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__Quantitative Aptitude Notes: Problems on Partnership__

__Quantitative Aptitude Notes: Problems on Partnership__

**What is Partnership?**

Persons Two
or more than two persons when start and run the new business

**jointly**of their own choice, the persons start they are called**Partners**and the deal is done between the partners is known as**Partnership.****Working and Sleeping Partners:**

A partner
who

**manages the business**is known as a**working partner**and the one who**simply invests**the money is a**sleeping partner**.**Ratio of Division of Gains**

**The amount investment of all the partners are for the same time period, the gain or loss amount is distributed among the partners in the ratio of their investments amounts.****Suppose, A and B invest some money Rs. R1 and R2 respectively for a year in a business, at the end of year profit will be distributed among them (A share of profit): (B share of profit) = A :B.****When investments are for different time periods, then equivalent capital are calculated a unit of time by taking (Capital x number of unit of time).****Suppose, A invest Rs. R1 for t1 months and B invest Rs. R2 for t2 months, then (A share of profit): (B share of profit) = Ax T1:B x T2.**

**Solved Examples:**

**Example:**Sara started a software business by investing Rs. 40,000 . After six months, Nitikajoined him with a capital of Rs. 60,000. After 3 years, they earned a profit of Rs .27,900 .What was Sara’s share in the profit?

**Answer:**

Short
trick- Sara : Nitika share of capital= ( 40,000 x 36 ) : ( 60,000 x 30 ) =
1440000 : 1800000 = 4 : 5 .

Sara’s
share is =

**Rs. 27900 x 4 / 9 = Rs. 12400.****Example:**Jasmin and Harish started a partnership business investing some amount of money in the ratio of 2 : 3. Ronit joined them after six months with an amount equal to that of Harish. In what proportion should the profit at the end of one year be distributed among Jasmin, Harish and Ronit?

**Answer:**

**Short trick-**

Let the
initial investment money ratio of Jasmin and Harish is 2x and 3x.So Jasmin ,
Harish and Ronit ratio of investment is (Jasmin : Harish : Ronit ) = (2x X 12 )
: ( 3x X 12 ) : ( 3x X 6 ) = 24 : 36 : 18 = 4 : 6 : 3.

**Example:**Jeenat started a business and she invested in 76000, After some month, Amrita came to join with her and invest 57000.The end of the year the total profit was divided among them into ratio form 2 : 1.Find after how many months Amrita join.

**Answer:**

**Step 1:**we can assume that Amrita join into business after x months. So Amrita’s money was invest into (12 – x) months.

**Step 2:**76000 x 12 / 57000 x (12 – x) = 2 / 1912000 = 114000 (12 – x) => 114 (12 – x) = 912=>x = 4

hence,
after 4 months Amrita join the business.

**Example:**Ashish, Mukta and Ridhi started a business each investing Rs.20,000. After 4 month Ashish withdraws Rs.6000, Mukta withdraws Rs.8000, Ridhi invest Rs.6000 more at the end of the years, a total profit was Rs.65600. Find the share of each.

**Answer:**

Ratio
capital of Ashish, Mukta and Ridhi= ( 20,000 x 4 + 14000 x 8 ) : ( 20,000 x 4 +
12000 x 8 ) : ( 20,000 x 4 + 26000 x 8 ) = 192000 : 176000 : 288000

Ashishshare
= (65600 x 192 / 656 ) = 192000

Mukta share
= (65600 x 176 / 656 ) = 176000

Ridhi share
= (65600 x 288 / 656 ) = 288000

__MEMORY BASED SOLVED PROBLEMS BASED ON VARIOUS TYPES__**TYPE 1: SIMPLE PARTNERSHIP**

**1. A, B, C subscribes together Rs. 50,000 for business. A subscribes Rs. 4000 more than B and B Rs. 5000 more than C. Out of a total profit Rs. 35000, A receives**

**Solution:**

Let C
Subscribe = x, then B = (x + 5000) and A = (x + 5000) + 4000

Total = x +
(x +5000) + (x + 5000) + 4000 = 50000

=> 3x +
14000 = 50,000 => 3x = 36,000 => x = 12000

=> Ratio
of shares of A: B: C = 21000: 17000: 12000 = 21: 17: 12

Therefore,
A’s share = 21/ 50 * 35000 = Rs. 14700

**2. A and B joined a partnership business by investing Rs. 30,000 and Rs. 50,000 respectively. If they earn a profit of Rs. 4000, find A’s share in the profit.**

**Solution:**

Since
periods for which the two amounts are invested, are same.

Therefore,
Ratio in which profit is to distributed between A and B is 30000: 50000 = 3: 5

Therefore,
A’s share in profit = (3/8) * 4000 = Rs. 1500

**3. A starts a business with Rs. 7000 and after 5 months, B joined as a partner. After a year, the profit is divided in the ratio 2: 3. The capital of B is:**

**Solution:**

(7000*12) /
(x*7) = 2/3 => x = 7000*3*12 / (7*2) = 18000

**Alternative Method:**

Profit –
Sharing ratio = 2: 3; Ratio of time = 12: 7

Therefore,
Ratio of capitals = 2/12 : 3/7 = 1/6 : 3/7 = 7: 18

Therefore,
A’s capital = 7 ratio = Rs. 7000 and B’s capital = 18 ratio = Rs. 18000

**4. A and B start a business jointly. A invests Rs 16000 for 8 month and B remains in the business for 4 months. Out of total profit, B claims 2/7 of the profit. How much money was contributed by B?**

**Solution:**

B claims
2/7 of the profit

A claims
remaining 5757 of the profit => A : B = 5/7 : 2/7 = 5 : 2

Let the
money contributed by B = b

Then A : B
=16000 * 8 : b * 4

Therefore,
16000 * 8: b * 4 = 5: 2

16000 * 8 *
2 = b * 4 * 5

16000 * 2 *
2 = b*5 => 3200 * 2 * 2 = b => b =12800

**5. x, y and z are partner in a business. B's capital is 1/6th of the total and A's capital equal to that of B and C together. How much does C receive out of a total annual profit of Rs 2400?**

**Solution:**

Let total
investment be x

Let
investment of A, B, C be a, b, c respectively

b = x/6; a
= b + c = x/6 + c; a + b + c=x

(x/6 + c)+
x/6 + c = x; x/3 + 2c = x

2c = 2x/3 =>
c = x/3

a = x/6 + c
= x/6 + x/3 = x/2

a: b: c =
x/2: x/6: x/3 = 1/2: 1/6: 1/3 = 3: 1: 2

Amount that
C receives = 2400 * 2/ (3+1+2) = 800

**TYPE 2: COMPOUND PARTNERSHIP**

**1. A and B joined a partnership and invested Rs. 50000 and Rs. 60000 respectively. After 8 months, B leaves and C joins with capital of Rs. 90000. If their profit of one year is Rs. 36000, find A’s share in the profit.**

**Solution:**

Capitals of
A, B and C are invested for 12, 8 and 4 months respectively.

Profit
sharing ratio = (50000*12) : (60000*8) : (90000*4) = 5: 4: 3

A’s share
in profit = 5/12 * 36000 = Rs. 15000

**2. A, B and C started a business with investment in the ratio 5: 6: 8 respectively. After one year C withdrew 50% of his capital and A increased his capital by 60% of his investment. After two years in what ratio should the earned profit be distributed among A, B and C respectively.**

**Solution:**

Investment
for the 1st year= 5: 6: 8

A’s capital
for second year = 5 + 60% of 5 = 5+ 3 = 8

C’s capital
for second year = 8 – 50% of 8 = 8 – 4 = 4

Therefore,
Required ratio = (5+8): (6+6): (8+4) = 13: 12: 12

**3. A began with Rs. 45000 and was joined afterwards by B with Rs. 54000. After how many months did B join if the profits at the end of the year were divided in the ratio 2: 1?**

**Solution:**

45000*12:
54000*x = 2: 1 => x = 45000*12*1 / (54000*2) = 5 => B joined after 7
months.

**Alternative Method:**

Ratio of
capitals = 45000: 54000 = 5: 6 => Ratio of profits = 2: 1

Therefore,
Ratio of periods = Ratio of profits/ Ratio of capitals = 2/5 : 1/6 = 12: 5 =>
B joined after 7 months.

**4. A and B entered into a partnership with capitals in the ratio 4 : 5. After 3 months, A withdrew 1414 of his capital and B withdrew 1515 of his capital. At the end of 10 months, the gain was Rs.760. What is A's share in the profit?**

**Solution:**

Ratio of
the initial capital of A and B = 4 : 5

Hence we
can take the initial capitals of A and B as 4x and 5x respectively

Ratio in
which profit will be divided

= (4x * 3)
+ 3/4 * 4x * 7: (5x * 3) + 4/5 * 5x * 7

= (12 + 21)
: (15 + 28) = 33: 43

A’s share =
760 * 33/76 = 330

**5. A and B started a business with Rs. 20000 and Rs. 35000 respectively. They agreed to share the profit in the ratio of their capital. C joins the partnership with the condition that A, B and C will share profit equally and pays Rs. 2,20,000 as premium for this, to be shared between A and B. This will be divided between A and B in the ratio of:**

**Solution:**

Old ratio
of A and B = 20000: 35000 = 4: 7 New ratio of A, B, C = 1: 1: 1

A’s share
is reduced by 4/11 – 1/3 = 1/33

B’s share
reduced by 7/11 – 1/3 = 10/33

Required
ratio = 1/33: 10/33 = 1: 10

**TYPE 3: PARTNERSHIP WITH RATIO**

**1. A, B and C shared profits in ratio of 5:7:8. They partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?**

**Solution:**

Simply
multiply profit sharing ratio with investment ratio to get investment amount
ratio.

Let X is
the total investment

⇒14 x = 5; 8
x = 7; 7x = 8

⇒ Final
investment ratio = 20: 49: 64

**2. A and B invest in a business in the ratio 3: 2. Assume that 5% of the total profit goes to charity. If A's share is Rs. 855, what is the total profit?**

**Solution:**

Assume that
the total profit is x

Since 5%
goes for charity, 95% of x will be divided between A and B in the ratio 3:2

Therefore,
A's profit = 95x/100 × 3/5

Given that
A's share is Rs. 855. Therefore,

95x/100 ×
3/5 = 855 => x = (855 × 100 × 5)/(3 × 95) = 1500

Hence the
total profit = 1500

**3. A, B and C invest in a partnership in the ratio: 7/2,4/3,6/5. After 4 months, A increases his share 50%. If the total profit at the end of one year is Rs.21,600, then what is B's share in the profit?**

**Solution:**

Ratio of
the initial investment =7/2:4/3:6/5 = 105:40:36

Therefore,
let the initial investments of A, B and C be 105x, 40x and 36x respectively

A increases
his share 50% after 4 months. Hence the ratio of their investments

B's share =
total profit × 10 / 54 =21600 × 10 / 54 = 4000

**4. In a business, A and C invested amounts in the ratio 2 : 1 , whereas the ratio between amounts invested by A and B was 3 : 2 . If Rs.157300 was their profit, how much amount did B receive?**

**Solution:**

Assume that
investment of C = x

Then,
investment of A = 2x

Investment
of B = 4x / 3

A:B:C =2x :
4x/3 : x => A: B: C=2: 4/3 :1 = 6 : 4 : 3

B's share
=157300× 4/(6+4+3) =157300 × 4/13 = 12100 × 4 = 48400

**5. A and B partner in a business. A contribute 1/4 of the capital for 15 months and B received 2/3 of the profit. For how long B's money was used?**

**Solution:**

B received
2/3 of the profit.

Then A
received remaining 1/3 of the profit.

A : B =1/3
: 2/3 = 1:2

Let the
total capital =x

Then A's
capital =x/4

B's capital
= x – x/4 = 3x / 4

Assume B's
money was used for b months

Then A : B
= x/4 × 15 : 3x/4 × b => 15 : 3b = 1 : 2 => b = 10

**6. A, B and C started a business with capitals in the ratio 5: 6: 8. If at the end of one year, they shared the profit in the ratio 5: 3 : 12, find the ratio of time for which they had contributed their capitals?**

**Solution:**

Ratio of
capitals = 5: 6: 8

Ratio of
share in profit = 5: 3: 12

Therefore,
Ratio of periods = Ratio of profits / Ratio of capitals = 5/5: 3/6: 12/8 = 1:
1/2: 3/2 = 2: 1: 3

**7. A and B started a partnership business investing capital in the ratio of 3 : 5. C joined in the partnership after six months with an amount equal to that of B. At the end of one year, the profit should be distributed among A, B and C in --- proportion.**

**Explanation:**

Initial
investment capital ratio of A and B = 3: 5

Let initial
capital of A and B be 3x3x and 5x5x respectively.

Amount that
C invested after 6 months =5x=5x (since it is equal to B's investment)

Ratio in
which profit should be distributed after 1 year

= 3x × 12 :
5x × 12 : 5x × 6 = 3 × 12 : 5 : 12 : 5 ×
6 = 6 : 10 : 5

**TYPE 4: PARTNERSHIP AND SHARES**

**1. Sethu and Siva started a business by investing Rs.4000 and Rs.3000 respectively. After 6 months, Ajay joined with them by investing Rs.4000. At the end of 2 years they earned a profit of Rs.5000 then what will be Siva's share?**

**Solution:**

Sethu
invests Rs.4000 for 24 months, Ajay invests Rs.3000 for 24 months and Ajay
invests Rs.4000 for 18 months.

Then, Sethu
: Siva : Ajay = (4000x24) : (3000x24) : (4000x18) = 4x24 : 3x24 : 4x18 = 4 : 3 :
3

Therefore,
Siva's share = Rs.5000 x 3/10 = Rs.1500.

**2. Sheela started a business in 2009 by investing Rs.50,000. She invested Rs. 20,000 as additional amount in 2010 and her friend Devi joined her with an amount of Rs.70,000. Sheela invested another Rs. 20,000 in 2011 and Anu joined them with Rs. 70,000. At the end of these 3 years, they earned a profit of Rs.3,00,000. Find Devi's share?**

**Solution:**

Sheela
invested Rs.50,000 for 12 months, Rs.(50000 + 20000) for 12 months and
Rs.(50000 + 20000 + 20000) for 12 months.

i.e., she
invested Rs.50,000 for 12 months, Rs.70000 for 12 months and Rs.90000 for 12
months.

Devi
invested Rs. 70000 for 2 years; i.e., Rs.70000 for 24 months

And, Anu
invested Rs.70000 for 1 year; i.e., Rs. 70000 for 12 months.

Their investing
ratio: Sheela : Devi : Anu = (50,000 x 12 + 70000 x 12 + 90000 x 12):(70000 x
24):(70000 x 12) = (25,20,000):(16,80,000):(8,40,000) = 252:168:84 = 3:2:1

Total
profit for 3 years = Rs.3,00,000

Therefore,
Devi's share = Rs.(3,00,000 x 2 /(3+2+1)) = Rs.(3,00,000 x 2/6) = Rs.1,00,000

**3. A started a business with a capital of Rs. 100000. One year later, B joined him with a capital of Rs. 200000. At the end of 3 years from the start of the business, the profit earned was Rs. 84000. The share of B in the profit exceeded the share of A by:**

**Solution:**

Ratio in
which profit is to distributed between A and B = 100000 * 3: 200000 * 2 = 3: 4

Therefore,
Difference in their share in profit = (4-3) / (3+4) * 84000 = Rs. 12000

**4. A, B and C start a business with each investing Rs 20,000. After 5 months A withdraws Rs 5000, B withdraws Rs 4000 and C invests Rs 6000 more. At the end of the year, a total profit of Rs 69900 was recorded. Find the share of A.**

**Solution:**

A: B: C =
(20000*5 + 15000*7): (20000*5 + 16000*7): (20000*5 + 26000*7)

A: B: C =
(20*5 + 15*7): (20*5 + 16*7): (20*5 + 26*7) = 205: 212: 282

A’s share =
69900* 205 / (205+212+282) = 69900 * 205 / 699 = 20500

**5. P, Q and R started a business by investing Rs.120000, Rs.135000 and Rs.150000 respectively. Find the share of each, out of the annual profit of Rs.56700.**

**Solution:**

P: Q: R =
120000:135000:150000 =120:135:150=24:27:30=8:9:10

Share of P
= 56700 × 8/27 = 2100 × 8 = 16800

Share of Q
= 56700 × 9/27 = 2100 × 9 = 18900

Share of R
= 56700 × 10/27 = 2100 × 10 = 21000