Three partners share the profit of a business in the ratio 4 : 6 : 8. They remain in the business for 12 months, 8 months and 6 months respectively. What was the ratio of their original investments?

Introduction / Context:
This question checks your understanding of how partners share profit in a business when their capitals are invested for different periods of time. The key idea is that profit sharing depends on the product of capital and time, not only on the capital or the time separately.

Given Data / Assumptions:

• The profit sharing ratio of three partners is 4 : 6 : 8.
• The time periods of investment are 12 months, 8 months and 6 months respectively.
• We have to find the ratio of their original investments.
• The total profit amount is not required for the solution.

Concept / Approach:
In partnership problems, each partner's share in profit is proportional to capital * time. If we know the final profit ratio and the time for each partner, we can reverse the process and find the investment ratio by dividing the profit ratio by the corresponding time periods. This effectively cancels out the effect of time and leaves only the ratio of capitals.

Step-by-Step Solution:
Step 1: Let the original investments of the three partners be I1, I2 and I3. Step 2: Their time periods are T1 = 12 months, T2 = 8 months and T3 = 6 months. Step 3: Since profit is proportional to capital * time, we have I1 * 12 : I2 * 8 : I3 * 6 = 4 : 6 : 8. Step 4: To find the investment ratio, divide each profit ratio term by the corresponding time. Step 5: So, I1 : I2 : I3 = 4 / 12 : 6 / 8 : 8 / 6. Step 6: Simplify each fraction: 4 / 12 = 1 / 3, 6 / 8 = 3 / 4, and 8 / 6 = 4 / 3. Step 7: Now express 1 / 3 : 3 / 4 : 4 / 3 in whole number form by taking the common denominator 12. Step 8: Multiply each term by 12: (1 / 3)*12 = 4, (3 / 4)*12 = 9, (4 / 3)*12 = 16. Step 9: Therefore, the ratio of their investments is 4 : 9 : 16.

Verification / Alternative check:
We can verify by checking that using 4, 9 and 16 as investment amounts with the given times gives back the profit ratio 4 : 6 : 8. Capital time products are 4 * 12 = 48, 9 * 8 = 72 and 16 * 6 = 96. The ratio 48 : 72 : 96 simplifies by dividing all terms by 12 to 4 : 6 : 8, which matches the profit ratio from the question. This confirms that the investment ratio 4 : 9 : 16 is correct.

Why Other Options Are Wrong:
4 : 6 : 8 simply repeats the profit ratio and ignores the unequal time periods, so it cannot be the investment ratio.
4 : 7 : 8 and 4 : 8 : 16 do not produce the required profit ratio when multiplied by the respective time periods. Only 4 : 9 : 16 gives the correct final profit ratio of 4 : 6 : 8.

Common Pitfalls:
A frequent mistake is to equate the investment ratio directly to the profit ratio without considering the different times for which the capitals were invested. Another error is to forget to use a common multiple when converting fractional ratios to whole numbers, which can lead to incorrect simplification. Always remember that in partnership questions, capital time products determine the true ratio of profit sharing.

Final Answer:
The ratio of their investments is 4 : 9 : 16.