A, B and C start a business and invest in the ratio 7 : 6 : 5. In the next year they increase their investments by 25%, 20% and 15% respectively. In what ratio should the profit earned only during the second year be distributed among them?

Introduction / Context:
This question is a classic partnership and ratio problem where partners change their capital in the second year. The key idea is that profit for a particular period is shared in proportion to the effective capital in that period. Since we are asked about profit only for the second year, we need to focus solely on the revised investments after the percentage increases are applied to the original ratio 7 : 6 : 5.

Given Data / Assumptions:
Initial investment ratio of A, B and C is 7 : 6 : 5. In the second year A increases capital by 25%. B increases capital by 20%. C increases capital by 15%. We assume time period for the second year is the same for all three partners.

Concept / Approach:
Profit in a partnership is proportional to capital multiplied by time. Here time for the entire second year is the same for all partners, so profit sharing is directly proportional to the capital invested in that year alone. We first calculate the increased capital of each partner relative to the original ratio using percentage increase formulas. Then we convert these new values into a simple ratio by removing decimals and simplifying to the lowest terms, which gives the required sharing ratio for the second year profit.

Step-by-Step Solution:
Step 1: Take initial capitals as 7, 6 and 5 units for A, B and C. Step 2: A increases capital by 25%, so new capital of A = 7 * (1 + 25 / 100) = 7 * 1.25 = 8.75 units. Step 3: B increases capital by 20%, so new capital of B = 6 * (1 + 20 / 100) = 6 * 1.20 = 7.2 units. Step 4: C increases capital by 15%, so new capital of C = 5 * (1 + 15 / 100) = 5 * 1.15 = 5.75 units. Step 5: To remove decimals, multiply all three by 100: 875 : 720 : 575. Step 6: Simplify by dividing each term by 5 to get 175 : 144 : 115. Step 7: Therefore the second year profit must be divided in the ratio 175 : 144 : 115 for A, B and C.

Verification / Alternative check:
One can also assume specific initial amounts, for example 7000, 6000 and 5000. Then after the given increases, the second year amounts become 8750, 7200 and 5750. Their greatest common divisor is 50, so the ratio is 175 : 144 : 115 which matches the earlier calculation. Any consistent scaling of initial capitals leads to the same simplified ratio, confirming that the result is correct and independent of the absolute amount chosen initially.

Why Other Options Are Wrong:
The ratio 155 : 144 : 175 does not match the percentage increase pattern and comes from miscalculating A and C. The ratio 155 : 124 : 95 underestimates all partners and particularly distorts B and C values. The option 135 : 147 : 152 is inconsistent with the fact that A had the largest percentage increase and should have the highest proportion in the second year. Only 175 : 144 : 115 correctly reflects the adjusted capitals based on 25%, 20% and 15% increments.

Common Pitfalls:
Learners sometimes add the same absolute increment to each partner instead of calculating percentage increases separately. Another common mistake is to include the first year capital in the calculation even though the question asks only about profit in the second year. Forgetting to simplify ratios by clearing decimals also leads to different but equivalent looking numbers that do not match the answer choices. Careful handling of percentages and focusing only on the required year avoids these errors.

Final Answer:
The profit earned only during the second year should be distributed among A, B and C in the ratio 175 : 144 : 115.