Partnership — Part 1 MCQs with Explanations

Let their investments be Rs. x for 14 months, Rs. y for 8 months and Rs. z for 7 months respectively. Then, 14x : 8y : 7z = 5 : 7 : 8. Now, $$\eqalign{ & \frac{{14x}}{{8y}} = \frac{5}{7} \Rightarrow 98x = 40y \cr & \Rightarrow y = \frac{{49x}}{{20}} \cr & {\text{And, }}\frac{{14x}}{{7z}} = \frac{5}{8} \cr & \Rightarrow 112x = 35z \cr & \Rightarrow z = \frac{{112x}}{{35}} = \frac{{16x}}{5} \cr & {\text{So}}\,x:y:z = x:\frac{{49x}}{{20}}:\frac{{16x}}{5} \cr & ({\text{Multiply by 20}}) \cr & \Rightarrow 20:49:64 \cr} $$

Suppose C's capital = x then B's capital = 4x (Since B's Capital is four times C's capital) A's capital = 6x ( Since twice A's capital is equal to thrice B's capital) A : B : C =6x : 4x : x = 6 : 4 : 1 B's share = $$16500 \times \frac{4}{{11}}$$ = 1500 × 4 = 6000.

Now as per question, Kamal invested for 12 months and Sameer invested for 7 months. So Kamal : Sameer = (9000 × 12) : (8000 × 7) = 108 : 56 = 27 : 14 Sameer Ratio in profit will be $$\eqalign{ & = {6970 \times \frac{{14}}{{41}}} \cr & = {\text{Rs}}{\text{. 2380}} \cr} $$

Let the amount invested by Q = q. $$\eqalign{ & 40000:q = 2:3 \cr & \Rightarrow \frac{{40000}}{q} = \frac{2}{3} \cr & \Rightarrow q = 40000 \times \frac{3}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 60000}} \cr} $$

Let P's capital = p, Q's capital = q and R's capital = r. Then $$\eqalign{ & 4p = 6q = 10r \cr & \Rightarrow 2p = 3q = 5r \cr & \Rightarrow q = \frac{{2p}}{3} \cr & \Rightarrow r = \frac{{2p}}{5} \cr & {\text{P}}:{\text{Q}}:{\text{R}} = p:\frac{{2p}}{3}:\frac{{2p}}{5} \cr & ({\text{Multiply by 15}}) \cr & = 15:10:6 \cr & {\text{R's share}} = 4650 \times \frac{6}{{31}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 150 \times 6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}900 \cr} $$

$$\eqalign{ & A:B:C \cr & = \left( {10 \times 7} \right):\left( {12 \times 5} \right):\left( {15 \times 3} \right) \cr & = 70:60:45 \cr & = 14:12:9 \cr & \therefore {\text{C's rent}} = {\text{Rs}}{\text{.}}\left( {175 \times \frac{9}{{35}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}45 \cr} $$

Assume that investment of C = x Then, investment of A = 2x Investment of B = $$\frac{{2{\text{A}}}}{3}$$ = $$\frac{{4{\text{x}}}}{3}$$ (since ratio of investment of A : B = 2 : 3 i.e B = $$\frac{{2{\text{A}}}}{3}$$) A : B : C = 2x : $$\frac{{4{\text{x}}}}{3}$$ : x = 2 : $$\frac{4}{3}$$ : 1 = 6 : 4 : 3 $$\eqalign{ & {\text{B's Share}} \cr & = 157300 \times \frac{4}{{6 + 4 + 3}} \cr & = 157300 \times \frac{4}{{13}} \cr & = 12100 \times 4 \cr & = {\text{Rs}}{\text{. 48400}} \cr} $$

A : B : C = (20,000 × 24) : (15,000 × 24) : (20,000 × 18) = 4 : 3 : 3. $$\eqalign{ & {\text{So B's Share}} \cr & = {\text{Rs}}{\text{. }}25000 \times \frac{3}{{10}} \cr & = {\text{Rs}}{\text{. 7500}} \cr} $$

P : Q : R = (25 × 1 + 35 × 2) : (35 × 2 : 25 × 1) : (30 × 3) = 95 : 95 : 90 = 19 : 19: 18

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

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{total}}\,{\text{profit}}\,{\text{be}}\,{\text{Rs}}{\text{.}}\,{\text{100}} \cr & {\text{After}}\,{\text{paying}}\,{\text{to}}\,{\text{charity,}}\,{\text{A's}}\,{\text{share}} \cr & = Rs.\,\left( {95 \times \frac{3}{5}} \right) = Rs.\,57 \cr & {\text{If}}\,{\text{A's}}\,{\text{share}}\,{\text{is}}\,Rs.\,57, \cr & {\text{Total}}\,{\text{profit}} = Rs.\,100 \cr & {\text{If}}\,{\text{A's}}\,{\text{share}}\,Rs.\,855, \cr & {\text{Total}}\,{\text{profit}} \cr & = {\frac{{100}}{{57}} \times 855} \cr & = 1500 \cr} $$

For managing, A received = 5% of Rs. 7400 = Rs. 370. Balance = Rs. (7400 - 370) = Rs. 7030. Ratio of their investments = (6500 x 6) : (8400 x 5) : (10000 x 3) = 39000 : 42000 : 30000 = 13 : 14 : 10 $$\eqalign{ & \therefore {\text{ B's Share}} \cr & = {\text{Rs}}{\text{. 7030}} \times \frac{{14}}{{37}} \cr & = {\text{Rs}}{\text{. 2660}} \cr} $$

Ratio of initial investments $$\eqalign{ & = {\frac{7}{2}:\frac{4}{3}:\frac{6}{5}} \cr & = 105:40:36 \cr} $$ Let the initial investments be 105x, 40x and 36x $$\left( {105x \times 4 + \frac{{150}}{{100}} \times 105x \times 8} \right)$$ $$:$$ $$\left( {40x \times 12} \right)$$ $$:$$ $$\left( {36x \times 12} \right)$$ $$\eqalign{ & = 1680x:480x:432x \cr & = 35:10:9 \cr & {\text{Hence,}}{\kern 1pt} {\text{B's}}{\kern 1pt} {\text{share}} \cr & = {\text{Rs}}{\text{.}}\left( {21600 \times \frac{{10}}{{54}}} \right) \cr & = {\text{Rs}}{\text{.}}\,4000 \cr} $$

$$\eqalign{ & {\text{Let}}\,C = x \cr & {\text{Then,}}\,{\text{B}} = x + 50000\,{\text{and}} \cr & {\text{A}} = x + 50000 + 40000 \cr & \,\,\,\,\,\,\,\, = x + 90000 \cr & So,\,x + x + 50000 + x + 90000 = 500000 \cr & \Rightarrow 3x = 360000 \cr & \Rightarrow x = 120000 \cr & {\text{A:B:C}} = 210000:170000:120000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 21:17:12 \cr & \therefore {\text{A's}}\,{\text{share}} \cr & = Rs.\,\left( {350000 \times \frac{{21}}{{50}}} \right) \cr & = Rs.\,1,47,000 \cr} $$

Let B's capital be Rs. x ∴ A's share in 12 months = 3500 × 12 And, B's share in 7 months = 7x $$\eqalign{ & {\text{Then}},\, {\frac{{3500 \times 12}}{{7x}} = \frac{2}{3}} \cr & \Rightarrow 14x = 126000 \cr & \Rightarrow x = 9000 \cr} $$

$$A:B$$ $$ = \left[ {4x \times 3 + \left( {4x - \frac{1}{4} \times 4x} \right) \times 7} \right]$$ $$:$$ $$\left[ {5x \times 3 + \left( {5x - \frac{1}{5} \times 5x} \right) \times 7} \right]$$ $$\eqalign{ & = \left( {12x + 21x} \right):\left( {15x + 28x} \right) \cr & = 33x:43x = 33:43 \cr & \therefore {\text{A's share}} = {\text{Rs}}{\text{.}}\left( {760 \times \frac{{33}}{{76}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,330 \cr} $$

Let initial investment of A is 3x and B is 5x, then C investment is also 5x, but most important to note in this question is the time duration of investment Like, A invested for 12 months, B invested for 12 months and C invested for 6 months. A : B : C = (3 x x 12) : (5 x x 12) : (5 x x 6) = 36 : 60 : 30 = 6 : 10 : 5.

$$\eqalign{ & {\text{Suppose}}\,{\text{B}}\,{\text{joined}}\,{\text{for}}\,x\,{\text{months}}. \cr & {\text{Then}}, \cr & {\frac{{85000 \times 12}}{{42500 \times x}} = \frac{3}{1}} \cr & \Rightarrow x = {\frac{{85000 \times 12}}{{42500 \times 3}}} = 8 \cr & {\text{So,}}\,{\text{B}}\,{\text{joined}}\,{\text{for}}\,{\text{8}}\,{\text{months}} \cr} $$

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

$$\eqalign{ & {\text{Arun}}\,{\text{:}}\,{\text{Kamal}}\,{\text{:}}\,{\text{Vinay}} \cr & = \left( {8000 \times 6} \right):\left( {4000 \times 8} \right):\left( {8000 \times 8} \right) \cr & = 48:32:64 \cr & = 3:2:4 \cr & \therefore {\text{Kamal's}}\,{\text{share}} \cr & = Rs.\,\left( {4005 \times \frac{2}{9}} \right) \cr & = Rs.\,890 \cr} $$

$$\eqalign{ & {\text{Simran}}\,{\text{:}}\,{\text{Nanda}} \cr & = \left( {50000 \times 36} \right):\left( {80000 \times 30} \right) \cr & = 3:4 \cr & \therefore {\text{Simran's}}\,{\text{share}} \cr & = Rs.\,\left( {24500 \times \frac{3}{7}} \right) \cr & = Rs.\,10,500 \cr} $$

Note: If you have clear concept of ratio and proportion chapter then it will really easy for you to solve partnership problems. P : Q = 85000 : 15000 = 17 : 3 Important to note there that if both have invested for different period of times then we had to multiply with number of months to get the desired ratio.

Let C = x Then B = $$\frac{{\text{x}}}{2}$$ and A = $$\frac{{\text{x}}}{4}$$ A : B : C = 1 : 2 : 4 C's share = $${\text{Rs}}{\text{.}}\left( {\frac{4}{7} \times 700} \right) = 400$$

Ratio of their shares = 22500 : 35000 = 9 : 14 Deepak's share $$\eqalign{ & = {\text{Rs}}{\text{.}}\left( {13800 \times \frac{{14}}{{23}}} \right) \cr & = {\text{Rs}}{\text{. 8}}400 \cr} $$

Suppose B joined after x months then, 21000 × 12 = 36000 × (12 - x) ⇒ 36x = 180 ⇒ x = 5

Nirmal : Kapil = 9000 × 12 : (12000 × 6 + 6000 × 6) = 1 : 1 Kapils share $$\eqalign{ & = {\text{Rs}}{\text{.}}\left( {4600 \times \frac{1}{2}} \right) \cr & = {\text{Rs}}{\text{. 2300}} \cr} $$

Suppose Ramesh invested Rs. x. Then, Manoj : Ramesh = 120000 × 6 : x × 12. $$\eqalign{ & \frac{{720000}}{{12{\text{x}}}}:\frac{{6000}}{{3000}} \cr & \Rightarrow {\text{x}} = \frac{{720000}}{{24}} \cr & \Rightarrow {\text{x}} = 30000 \cr} $$

Just take care of the months of investment, rest all will be simple. Yogesh : Pranab : Atul = 45000 × 12 : 60000 × 9 : 90000 × 3 = 2 : 2 : 1 Atul's share $$\eqalign{ & = {\text{Rs}}{\text{. }}20000 \times \frac{1}{5} \cr & = {\text{Rs}}{\text{. }}4000 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{total}}\,{\text{profit}}\,{\text{be}}\,Rs.\,Z \cr & {\text{Bharti's}}\,{\text{share}} = {\frac{2}{3}} Z \cr & {\text{Rahul's}}\,{\text{share}} = Z - {\frac{2}{3}} Z \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{Z}{3} \cr & {\text{Rahul:Bharti}} \cr & = \frac{Z}{3}: {\frac{2}{3}} Z = 1:2 \cr & {\text{Now}}\,{\text{let}}\,{\text{the}}\,{\text{total}}\,{\text{capital}}\,{\text{be}}\,Rs\,X, \cr & {\text{and}}\,{\text{Bharti}}\,{\text{capital}}\,{\text{was}}\,{\text{used}}\,{\text{for}}\,Y\,{\text{months}}. \cr & {\text{then,}}\,{\text{Rahul}}\,{\text{capital}}\,{\text{will}}\,{\text{be}}\, {\frac{1}{4}} X \cr & {\text{and}}\,{\text{Bharti}}\,{\text{capital}}\,{\text{will}}\,{\text{be}}\,X - {\frac{1}{4}} X \cr & = \frac{{3X}}{4} \cr & \frac{{\frac{1}{4}X*15}}{{\frac{3}{4}X*Y}} = \frac{1}{2} \cr & Y = \frac{{15*2}}{3} = 10 \cr & {\text{So,}}\,{\text{Bharti's}}\,{\text{money}}\,{\text{was}}\,{\text{used}}\,{\text{for}}\,{\text{10}}\,{\text{months}} \cr} $$

In this type of question as time frame for both investors is equal then just get the ratio of their investments. P : Q = 85000 : 15000 = 85 : 15 = 17 : 3

$$\eqalign{ & A:B:C \cr & = 35000:45000:55000 \cr & = 7:9:11 \cr & {\text{Now}}\,{\text{we}}\,{\text{are}}\,{\text{having}}\,{\text{the}}\,{\text{ratio}}{\text{.}} \cr & {\text{to}}\,{\text{get}}\,{\text{the}}\,{\text{share,}}\,{\text{first}}\,{\text{make}}\,{\text{total}}\,{\text{of}}\,{\text{above}}\,{\text{ratio}}{\text{.}} \cr & {\text{then}}\,{\text{get}}\,{\text{each}}\,{\text{share}}{\text{.}} \cr & {\text{A's}}\,{\text{Share}} = 40500*\frac{7}{{27}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,10500 \cr & {\text{B's}}\,{\text{Share}} = 40500*\frac{9}{{27}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,13500 \cr & {\text{C's}}\,{\text{Share}} = 40500*\frac{{11}}{{27}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,16500\, \cr} $$

Let total capital be Rs. x $$\eqalign{ & {\text{Then,}} \cr & {\text{Rahul's share}} \cr & = {\text{Rs}}{\text{.}}\frac{x}{2} \cr & {\text{Arun's share}} = {\text{Rs}}{\text{.}}\frac{x}{3} \cr & {\text{Sumit's share}} \cr & = {\text{Rs}}{\text{.}}\left[ {x - \left( {\frac{x}{2} + \frac{x}{3}} \right)} \right] \cr & = {\text{Rs}}{\text{. }}\frac{x}{6} \cr & \therefore {\text{Required ratio}} \cr & = \frac{x}{2}:\frac{x}{3}:\frac{x}{6} \cr & = \frac{1}{2}:\frac{1}{3}:\frac{1}{6} \cr & = 3:2:1 \cr} $$

$$\eqalign{ & {\text{Ratio of shares}} \cr & = 8:12:14 \cr & = 4:6:7 \cr & \therefore {\text{Kiara's share}} \cr & = {\text{Rs}}{\text{.}}\left( {578 \times \frac{7}{{17}}} \right) \cr & = {\text{Rs. }}{\text{238}} \cr} $$

$$\eqalign{ & \because {\text{P}}:{\text{Q}}:{\text{R}} \cr & = 45000:70000:90000 \cr & = 9:14:18 \cr & \therefore \,{\text{Q's share}} \cr & = {\text{Rs}}{\text{.}}\left( {164000 \times \frac{{14}}{{41}}} \right) \cr & = {\text{Rs. }}{\text{56000}} \cr} $$

Let the total profit = Rs. k According to the question, Remaining profit after paying 20% working Partner's commission = (k - 8000) $$\eqalign{ & \therefore \,\left( {k - 8000} \right) \times \frac{{20}}{{100}} = 8000 \cr & \Leftrightarrow k = 48000 \cr & \therefore {\text{ Total profit}} = {\text{Rs}}{\text{. 48000}} \cr} $$

Lert the share of A = Rs. x According to the question, $$\eqalign{ & {\text{A}}:{\text{B}}:{\text{C}} \cr & {\text{Capital}} \to x:2x:\left( {4x - 50} \right) \cr & \Rightarrow \left( {x + 2x + 4x - 50} \right) = 13950 \cr & \Rightarrow 7x - 50 = 13950 \cr & \Rightarrow 7x = 14000 \cr & \Rightarrow x = 2000 \cr & {\text{Share of A}} = {\text{Rs}}{\text{. 2000}} \cr} $$

$$\eqalign{ & \because {\text{Ratio of shares}} \cr & = 11:16.5:8.25 \cr & = 4:6:3 \cr & {\text{Anil's share}} \cr & = {\text{Rs}}{\text{.}}\left( {19.5 \times \frac{3}{{13}}} \right) \cr & = {\text{Rs. }}{\text{4}}{\text{.5 lakh}} \cr & \therefore {\text{Required amount}} \cr & = 50\% {\text{ of 4}}{\text{.5 lakh}} \cr & = {\text{Rs. }}{\text{2}}{\text{.25 lakh}} \cr} $$

$$\eqalign{ & {\text{Ratio of shares}} \cr & = 1500:2500 \cr & = 3:5 \cr & {\text{Share of first friend}} \cr & = {\text{Rs}}{\text{.}}\left[ {\frac{{400}}{2} + \left( {400 \times \frac{3}{8}} \right)} \right] \cr & = {\text{Rs}}.\left( {200 + 150} \right) \cr & = {\text{Rs}}{\text{. 350}} \cr & {\text{Share of second friend}} \cr & = {\text{Rs}}{\text{.}}\left[ {\frac{{400}}{2} + \left( {400 \times \frac{5}{8}} \right)} \right] \cr & = {\text{Rs}}.\left( {200 + 250} \right) \cr & = {\text{Rs}}{\text{. 450}} \cr} $$

Let A's capital = Rs. x Then, B's capital = Rs. (x - 600) C's capital = Rs. (x - 600) - 300 = Rs. (x - 900) ∴ x + (x - 600) + (x - 900) = 3000 ⇒ 3x = 4500 ⇒ x = 1500 $$\eqalign{ & {\text{So, A}}:{\text{B}}:{\text{C}} \cr & = 1500:900:600 \cr & = 5:3:2 \cr & {\text{Hence, B's share}} \cr & = {\text{Rs}}{\text{.}}\left( {886 \times \frac{3}{{10}}} \right) \cr & = {\text{Rs}}{\text{. 265}}{\text{.80}} \cr} $$

Let Y's investment is used for T months → Now by using formula $$\eqalign{ & \frac{{5 \times 8}}{{6 \times {{\text{T}}_2}}} = \frac{5}{9} \cr & {\text{T}} = 12{\text{ months}} \cr} $$