Partnership – Concepts, Types and Solved Problems

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What is partnership?

Partnership is kind of group business.

In partnership profit (P) is shared according to the investment (I) and time (T).

The more is investment, the more profit.

Hence, P ∝ I

Again, the more is time of investment, the more is profit,

Hence, P ∝ I

By combining both, P ∝ IT

Taking proportionality constant as 1

P = IT

From equation we could easily find I or T if P is given.

\text{I} = \displaystyle \frac{\text{P}}{\text{T}} and

\text{T} = \displaystyle \frac{\text{P}}{\text{I}}

Question: Three partners A, B and C has the following investment
capital investment = 1 : 2 : 3
time period = 8 : 12 : 6
Profit ratio?

I ⟶ 1 : 2 : 3
T ⟶ 8 : 12 : 6 = 4 : 6 : 3
_____________________________________
P ⟶ 1×4 : 2×6 : 3×3= 4 : 12 : 9

Here, P is compound ratio of I and T

Question: Profit sharing ratio = 3 : 2 : 1, time = 6 months : 8 months : 12 months, capital investment ratio?

I ⟶
T ⟶ 6 months : 8 months : 12 months = 3 : 4 : 6
___________________________________________________
P ⟶ 3 : 2 : 1

\text{P} = \text{IT} ⇒ \text{I} = \displaystyle \frac{\text{P}}{\text{T}}

I ⟶ \displaystyle \frac{3}{3} : \displaystyle \frac{2}{4}: \displaystyle \frac{1}{6} = 1 : \displaystyle \frac{1}{2}: \displaystyle \frac{1}{6} = 6:3:1

Types of Partnership

• Simple Partnership
• Compound Partnership
• Sleeping partner
• Multiple investment for multiple time

Simple Partnership

In this kind of partnership, partners invest their money for same period of time and therefore profit is divided as per their investment or capital ratio.

Question: A and B started a partnership by investing Rs. 10,000 and Rs. 15,000 respectively. How much will each get out of profit of Rs. 60,000 at the end of the year.

Investment ratio = 100000 : 15000 = 2 : 3

A will get = \displaystyle \frac{2}{5}\times 60000 = 24000

B will get =\displaystyle \frac{3}{5}\times 60000 = 36000

Compound Partnership

In this partnership, partners invest their money for different period of time and therefore profit is divided as per investment × time ratio.

Note: If total time of investment is n months and someone joins after x months, his/her investment will work for ( n – x ) months. This concept will be useful for finding time ratio.

Question: A invested Rs. 10,000 for 7 months and B invested Rs. 12,000 for 5 months in a partnership, How much will each get out of profit of Rs. 6,500.

I ⟶ 10000 : 12000 = 5 : 6
T ⟶ 7 months : 5 months = 7 : 5
_____________________________________
P ⟶ 35 : 30 = 7 : 6

A’s share = \displaystyle \frac{7}{13}\times 6500 = 3500

B’s share = \displaystyle \frac{6}{13}\times 6500 = 3000

Question: Tanvi started a business investing ₹45000. After 8 moths Anisha joined her with a capital of ₹52000. At the end of the year the total profit was ₹56165. What is the share of profit of Anisha?
(a) ₹21450
(b) ₹24440
(c) ₹27635
(d) None of these

Ans: (d) None of these

Explanation:

In the ratios first term is for Tanvi and 2nd term is for Anisha

I ⟶ 45000 : 52000 = 45 : 52
T ⟶ 12 : 4 = 3 : 1
_____________________________________
P ⟶ 135 : 52

Anisha’s share = \displaystyle \frac{52}{187}\times 56165 = 15618.0748

Question: Mrudul invested an amount of ₹29500 in order to start a business. Shalaka joins her 4 months later by investing an amount of ₹33500. If the business earns a profit of ₹120575 at the end of two years, what was Mrudul’s share in the profit?
(a) ₹60725
(b) ₹61950
(c) ₹59250
(d) ₹58625

Ans: (b) ₹61950

Explanation:

I ⟶ 29500 : 33500 = 59 : 67
T ⟶ 24 : 20 = 6 : 5
_____________________________________
P ⟶ 354 : 335

Mrudul’s share = \displaystyle \frac{354}{689}\times 120575 = 61950

Sleeping Partners

A partner who only invests money but does not take part in the business is called sleeping partners. The partners who not only invests money but also takes part in the business receives some incentives out of the total profit for managing the business. The remaining profit is divided as per profit sharing ratio.

Question: A and B started a business by investing Rs. 20,000 and Rs. 30,000 respectively. B is a sleeping partner and therefore A receives 10% for managing the business. How much will each get out of profit of Rs. 5,000 at the end of the year.

A get for managing business = 5000 × 10% = 500

Remaining profit = 5000 – 500 = 4500

I ⟶ 20000 : 30000 = 2 : 3

So, P ⟶ 20000 : 30000 = 2 : 3

A’s share = \displaystyle \frac{2}{5}\times 4500 + 500 = 2300

B’s share = \displaystyle \frac{3}{5}\times 4500 = 2700

Multiple Investment for Multiple Time

In this type of investment partners are free to add more investment or withdraw some or full previous investment. For this type of investment profit is shared among partners as per \sum \text{IT} ratio.

For the calculation purpose effective investment (I) will be multiplied with time (T) for what the investment works. See the following problem for clarification of the required concept.

Question: A and B entered into a partnership with capital in the ratio 4 : 5. After 3 months, A withdrew 1/4^th of his capital and B withdrew 1/5^th of capital. The gain at the end of 10 months was 760. A’s share in the profit
(a) 330
(b) 360
(c) 380
(d) 430

Ans: (a) 330

Explanation:

         A    :    B
IT⟶ 4×3 : 5×3
IT⟶ 3×7 : 4×7
______________________________
P⟶ 12+21 : 15+28
     = 33 : 43

How ITs are calculated?

Capital ratio of A to B is given as 4 : 5.
We have assumed that investment by A = 4
And Investment by B = 5
Total investment time = 10 months

First IT calculation:

This 4 and 5 work for 3 months since they are withdrawing after 3 months.
For A, IT = 4 × 3
For B, IT = 5 × 3

Second IT Calculation:

Now, after 3 months A withdrew 1/4^th of his capital.

So, he actually withdrew = 4\times \displaystyle \frac{1}{4} = 1

Hence his effective Investment right now is 4 – 1 = 3 and this will work for rest 7 months
This time for A, IT = 3 × 7

Now, after 3 months B withdrew 1/5^th of his capital.

So, he actually withdrew = 5\times \displaystyle \frac{1}{5} = 1

Hence, his effective Investment right now is 5 – 1 = 4 and this will work for rest 7 months
This time for B, IT = 4 × 7

So, \sum \text{IT} for A = 4 × 3 + 3 × 7 = 12 + 21 = 33

And \sum \text{IT} for B = 5 × 3 + 4 × 7 = 15 + 28 = 43

∴ A’s share = \displaystyle\frac{33}{76}\times 760 = 330

For Practice questions you may refer the post mentioned below.
Kindly note the question number first – 2019(Q2, Q3) ; 2018(Q5) ; 2016(Q2) ; 2014(Q1)
Read this post.

Practice Problems on Partnership

Question: Prakash, Sunil and Anil started a business jointly investing ₹11 lakhs, ₹16.5 lakhs and ₹8.25 lakhs respectively. The profit earned by them in the business at the end of three years was ₹19.5 lakhs. What will be the 50% of Anils’ share in the profit?
(a) ₹4.5 lakhs
(b) ₹2.25 lakhs
(c) ₹2.5 lakhs
(d) ₹3.75 lakhs

Ans: (b) ₹2.25 lakhs

Explanation:

Time is same for everyone, so profit will shared as per investment ratio

Prakash : Sunil : Anil
= 11 : 16.5 : 8.25
= 1100 : 1650 : 825 (multiply by 100)
= 44 : 66 : 33 (divide by 25)
= 4 : 6 : 3 (divide by 11)

50% of Anil’s share = \displaystyle \frac{1}{2} \times \left(\displaystyle \frac{3}{13}\times 19.5\right) = 2.25

Question: Karan starts a business by investing ₹60000. Six months later Shirish joins him by investing ₹100000. At the end of one-year from commencement of the business, they earn a profit of ₹151800. What is the Shirish’s share in the profit?
(a) ₹55200
(b) ₹82800
(c) ₹62500
(d) None of these

Ans: (d) None of these

Explanation:

Ratios are Karan : Shirish

I ⟶ 60k : 100k = 3 : 5
T ⟶ 12 : 6 = 2 : 1
_____________________________________
P ⟶ 6 : 5

A’s share = \displaystyle \frac{5}{11}\times 151800 = 69000

To practice SSC CGL MAINS questions on partnership visit here

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