**Partnership Problems with Solutions :**

In this section, we are going to learn, how to solve problems on partnership step by step.

**Partnership : **

When a business is run by two or more persons, it is known as partnership and the people who are running the business are called partners.We have the following four cases in partnership

**Case 1 : **

If all the partners invest equal amount for same time period then the profit is divided equally among them.

**Case 2 :**

If all the partners invest equal amount for different time period, then the profit is divided among them in the ratio of time period.

**Case 3 :**

If all the partners invest different amount for same time period, then the profit is divided among them in the ratio of amount invested.

**Case 4 :**

If all the partners invest different amount for different time period, then the profit is divided among them in the ratio of the product of investment and time period for each partner.

**Problem 1 :**

A invested 125% as much money as B, C invested 80% as much money as B. The total of all the three is $61,000. How much did C invest ?

**Solution : **

Because both A and C are compared to B, let us assume the investment of B as "x".

**Given :** A invested 125% as much money as B.

Investment of A = 125% of x = 1.25x

**Given : **C invested 80% as much money as B.

Investment of C = 180% of x = 0.8x

**Given : **The total of all the three is $61,000

So, we have

1.25x + x + 0.8x = 61000

3.05x = 61000

Divide both sides by 3.05

x = 61000/3.05

x = 20000

Investment of C is

= 0.8x

= 0.8 ⋅ 20000

= 16000

Hence, C invested $16000.

**Problem 2 :**

Daniel started a business with a capital of $ 8000. After six months, David joined him with investment of some capital. If at the end of the year, each of them gets equal amount as profit, how much did David invest ?

**Solution : **

Let "x" be the investment of David.

David's investment was in the business for six months.

Daniel invested $8000 and his investment was in the business for 12 months.

Then, the profit sharing ratio of David and Daniel is

6x : 12 ⋅ 8000

**Given : ** At the end of the year, each of them gets equal amount as profit.

So, we have

6x = 12 ⋅ 8000

Divide both sides by 6.

x = 2 ⋅ 8000

x = 16000

Hence, David's invested $16,000.

**Problem 3 :**

A, B and C start a partnership.The capitals of A, B and C are in the ratio 10 : 9 : 6 and the time period of A and B is in the ratio 2 : 3. B gets $10,800 as his share out the of a total profit of $26,000. If A's capital was there is in the business for 8 months, for how many months was C's capital in the business ?

**Solution : **

Let C's capital be in the business for "x" months.

Ratio of capitals is

A : B : C = 10 : 9 : 6

Ratio of time period is

A : B : C = 2 : 3 : x

As per case 4, ratio of profit share is

A : B : C = 20 : 27 : 6x

From the above ratio, we have

A's share = 20k

B's share = 27k

C's share = 6x ⋅ k

**Given :** B's share is $10,800.

Then, we have

27k = 10800

k = 400

So, we have

A's share = 20 ⋅ 400 = 8000

C's share = 6x ⋅ 400 = 2400x

**Given :** Total Profit is $26000

So, we have

8000 + 10800 + 2400x = 26000

18800 + 2400x = 26000

2400x = 7200

x = 3

Then, ratio of time period is

A : B : C = 2 : 3 : 3

From the above ratio, we have

A's time period = 2y

B's time period = 3y

C's time period = 3y

**Given : **A's capital was there is in the business for 8 months.

So, we have

2y = 8

y = 4

Then, C's time period is

= 3y

= 3 ⋅ 4

= 12

Hence, C's capital was in the business for 12 months.

**Problem 4 :**

A and B start a partnership by investing $24,000 and $36,000 respectively. Their agreement is to share half of the total profit equally and then share the remaining half in the ratio of their capital. If they share the entire profit in the ratio of their capitals, B would have got $2500 more than what she would have got otherwise. What is the total profit ?

**Solution : **

**Given : **A and B start a partnership by investing $24,000 and $36,000 respectively.

Ratio of capitals A and B is

= 24000 : 360000

= 2 : 3

Let the total profit be "x"

Half of the total profit = x / 2.

**Given :** A and B share half of the total profit equally.

Then,

B's share = (x/2) / 2 = x / 4

**Given :** A and B share remaining half of the total profit in the ratio of their capitals.

Ratio of their capitals is 2 : 3.

So, we have

B's share = 3/5 ⋅ x/2 = 3x / 10

Total profit of B is

= x/4 + 3x/10

= 5x/20 + 6x/20

= 11x / 20 -----(1)

Sharing the entire profit "x" in the ratio of their capitals (That is 2 : 3) :

B's share = 3x / 5 -----(2)

**Given :** In (2), B would get $2500 more than what she gets in (1).

So, we have

3x/5 - 11x/20 = 2500

12x/20 - 11x/20 = 2500

x/20 = 2500

x = 50000

Hence, the total profit is $50,000

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