The problem requires calculating the difference in profit between partners X and Y, considering their initial investment ratio, a change in X's capital, and the total annual profit.
Investment Ratio and Initial Capital
The initial investment ratio of the three friends X, Y, and Z is given as $6:3:5$.
We are given that Z's investment is ₹12,000. Since Z's share in the ratio is 5 parts, we can determine the value of one part:
- Value of 1 part: $1 \text{ part} = \frac{₹12,000}{5} = ₹2,400$
- X's initial investment (6 parts): $X_{\text{initial}} = 6 \times ₹2,400 = ₹14,400$
- Y's investment (3 parts): $Y_{\text{initial}} = 3 \times ₹2,400 = ₹7,200$
Profit Sharing Ratio Calculation
Profit is distributed based on the investment amount and the duration. The total profit is annual (12 months).
- X's Profit Share: X invested his full capital for the first 6 months and then withdrew half (invested $\frac{1}{2}$ of his capital for the remaining 6 months).
X's effective investment ratio for the year is calculated considering time: $(6 \text{ months} \times 6 \text{ parts}) + (6 \text{ months} \times \frac{6}{2} \text{ parts}) = (6 \times 6) + (6 \times 3) = 36 + 18 = 54 \text{ (capital-time units)}$ - Y's Profit Share: Y invested his capital for the entire 12 months.
Y's effective investment ratio for the year is: $12 \text{ months} \times 3 \text{ parts} = 36 \text{ (capital-time units)}$ - Z's Profit Share: Z invested his capital for the entire 12 months.
Z's effective investment ratio for the year is: $12 \text{ months} \times 5 \text{ parts} = 60 \text{ (capital-time units)}$
The ratio for profit sharing is $X:Y:Z = 54:36:60$.
To simplify this ratio, we divide each term by their greatest common divisor, which is 6:
$ \frac{54}{6} : \frac{36}{6} : \frac{60}{6} = 9:6:10 $The final profit sharing ratio is $9:6:10$.
Individual Profit Calculation
The total annual profit is ₹69,000.
The sum of the parts in the simplified profit sharing ratio is $9 + 6 + 10 = 25$.
- X's Profit: $ \text{X's Profit} = \left(\frac{9}{25}\right) \times ₹69,000 $ $ \text{X's Profit} = 9 \times ₹2,760 = ₹24,840 $
- Y's Profit: $ \text{Y's Profit} = \left(\frac{6}{25}\right) \times ₹69,000 $ $ \text{Y's Profit} = 6 \times ₹2,760 = ₹16,560 $
Final Profit Difference
The question asks for the difference between X's profit and Y's profit.
$ \text{Difference} = \text{X's Profit} - \text{Y's Profit} $ $ \text{Difference} = ₹24,840 - ₹16,560 = ₹8,280 $Important Questions from Partnership
A, B and C invest in a business in the ratio 4 ∶ 5 ∶ 7. C is a sleeping partner, so his share of profits will be half of what it would have been if he were a working partner. If they make Rs 36,000 profit of which 25% is reinvested in the business, how much does B get (in Rs)?
Sumit, Ravi and Puneet invest Rs. 45000, Rs. 81000 and Rs. 90000 respectively to start a business. At the end of the year the total profit is Rs. 4800. 30% of the total profit gives in charity and rest is divided among them. What will be the share of Sumit?
A sum of ₹ 159250 is divided among A, B, C, and D such that the ratio of the shares of A and B is 1 : 3, that of B and C is 2 : 5, and that of C and D is 2 : 3. The share (in ₹) of A is:
A and B start a business by investing Rs. 1,00,000 and Rs. 1,50,000 respectively. Find the respective share of each out of a total profit of Rs. 24, 000.
Two partners A and B have started business with the capitals of Rs. 6,000 and Rs. 8,000 respectively. If they made profit of Rs. 5,600 then the share (in Rs.) of A is:
