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## PARTNERSHIP -> IMPORTANT FACTS AND FORMULAE

 I. Partnership : When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership. II. 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 A and B invest 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 * 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 Rs. y for q months, then (A’s share of profit) : (B’s share of profit) = xp : yq. III. Working and Sleeping Partners : A partner who manages the business is known as working partner and the one who simply invests the money is a sleeping partner.

## PARTNERSHIP -> SOLVED EXAMPLES

1. A, B and C start a business each investing Rs. 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. Sol. Ratio of the capitals of A, B and C = 20000 * 5 + 15000 * 7 : 20000 * 5 + 16000 * 7 : 20000 * 5 + 26000 * 7 205000 : 212000 : 282000 = 205 : 212 : 282. ∴ A’s share = Rs. [69900 * 205/699] = Rs. 20500; B’s share = Rs. [69900 * 212/699] = Rs. 21200; C’s share = Rs. [69900 * 282/699] = Rs. 28200. Four milkman 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. Sol. Ratio of shares of A, B, C, D = (24 * 3) : (10 * 5) : (35 * 4) : (21 * 3) = 72 : 50 : 140 : 63. Let total rent be Rs. x Then, A’s share = Rs. 72x / 325 ∴ 72x / 325 = 720 ⇔ x = 720 * 325 / 72 = 3250. 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. 56,700. Sol. Ratio of shares of A, B and C = Ratio of their investments = 120000 : 135000 : 150000 = 8 : 9 : 10. ∴ A’s share = Rs. [56700 * 8/27] = Rs. 16800. B’s share = Rs. [56700 * 9/27] = Rs. 18900. C’s share = Rs. [56700 * 10/27] = Rs. 21000.

## PARTNERSHIP -> EXERCISE

1. Albert 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.

• A. 3300
• B. 4400
• C. 5500
• D. 6600
 Ans: A. Sol. Clearly, Albert invested his capital for 12 months, Peter for 9 months and Ronald for 3 months. So, ratio of their capitals = (45000 * 12) : (60000 * 9) : (90000 * 3) = 540000 : 540000 : 270000 = 2:2:1. ∴ Alfred’s share = Rs. [16500 * 2/5] = Rs. 6600; Peter’s share = Rs. [16500 * 2/5] = Rs. 6600; Ronald’s share = Rs. [16500 * 1/5] = Rs. 3300.

2. A and B are partners in a business. A contributes 1/4 of the capital for 15 months and B received 2/3 of the profit. For how long B’s money was used?

• A. 4 months
• B. 6 months
• C. 8 months
• D. 10 months
 Ans: D. Sol. Let the total profit be Rs. z. Then, B’ share = Rs. 2z/3, A’s share = Rs. [z - 2z/3] = Rs. z/3 ∴ A : B = z/3 : 2z/3 = 1 : 2. Let the total capital be Rs. x and suppose B’s money was used for x months. Then, [1/4 x * 15] / [3/4 x * y] = 1/2 ⇔ y = [15 * 2 / 3] = 10. Thus, B’s money was used for 10 months.

3. A and B started a business with initial investments in the ratio 14 : 15 and their annual profits were in the ratio 7 : 6. If A invested the money for 10 months, for how many months did B invest his money?

• A. 2
• B. 4
• C. 6
• D. 8
 Ans: D. Sol. Suppose A invested Rs. 14x for 10 months and B invested Rs. 15x for y months. Then, 14x * 10 / 15x * y = 7/6 ⇒ y = 840 / 105 = 8 Hence, B invested the money for 8 months.

4. A starts business with Rs.3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in the ratio 2 : 3. What is B’s contribution in the capital?

• A. 6000
• B. 9000
• C. 9500
• D. 12000
 Ans: B. Sol. Let B’s capital be Rs. x. Then, 3500 * 12 / 7x = 2/3 ⇔ 14x = 126000 ⇔ x = 9000.

5. A began a business with Rs. 85,000. He was joined afterwards by B with Rs. 42,500. For how much period does B join, if the profits at the end of the year are divided in the ratio of 3 : 1?

• A. 8 months
• B. 6 months
• C. 4 months
• D. 2 months
 Ans: A. Sol. Suppose B joined for x months. Then, 85000 * 12 / 42500 * x = 3/1 or x = 85000 * 12 / 42500 * 3 = 8. So, B joined for 8 months.