Partnership - 02

Explanation:-

Let the total investment be Rs. z
Then,
$$\eqalign{ & 20\% {\text{ of }}z = 98000 \cr & \Leftrightarrow z = \left( {\frac{{98000 \times 100}}{{202}}} \right) \cr & \Leftrightarrow z = 490000 \cr}$$
Let the capital of P, Q and R be
Rs. 5x, Rs. 6x and Rs. 6x respectively
Then,
$$\Leftrightarrow \left( {5x \times 12} \right)$$   + $$\left( {6x \times 12} \right)$$   + $$\left( {6x \times 6} \right)$$   = $$490000 \times 12$$
$$\eqalign{ & \Leftrightarrow 168x = 490000 \times 12 \cr & \Leftrightarrow x = \left( {\frac{{490000 \times 12}}{{168}}} \right) \cr & \Leftrightarrow x = 35000 \cr & \therefore {\text{R's investment}} \cr & = {\text{Rs}}{\text{. }}6x \cr & = {\text{Rs}}{\text{.}}\left( {6 \times 35000} \right) \cr & = {\text{Rs}}{\text{. 210000}} \cr}$$

Explanation:-

Clearly, A invested his capital for 12 months
while each one of B and C invested his capital for (12 - x) months
Ratio of profits os A, B, C
$$= \left( {50000 \times 12} \right)$$   : $$\left[ {60000 \times \left( {12 - x} \right)} \right]$$     : $$\left[ {70000 \times \left( {12 - x} \right)} \right]$$
$$\eqalign{ & = 60:6\left( {12 - x} \right):7\left( {12 - x} \right) \cr & {\text{But ratio of profits}} \cr & = 20:18:21 \cr & = 60:54:63 \cr}$$
$$\therefore 60:\left( {72 - 6x} \right):\left( {84 - 7x} \right)$$     = $$60$$ : $$54$$ : $$63$$
$$\eqalign{ & {\text{So, }}72 - 6x = 54 \cr & \Rightarrow 6x = 18 \cr & \Rightarrow x = 3 \cr}$$

Explanation:-

	A	:	B	:	C
Ratio of Profit →	2	:	3	:	7

$$\eqalign{ & {\text{Average gain}} = \frac{{2 + 3 + 7}}{3} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{\text{ units}} \cr & {\text{According to the question,}} \cr & {\text{4 units}} = {\text{Rs}}{\text{. 8000}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. 2000}} \cr & {\text{3 units}} = 3 \times 2000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 6000}} \cr & \therefore {\text{Share of B}} = {\text{Rs}}{\text{. 6000}} \cr}$$

Explanation:-

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

Explanation:-

Suppose B invested Rs. x for y months
Then, A invested Rs. 3x for 2y months
$$\eqalign{ & {\text{So, A}}:{\text{B}} \cr & = \left( {3x \times 2y} \right):\left( {x \times y} \right) \cr & = 6xy:xy \cr & = 6:1 \cr & \therefore {\text{B's profit}}:{\text{Total profit}} = 1:7 \cr & {\text{Let the total profit is Rs}}{\text{. }}x \cr & {\text{Then,}}\frac{1}{7} = \frac{{4000}}{x} \cr & \Leftrightarrow x = 28000 \cr}$$

Explanation:-

$$\eqalign{ & {\text{Let total capital}} = {\text{Rs}}{\text{.}}\,{\text{x}} \cr & {\text{Then, A's capital}} = {\text{Rs}}{\text{.}}\,\frac{x}{3} \cr}$$
B's capital = (A + C)'s capital
(B's capital) = (A + B + C)'s capital = Rs. x
$$\eqalign{ & \Rightarrow {\text{B's capital}} = {\text{Rs}}{\text{.}}\frac{x}{2} \cr & \Rightarrow {\text{C's capital}} \cr & = {\text{Rs}}{\text{.}}\left[ {x - \left( {\frac{x}{3} + \frac{x}{2}} \right)} \right] \cr & = {\text{Rs}}{\text{.}}\,\frac{x}{6} \cr & \therefore {\text{A}}:{\text{B}}:{\text{C}} \cr & = \frac{x}{3}:\frac{x}{2}:\frac{x}{6} \cr & = 2:3:1 \cr & {\text{So C's share}} \cr & = {\text{Rs}}{\text{.}}\left( {900 \times \frac{1}{6}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{150}} \cr}$$

Explanation:-

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

Explanation:-

	A	B	C
Amounts invested	14000
Time (in months)	12	7	5
	168000

$$\eqalign{ & {\text{Ratio of profits }}4:3:2 \cr & {\text{Let their profits }}4x:3x:2x \cr & \Leftrightarrow 4x = 168000 \cr & \Leftrightarrow x = 42000 \cr & \Rightarrow {\text{Profit share of C}} \cr & = 2x \cr & = 2 \times 42000 \cr & = {\text{Rs}}{\text{. 84000}} \cr & \Rightarrow {\text{Capital invested by C}} \cr & = \frac{{84000}}{5} \cr & = {\text{Rs. }}16800 \cr}$$

Explanation:-

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

Explanation:-

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