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A, B and C start a business with initial investments in the ratio of `3 : 4 : 2.`Four months after the start of the business, B halves his investment. What was C’s investment?
I The difierence between B’s and C’s share of the annual profit was ₹600
II The ratio of the annual investments by A, B and C was `9:8:6`

A

if the data in statement n alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question

B

if the data either in statement I alone or in statement II alone are sufficient to answer the question

C

if the data in both the statements I and II together are not sufficient to answer the question

D

if the data in both the statements I and II together are necessary to answer the question

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will analyze the information given and derive C's investment. ### Step 1: Understand the initial investment ratios A, B, and C start a business with investments in the ratio of 3:4:2. Let’s denote their investments as: - A's investment = 3X - B's investment = 4X - C's investment = 2X ### Step 2: Calculate the investment after 4 months Since the problem states that B halves his investment after 4 months, we need to calculate the effective investment for each partner over a year (12 months). - A's investment remains the same for 12 months: - A's total investment = 3X * 12 = 36X - B's investment for the first 4 months is 4X, and for the remaining 8 months, it is halved (2X): - B's total investment = (4X * 4) + (2X * 8) = 16X + 16X = 32X - C's investment remains the same for 12 months: - C's total investment = 2X * 12 = 24X ### Step 3: Set up the ratio of annual investments Now we have the total investments for A, B, and C: - A's total investment = 36X - B's total investment = 32X - C's total investment = 24X The ratio of their annual investments is: - A : B : C = 36X : 32X : 24X ### Step 4: Simplify the ratio To simplify the ratio: - Divide each term by 4: - A : B : C = 9 : 8 : 6 This matches the second statement given in the problem. ### Step 5: Use the first statement about profit difference According to the first statement, the difference between B's and C's share of the annual profit is ₹600. The profit share is proportional to their investments. Let’s denote the total profit as P. The shares of B and C can be expressed as: - B's share = (B's investment / Total investment) * P = (32X / (36X + 32X + 24X)) * P - C's share = (C's investment / Total investment) * P = (24X / (36X + 32X + 24X)) * P The total investment = 36X + 32X + 24X = 92X. ### Step 6: Calculate the profit shares - B's share = (32X / 92X) * P = (32/92) * P - C's share = (24X / 92X) * P = (24/92) * P The difference between B's and C's share is: - Difference = B's share - C's share = [(32/92) * P - (24/92) * P] = [(32 - 24) / 92] * P = (8/92) * P = (2/23) * P According to the problem, this difference is ₹600: - (2/23) * P = 600 - P = 600 * (23/2) = ₹6900 ### Step 7: Calculate C's investment Now we can find C's investment: - C's investment = 2X - Total investment = 92X - C's share of the profit = (24/92) * P = (24/92) * 6900 Calculating C's share: - C's share = (24/92) * 6900 = ₹1800 ### Step 8: Find the value of X Since C's investment is proportional to his share of the profit: - C's profit share = (C's investment / Total investment) * Total profit - 1800 = (2X / 92X) * 6900 - Simplifying gives us: 1800 = (2/92) * 6900 - 1800 = (13800/92) - Thus, 2X = 1800 * (92/6900) - Therefore, C's investment = 2X = ₹3600. ### Final Answer C's investment is ₹3600.
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