Dear reader,
Below are 3 problems which deals with ratios between partners' share of capital, profit etc.

Formulas to Remember:

i) When investments of all the partners are for the same duration, the profit 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 x 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 = Total investment of A : Total investment of B

Question 1

A starts a project of duration of 15 months with capital Rs.60000. 3 months after the start, B joins with A and invests one fourth of that of A. After another 6 months, C joins with capital Rs. 90000. At the end of the year, A withdrew the partnership leaving the firm to be run by B and C. What will be the shares of B and C if total profit is Rs.50000?

a)Rs.25000, Rs.6250 b)Rs.6250,Rs.18750 c)Rs.18750, Rs.6250 d)Rs.25000,Rs.18750

Answer : b)Rs.6250,Rs.18750.

Solution:

A works since start of the project but withdrew 3 months before completion. Therefore, he works for 15 - 3 = 12 months
B works 3 months after start of the project and continues till end. Therefore, his duration of partnership = 15 - 3 = 12 months
C joins 9 months after A started business (6 months after B who was already late by 3 months) and works till completion. Therefore, his duration = 15 - 9 = 6

The ratio of their profits =(60000 x 12):(1/4 x 60000 x 12):(90000 x 6)
=720000:180000:540000
=4:1:3

Now let us find the respective share of profits out of total profit Rs.50,000

A's share= 4/8 x 50000 = 25000
B's share = 1/8 x 50000 = 6250
C's share = 3/8 x 50000 = 18750.

Based on above results, our answer is option b.

Question 2

A starts a business and invests Rs.45,000 per month for two months. After few months, B joins with a planned monthly investment of Rs. 35,000. After 2 months A steps up his monthly investment by another Rs.2000. At the end of the year, the total profit was divided in the ratio 2 : 1. After how many months since start of business did B join?

a)1 b)2 c)3 d)4

Answer : d) 4

Solution:

Suppose B joined after X months.

Then, B's money is being invested for (12 - X) months. He invests Rs.35,000 per month.
Therefore, his total investment = [35000 x (12 - X)]

The investment of A = 45,000 per month for first two months + (45000 + 2000)per month for the next 10 months
=[(45000 x 2) + (45000 + 2000) x 10]
= [90000 + 470000]
= 560000.

Ratio of profit of A to that of B = 2 : 1

By using the formula in the introductory paragraph, we will get,

560000 : [35000 x (12 - X)] = 2 : 1.
560000 / [35000 x (12 - X)] = 2/1
2 x [35 x (12 - X)] = 560
840 - 560 = 70X
280 = 70X
X = 4

Hence, B joined after 4 months.

Question 3

A,B,C were getting into a business for 3 years. After 1 year C withdrew Rs.5000 and after another 6 months B invested Rs.15000. Also at the end of 2 years A invested Rs.5000. Finally they shared the profit in the ratio of 2:3:4. Now suppose 2:1:4 is the ratio of their initial investments then what is the contribution of C in capital?

a)Rs.15000 b)Rs.450000 c)Rs.10000 d)Rs.20000

Answer : c) Rs.10000

Solution:

Let the initial investments of A,B,C be 2X, X, 4X respectively. (since the ratio of initial investment amount is :: 2:1:4).

Part 1: To find the value of x.

Based on data in the question, we can calculate the individual investments as follows.

Total investment of A = [2X x 24 ]+[2X + 5000] x 12.
= 48X + 24X + 60000
= 72X +60000.
Total investment of B = [X x 18] + [(X + 15000) x 18]
= 18X + 18X + 270000
= 36X + 270000
Total investment of c = [4X x 12] + [(4X - 5000) x 24]
= 48X + 96X - 120000
= 144X - 120000

Now using the formula given in the introductory paragraph.

A's profit : B's profit : C's profit = A's total investment : B's total investment : C's total investment

Or
2 : 3 : 4 = [72X + 60000]:[36X + 270000]:[144X - 120000].

by using the fact if a:b:c = x:y:z then a/b = x/y
we can write, 2/3 = [72X + 60000] / [36X + 270000]

3 x [72X + 60000] = 2 x [36X + 270000]
216X + 180000 = 72X + 540000
144X = 360000
X = 2500.

Part 2: To find the initial investment of C

Since the ratio of their initial investments is 2:1:4, the initial investment of C is X

4X = 4 x 2500
=10000.
Hence the answer is Rs.10000