & = {\text{Rs}}{\text{.}}\left( {14000 \times \frac{9}{{56}}} \right) \cr & = {\text{Rs. }}{\text{2250}} \cr & {\text{B's investment}} \cr & = {\text{Rs}}{\text{.}}\left( {14000 \times \frac{{30}}{{56}}} \right) \cr & = {\text{Rs. }}{\text{7500}} \cr & {\text{Hence, required difference}} \cr & = {\text{Rs}}{\text{.}}\left( {7500 - 2250} \right) \cr & = {\text{Rs. }}{\text{5250}} \cr} $$

$$\eqalign{ & {\text{A}}:{\text{B}} \cr & 4:{\text{ }}5 \cr & {\text{According to the question,}} \cr & \left( {4 + 5} \right){\text{units}} = {\text{Rs}}{\text{. 14130}} \cr & {\text{1 unit}} = \frac{{14130}}{9} = {\text{Rs}}{\text{. }}1570 \cr & 5{\text{ units}} = 5 \times 1570 = {\text{Rs}}{\text{. 7850}} \cr & \therefore {\text{Hence share of B}} = {\text{Rs}}{\text{. 7850}} \cr} $$

Initial ratio of investments by A and B = 2 : 3 Let their respective investments are 2x and 3x According to the question, If A added Rs. 10000 to his investment Then the new ratio = 3 : 2 $$\eqalign{ & \Rightarrow \frac{{2x + 10000}}{{3x}} = \frac{3}{2} \cr & \Rightarrow 4x + 20000 = 9x \cr & \Rightarrow 5x = 20000 \cr & \Rightarrow x = 4000 \cr} $$ Original investment by A = 2 × 4000 = Rs. 8000

Let the initial investments of A, B and C be Rs. 5x, Rs. 6x and Rs. 8x respectively Then, A : B : C [5x × 12 + (160% of 5x) × 12] : (6x × 24) : (8x × 12 + 4x × 12) = 156x : 144x : 144x = 13 : 12 : 12

Fraction of profit received by each one of Jhon and Mona $$\eqalign{ & = \frac{{\left( {1 - \frac{2}{3}} \right)}}{2} \cr & = \frac{1}{6} \cr} $$ Ratio of capitals of Jhon, Mona and Gordon = Ratio of their profits $$\eqalign{ & \frac{1}{6}:\frac{1}{6}:\frac{2}{3} = 1:1:4 \cr & {\text{Let the total capital be Rs}}{\text{.}}x \cr & {\text{Then,}} \cr & \frac{1}{6}{\text{of }}\left( {7\% {\text{ of }}x - 4\% {\text{ of }}x} \right) \cr & = \$ {\text{ 60 million}} \cr & \Rightarrow 3\% \,{\text{of }}x = \$ {\text{ }}3{\text{60 million}} \cr & \Rightarrow x = \$ \left( {\frac{{360 \times 100}}{3}} \right){\text{million}} \cr & \Rightarrow x = \$ {\text{ 12000 million }} \cr & \therefore {\text{ Mona's capital }} \cr & {\text{ = }}\left( {\frac{1}{6} \times \$ {\text{12000 million }}} \right) \cr & = \$ \,200{\text{0 million }} \cr} $$

$$\eqalign{ & = {\text{Anu}}:{\text{Bimla}} \cr & = 5000:6000 \cr & = 5:6{\text{ }} \cr} $$ Anu's share for managing business $$\eqalign{ & = 12.5\% {\text{ of Rs}}{\text{.}}\,880 \cr & = {\text{Rs}}{\text{.}}\,110 \cr & {\text{Net profit}} \cr & = {\text{Rs}}{\text{.}}\left( {880 - 110} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{770}} \cr & {\text{Anu's share}} \cr & = {\text{Rs}}{\text{.}}\left( {770 \times \frac{5}{{11}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{350}} \cr & {\text{Anu's total share}} \cr & = {\text{Rs}}{\text{.}}\left( {110 + 350} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{460}} \cr & {\text{Bimla's share}} \cr & = {\text{Rs}}{\text{.}}\left( {770 \times \frac{6}{{11}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{420}} \cr} $$

Let capital of X be Rs. 11x and Y's capital be Rs. 12x and let time for which Y invested capital is T 2 months by using formula, $$\eqalign{ & \Leftrightarrow \frac{{{{\text{C}}_1} \times {{\text{T}}_1}}}{{{{\text{C}}_2} \times {{\text{T}}_2}}} = \frac{{{{\text{P}}_1}}}{{{{\text{P}}_2}}} \cr & \Leftrightarrow \frac{{11x \times 8}}{{12x \times {{\text{T}}_2}}} = \frac{2}{3} \cr & \Leftrightarrow {{\text{T}}_2} = 11{\text{ months}} \cr} $$ Hence the time for which Y invested his capital is 11 months

Let A's capital = Rs. x Let B's capital = Rs. y Now, according to the question, A B Capital → x y Time (in month) → (9 + 1) = 10 9 Ratio of profit → 5 6 $$\eqalign{ & {\text{We know,}} \cr & \frac{{10 \times x}}{{x \times y}} = \frac{5}{6} \cr & \Leftrightarrow \frac{x}{y} = \frac{3}{4} \cr} $$ Hence the required ratio of capital of A and B is = 3 : 4

$$\eqalign{ & {\text{Let the total profit be Rs}}{\text{. }}x \cr & {\text{Then, }}60\% {\text{ of the profit}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{60}}{{100}} \times x} \right) \cr & = {\text{Rs}}{\text{.}}\,\frac{{3x}}{5} \cr} $$ From this part of the profit each gets = $${\text{Rs}}{\text{.}}\,\frac{{3x}}{{10}}$$ $$\eqalign{ & 40\% {\text{ of total profit }} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{40}}{{100}} \times x} \right) \cr & = {\text{Rs}}{\text{.}}\,\frac{{2x}}{5} \cr} $$ Now, this amount of Rs. $$\frac{2x}{5}$$ has been divided in the ratio of capitals, which is 125000 : 85000 or 25 : 17 as interests ∴ Interest on first capital $$\eqalign{ & = {\text{ Rs}}{\text{.}}\left( {\frac{{2x}}{5} \times \frac{{25}}{{42}}} \right) \cr & {\text{ = Rs}}{\text{.}}\,\frac{{5x}}{{21}} \cr} $$ Interest on second capital $$\eqalign{ & = {\text{Rs}}{\text{.}}\left( {\frac{{2x}}{5} \times \frac{{17}}{{42}}} \right) \cr & {\text{ = Rs}}{\text{.}}\,\frac{{17x}}{{105}} \cr} $$ Total money received by first partner $$\eqalign{ & = {\text{Rs}}{\text{.}}\left( {\frac{{3x}}{{10}} + \frac{{5x}}{{21}}} \right) \cr & {\text{ = Rs}}{\text{.}}\,\frac{{113x}}{{210}}{\text{ }} \cr} $$ Total money received by second partner $$\eqalign{ & = {\text{Rs}}{\text{.}}\left( {\frac{{3x}}{{10}} + \frac{{17x}}{{105}}} \right) \cr & {\text{ = Rs}}{\text{.}}\,\frac{{97x}}{{210}} \cr & \therefore \frac{{113x}}{{210}} - \frac{{97x}}{{210}} = 3000 \cr & \Rightarrow x = 39375{\text{ }} \cr & {\text{Hence,}} \cr & {\text{Total profit Rs}}{\text{. 39375}} \cr} $$

$$\eqalign{ & {\text{Let Ninad's investment be Rs}}{\text{.}}x \cr & {\text{Then, ratio of capitals}} \cr & = \left( {x \times 12} \right):\left( {2x \times 6} \right):\left( {3x \times 4} \right) \cr & = 12x:12x:12x \cr & = 1:1:1 \cr & \therefore {\text{Manav's share}} \cr & = {\text{Rs}}{\text{.}}\left( {45000 \times \frac{1}{3}} \right) \cr & = {\text{Rs}}{\text{.15000}} \cr} $$

$$\eqalign{ & = {\text{A}}:{\text{B}}:{\text{C}}:{\text{D}} \cr & = 15 \times 4:12 \times 2:18 \times 6:16 \times 5 \cr & = 60:24:108:80 \cr & = 15:6:27:20 \cr & {\text{Let the total rent be Rs}}{\text{.}}x \cr & {\text{Then,}} \cr & {\text{A's share}} = {\text{Rs}}{\text{.}}\left( {\frac{{15x}}{{68}}} \right) \cr & \therefore \,\frac{{15x}}{{68}} = 1020 \cr & \Rightarrow x = \left( {\frac{{1020 \times 68}}{{15}}} \right) \cr & \Rightarrow x = 4624 \cr & {\text{Hence, C's share}} \cr & = {\text{Rs}}{\text{.}}\left( {4624 \times \frac{{27}}{{68}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1836}} \cr} $$

$$\eqalign{ & {\text{Let total profit}} = {\text{16 units }} \cr & {\text{According to the question,}} \cr & {\text{Profit share of A}} \cr & = \frac{3}{{16}} \times 16{\text{ units}} \cr & = 3{\text{ units}} \cr & {\text{profit share of B }} \cr & {\text{ = }}\frac{1}{4} \times 16 = 4{\text{ units}} \cr & {\text{then profit share of C}} \cr & = \left[ {16 - \left( {4 + 3} \right)} \right] = 9{\text{ units}} \cr & {\text{But profit of C}} = {\text{Rs}}{\text{. 243 (given)}} \cr & {\text{9 units}} = {\text{Rs}}{\text{. 243}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. 27}} \cr & {\text{Profit share of B}} \cr & = 4{\text{ units}} \cr & = 27 \times 4 \cr & = {\text{Rs}}{\text{. 108}} \cr} $$

Total profit = Rs. 880 Since A gets 15% of total profit for management ∴ Remaining profit $$\eqalign{ & = 880 - \frac{{880 \times 15}}{{100}} \cr & = {\text{Rs}}.\,748 \cr} $$ A B Amount 5000 6000 Ratio of Capital 5 : 6 The remaining profit is being divided in the ratio of their capital A's share of profit $$\eqalign{ & = \frac{{748}}{{\left( {5 + 6} \right)}} \times 5 \cr & = {\text{Rs}}.\,340 \cr} $$ Total profit received by A = 340 + 132 = Rs. 472

Ratio of initial investments $$\eqalign{ & = \frac{1}{3}:\frac{1}{4}:\frac{1}{5} \cr & = 20:15:12 \cr} $$ Let their initial investments be 20x, 15x and 12x respectively = A : B : C = (20x × 15 + 10x × 15) : (15x × 30) : (12x × 30) = 450x : 450x : 360x = 5 : 5 : 4 $$\eqalign{ & \therefore {\text{C's share}} \cr & = {\text{Rs}}{\text{.}}\left( {4340 \times \frac{4}{{14}}} \right) \cr & = {\text{Rs}}{\text{.1240}} \cr} $$

Let their initial investments be x, 3x and 5x respectively Then, $$ = {\text{A}}:{\text{B}}:{\text{C}}$$ $$ = \left( {x \times 4 + 2x \times 8} \right)$$ : $$\left( {3x \times 4 + \frac{{3x}}{2} \times 8} \right)$$ : $$\left( {5x \times 4 + \frac{{5x}}{2} \times 8} \right)$$ $$\eqalign{ & = 20x:24x:40x \cr & = 5:6:10 \cr} $$

$$\eqalign{ & {\text{Suppose, }} \cr & {\text{A invests Rs}}{\text{.}}\frac{x}{6}{\text{ for }}\frac{y}{6}{\text{ months}} \cr & {\text{Then,}} \cr & {\text{ B invests Rs}}{\text{.}}\frac{x}{3}{\text{ for }}\frac{y}{3}{\text{ months}} \cr & {\text{C invests}}\left[ {x - \left( {\frac{x}{6} + \frac{x}{3}} \right)} \right]i.e.,{\text{ Rs}}{\text{.}}\frac{x}{2}{\text{ for }}y{\text{ months}} \cr & \therefore {\text{A}}:{\text{B}}:{\text{C}} \cr & {\text{ = }}\left( {\frac{x}{6} \times \frac{y}{6}} \right):\left( {\frac{x}{3} \times \frac{y}{3}} \right):\left( {\frac{x}{2} \times y} \right) \cr & = \frac{1}{{36}}:\frac{1}{9}:\frac{1}{2} \cr & = 1:4:18 \cr & {\text{Hence, B's share}} \cr & = {\text{Rs}}{\text{. }}\left( {4600 \times \frac{4}{{23}}} \right) \cr & = {\text{Rs}}{\text{. 800}} \cr} $$

A B C Amounts invested 14000 Time (in months) 12 7 5 168000 $$\eqalign{ & {\text{Ratio of profits }}4:3:2 \cr & {\text{Let their profits }}4x:3x:2x \cr & \Leftrightarrow 4x = 168000 \cr & \Leftrightarrow x = 42000 \cr & \Rightarrow {\text{Profit share of C}} \cr & = 2x \cr & = 2 \times 42000 \cr & = {\text{Rs}}{\text{. 84000}} \cr & \Rightarrow {\text{Capital invested by C}} \cr & = \frac{{84000}}{5} \cr & = {\text{Rs. }}16800 \cr} $$

Let the total capital of A, B and C = 15 units Let total time for investment = 12 units Now, According to the question, Capital Investment of A = $$\frac{1}{3} \times 15$$ = 5units Capital Investment of B = $$\frac{1}{5} \times 15$$ = 3units Capital Investment of C = 15 - (5 + 3) = 7 A's Capital Invested for time = $$\frac{1}{4} \times 12$$ = 3units B's Capital Invested for time = $$\frac{1}{6} \times 12$$ = 2units C's Capital Invested all the time. i.e = 12units ∴ Profit ratio of A : B : C = (5 × 3) : (3 × 2) : (7 × 12) = 15 : 6 : 84 = 5 : 2 : 28 Total profit = 5 + 2 + 28 = 35 units Also, total profit = Rs. 1820 (given) $$\eqalign{ & {\text{35 units}} = {\text{Rs}}{\text{. 1820}} \cr & {\text{1 unit}} = \frac{{1820}}{{35}} = {\text{Rs}}.52 \cr & {\text{Hence A's share in profit}} \cr & = 5{\text{ units}} \cr & = 52 \times 5 \cr & = {\text{Rs}}{\text{. 260}} \cr} $$

Let C's capital be Rs. x Then, A : B : C = 2560 : 2000 : x $$\eqalign{ & {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left( {1105 \times \frac{{2560}}{{4560 + x}}} \right) \cr & \therefore 1105 \times \frac{{2560}}{{4560 + x}} = 320 \cr & \Rightarrow 320x + 1459200 = 2828800 \cr & \Rightarrow 320x = 1369600 \cr & \Rightarrow x = 4280 \cr} $$