This is a very prominent part of most of the exams. These questions are really scoring as well because these questions are really scoring. Check below what this topic is about and how are typical questions solved.

**What is Partnership?**

When two or more people join hands to create a start-up or some business it’s called a ‘partnership’ business. Typically, they invest some capital and earn some profit. And, this profit is distributed among partners either in some prefixed ratio or the *ratio of their investment. In exams*, mostly profit distribution is based on later type. The calculation for profit based on investment is done as shown below:**P _{1}: P_{2} = C_{1}×T_{1}: C_{2}×T_{2}**

_{}Here,

**P**= Profit for Partner 1.

_{1}**C**= Capital by Partner 1.

_{1}**T**= Time period for which Partner 1 invested his capital.

_{1}**P**= Profit for Partner 2.

_{2}**C**= Capital by Partner 2.

_{2}**T**= Time period for which Partner 2 invested his capital.

_{2}**Typical Question Methodology:**

**1)** We are given the amount of investment by two partners for particular time like A and B invested Rs.1500 and Rs.2000 respectively. **After 4 months**, they admit C with contribution of Rs.2250. And, B withdraws his contribution after 9 months ⇒ A invested his capital for 12 months while C invested his capital for (12 – 4=) 8 months and B invested for 9 months.**So, according to ratio formula, A: B: C** = 1500×12: 2000×9: 2250×8 = 180: 180: 180 (We’ve cancelled out the equal number of zeroes from all the numbers)

⇒ A: B: C = 1: 1: 1**2)** Second, we are given the profit at the end of year was Rs 900. What’d B get?**Now, A: B: C =** 1: 1: 1**⇒ B’s share = **{1/(1+1+1)} =1/3rd of profits = (1/3)*900 = Rs. 300

We can see that calculating profit isn’t that tricky so we’ll be solving sample questions in a way that would help you in calculating the ratio of profits among partners.

**Sample Questions:**

**Q1.** A contributes one-forth capital for one fourth of time. B contributes one fifth of capital for half of the time. C contributes the remaining capital for the whole time. Find their profit-sharing ratio.**Sol.** Part of Capital by: A = 1/4 ; B = 1/5 ; C gave remaining ⇒ C’s share in capital = 1 – (1/4 + 1/5) = 11/20

Time investment by: A = 1/4; B = 1/2; C = 1 (whole time means full one 1 year)

Using, **P _{1}: P_{2} = C_{1}×T_{1}: C_{2}×T**

_{2}

⇒ A: B : C =

Now we’ll take LCM of 16, 10 and 20 which is 80 and divided it by each denominator and multiply number thus obtained by each of the respective denominator.

⇒ **A: B: C = 5: 8: 44.**

**Q2.** A begun a business with Rs.450, after some months B joined him with capital of Rs.300. In the end, the profits were distributed in the ratio of 2:1. After how long did B join?**Sol.** Capital by: A = 450, B = 300

Time capital stay invested by: A = 12, B = n(say)

Using, **P _{1}: P_{2} = C_{1}×T_{1}: C_{2}×T_{2}**

_{(2/1) = (450 ×12)/(300n)}

⇒ n = 9 months

So, B’s capital was invested for 9 months and he joined after (12 – 9 =) 3 months.

**Q3.** A and B rent a pasture for 10 months. A puts in 100 cows for 8 months. How many can B put in for remaining 2 months, if he pays half as much again as A?**Sol.** Here, B has to pay half as much again as A ⇒ If A pays 1 then B pays (1+1/2=) 3/2

So, required ratio = 1: 3/2 ⇒ A: B = 2: 3

Also, number of cows is considered as capital. Let cows brought in by B be ‘n’

So, using, **P _{1}: P_{2} = C_{1}×T_{1}: C_{2}×T**

_{2}2: 3 = 100×8: n×2

⇒ n = 600 cows

**Q4.** A, B and C invested capitals in the ratio of 5: 6: 8.At the end of business term, they received the profits in the ratio of 5: 3: 12. Find the ratio of time for which they contributed their capital.

Sol. To solve such questions use formula below:**T _{A}: T_{B}: T_{C} = P_{A}÷C_{A}: P_{B}÷C_{B}: P_{C}÷C**

_{C}Here, T

_{A}= Time by A, P

_{A}= Profit for A, C

_{A}= Capital by A.

T

_{B}= Time by B, P

_{B}= Profit for B, C

_{B}= Capital by B.

T

_{C}= Time by C, P

_{C}= Profit for C, C

_{C}= Capital by C.

⇒ T

_{A}: T

_{B}: T

_{C}= 5÷5: 3÷6: 12÷8 = 1: 1/2: 3/2 = 2: 1: 3

**Q5.** Three partners A,B and C had business. A, whose money had been used for 4 months claimed 1/8 profit. B whose money was used for 6 month claimed 1/3 of profit. C had invested Rs. 1560 for 8 months. How much did A and B contribute?**Sol.**⇒ n = **Rs. 720**

So, A invested Rs 720. Similarly, we can find the amount of investment by B.

**Q6.** A and B invested in the ratio of 3: 2. If 5% of total profit goes to charity and A got Rs 855 as his share then find the total profit?**Sol. In such questions, use this formula,**

So, the total profit is **Rs. 1500.**** **

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