Contents
Chapter 2 Partnership
2.1 Important Facts and Formulae
1. Partnership : When two or more than two persons run a business joined they are called partners and the deal is known a partnership.
2. Ratio of Division of Gains :
(i) When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.
Suppose \(\mathrm {A}\) and \(\mathrm {B}\) invest \(\mathrm {Rs}\). \(x\) and Rs. \(y\) respectively for a year in a business, then at the end of the year :
(A’s share of profit) : (B’s share of profit) \(=x: y\).
(ii) When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital \(\times \) number of units of time). Now, gain or loss is divided in the ratio of these capitals.
Suppose A invests Rs. \(x\) for \(p\) months and B invests Re \(y\) for \(q\) months, then
(A’s share of profit) : (B’s share of profit) \(=x p : yq\).
3. Working and Sleeping Partners : A partner who manages the buinew is known as a working partner and the one who simply invests the money is a sleeping partner:
2.2 Solved Examples
1. \(A\), \(B\) and \(C\) started a business by investing Rs. 1,20,000, Rs. 1,35,000 and Rs. 1,50,000 respectively. Find the share of each, out of an annual profit of Rs. 56,7600
Solution :
Ratio of shares of \(\mathrm {A}, \mathrm {B}\) and \(\mathrm {C}=\) Ratio of their investments \(=120000: 135000: 150000=8: 9: 10 . \)
\begin{align*} & \text { A's share = Rs. }\left (56700 \times \dfrac {8}{27}\right )=\text { Rs. } 16800 . \\ & \text { B's share = Rs. }\left (56700 \times \dfrac {9}{27}\right )=\text { Rs. } 18900 . \\ & \text { Cs share }=\text { Rs. }\left (56700 \times \dfrac {10}{27}\right )=\text { Rs. } 21000 . \end{align*}
2. Alfred started a business investing Rs. 45,000. After 3 months, Peter joined him with a capital of Rs. 60,000 . After another 6 months, Ronald joined them with a capital of Rs. 90,000 . At the end of the year, they made a profit of Rs. 16,500. Find the share of each.
Solution :
Clearly, Alfred invested his capital for 12 months, Peter for 9 months and Ronald for 3 months.
\begin{align*} \text {So, ratio of their capitals} & =(45000 \times 12):(60000 \times 9):(90000 \times 3) \\ & =540000: 540000: 270000=2: 2: 1 . \end{align*} : Alfred’s share \(=\) Rs. \(\left (16500 \times \dfrac {2}{5}\right )=\) Rs. \(6600 ;\)
Peter’s share \(=\) Rs. \(\left (16500 \times \dfrac {2}{5}\right )=\) Rs. 6600
Ronald’s share \(=\) Rs. \(\left (16500 \times \dfrac {1}{5}\right )=\) Rs. 3300.
3. A, B and C start a business each investing Re. 20,000, After \(5\) months A withdrew Rs. 5000, B withdrew Rs. 4000 and C invests Rs.6000 more. At the end of the year, a total profit of Rs. 69,900 was recorded. Find the share of each.
Solution :
Ratio of the capitals of \(A, B\) and C
\begin{align*} & = 20000 \times 5 + 15000\times 7 : 20000 \times 5 + 16000 \times 7 : 20000 \times 5 + 26000 \times 7 \\ & = 205000 : 212000 : 282000 = 205 : 212 : 282 \end{align*}
\(\seteqnumber{0}{2.}{0}\)\begin{align*} \therefore \quad \text { A's share }=\text { Rs. }\left (69900 \times \dfrac {205}{699}\right ) & =\text { Rs. } 20500 ; \\ \text { B's share }=\text { Rs. }\left (69900 \times \dfrac {212}{699}\right ) & =\text { Rs. } 21200 ; \\ \text { C's share }=\text { Rs. }\left (69900 \times \dfrac {282}{699}\right ) & =\text { Rs. } 28200 . \end{align*}
4. A, B and C enter into partnership. A invests 3 times as much as B invests and B invests two-third of what C invests. At the end of the year, the profit earned in Ra. 6600 . What is the share of B ?
Solution :
Let C’s capital \(=\) Rs. \(x\). Then, B’s capital \(=\) Rs. \(\dfrac {2}{3} x\).
\begin{align*} & \quad \text { A's capital }=\text { Rs. }\left (3 \times \dfrac {2}{3} x\right )=\text { Rs. } 2 x . \\ & \therefore \quad \text { Ratio of their capitals }=2 x: \dfrac {2}{3} x: x=6: 2: 3 . \\ & \text { Hence, B's share }=\mathrm {Rs} .\left (6600 \times \dfrac {2}{11}\right )=\text { Rs. } 1200 . \end{align*}
5. Four milkmen rented a pasture. A grazed 24 cows for 3 months, B 10 cows for 5 months, C 35 cows for 4 months and D 21 cows for 3 months. If A’s share of rent is Rs. 720, find the total rent of the field.
Solution :
Ratio of shares of
\begin{align*} A, B, C, D &=(24 \times 3):(10 \times 5):(35 \times 4):(21 \times 3)\\ &=72: 50: 140: 63 \end{align*} Let total rent be Rs, \(x\). Then, A’s share \(=\) Rs. \(\dfrac {72 x}{325}\).
\[ \therefore \quad \dfrac {72 x}{325}=720 \Rightarrow x=\dfrac {720 \times 325}{72}=3250 . \]
Hence, total rent of the field is Rs. 3250 .
6. A invested Rs. 76,000 in a business. After few months, \(B\) joined him with Rs. 57,000. At the end of the year, the total profit was divided between them in the ratio 2: 1. After how many months did \(B\) join?
Solution :
Suppose B joined after \(x\) months. Then, B’s money was invested for \((12-x)\) months
\begin{align*} \therefore \quad \dfrac {76000 \times 12}{57000 \times (12-x)}=\dfrac {2}{1} \quad & \Rightarrow \quad 912000=114000(12-x) \\ & \Rightarrow 114(12-x)=912 \Rightarrow (12-x)=8 \Rightarrow x=4 \end{align*} Hence, B joined after 4 months.
7. A, B and C enter into a partnership by investing in the ratio of \(3: 2: 4\). After one year, B invests another Rs. 2,70,000 and C, at the end of 2 years, also invests Rs. 2,70,000 At the end of three years, profits are shared in the ration of \(3:4:5\). Find the initial investment of each.
Solution :
Let initial investments of \(A, B\) and \(C\) be Rs. \(3 x\), Rs. \(2 x\) and Rs. \(4 x\) respectively. Then,
\begin{align*} (3x \times 36 ) :[(2x \times 12) +(2x + 270000) \times 24 ] : [4x \times 24 )(4x + 270000) \times 12] = 3 : 4 :5 \end{align*}
\(\seteqnumber{0}{2.}{0}\)\begin{align*} \Rightarrow \quad 108 x:(72 x+6480000):(144 x+3240000) & =3: 4: 5 \\ \therefore \dfrac {108 x}{72 x+6480000}& =\dfrac {3}{4} \Rightarrow 432 x=216 x+19440000 \\ \Rightarrow \quad 216 x& =19440000 \Rightarrow x=90000 . \end{align*}
\(\seteqnumber{0}{2.}{0}\)\begin{align*} \text { Hence, A's initial investment }&=3 x=\text { Rs. } 2,70,000 \text {; } \\ \text { B's initial investment }&=2 x=\text { Rs. } 1,80,000 \text {; } \\ \text { C's initial investment }&=4 x=\text { Rs. } 3,60,000. \end{align*}
0 Comments