A, B and C invest amounts in the ratio 3 : 4 : 5 respectively. The three schemes offer compound interest at 20% per annum, 15% per annum and 10% per annum respectively. What will be the ratio of their amounts after 1 year?

Introduction / Context:
This question combines ratio concepts with simple compound interest for a single year. Although the schemes are described as compound interest schemes, for exactly one year the amount is simply principal multiplied by 1 plus the interest rate. Our task is to find the ratio of the final amounts after one year when three different interest rates are applied to capitals in an initial ratio of 3 : 4 : 5.

Given Data / Assumptions:
Initial investment ratio A : B : C = 3 : 4 : 5. A earns compound interest at 20% per annum. B earns compound interest at 15% per annum. C earns compound interest at 10% per annum. We consider only 1 year duration for all three investments.

Concept / Approach:
For compound interest, amount after 1 year is given by A = P * (1 + R / 100). Since time is 1 year for all, we do not need higher powers. We treat the ratio capitals as actual principal values, multiply each by the appropriate factor and then form a ratio of the resulting amounts. Finally, we remove any decimals by scaling and reduce to the simplest ratio which matches one of the answer options.

Step-by-Step Solution:
Step 1: Assume investments are 3, 4 and 5 units for A, B and C respectively. Step 2: Amount for A after 1 year at 20% = 3 * (1 + 20 / 100) = 3 * 1.20 = 3.6 units. Step 3: Amount for B after 1 year at 15% = 4 * (1 + 15 / 100) = 4 * 1.15 = 4.6 units. Step 4: Amount for C after 1 year at 10% = 5 * (1 + 10 / 100) = 5 * 1.10 = 5.5 units. Step 5: Now we have ratio 3.6 : 4.6 : 5.5. Step 6: Multiply each term by 10 to remove decimals: 36 : 46 : 55. Step 7: There is no common factor among 36, 46 and 55, so 36 : 46 : 55 is already in simplest form.

Verification / Alternative check:
We can verify by choosing actual principal amounts, for example Rs 3000, Rs 4000 and Rs 5000. After one year their amounts would be 3000 * 1.20 = 3600, 4000 * 1.15 = 4600 and 5000 * 1.10 = 5500. The ratio 3600 : 4600 : 5500 simplifies by dividing all terms by 100 to 36 : 46 : 55, which matches the derived ratio. This confirms that the calculation is correct and independent of the chosen base principal units.

Why Other Options Are Wrong:
The ratio 3 : 15 : 25 does not reflect the effect of different interest rates and seems to treat interest as simple additions to the original ratio. The option 6 : 6 : 5 suggests B and C amounts are almost equal, which contradicts a higher initial capital for C. The ratio 12 : 23 : 11 bears no relation to the percentage increments 20%, 15% and 10%. Only 36 : 46 : 55 is consistent with multiplying each original capital by its correct amount factor.

Common Pitfalls:
Students may mistakenly add or subtract the percentage directly to the ratio values instead of using the multiplicative factor 1 + R / 100. Another common error is to convert interest to simple interest for multiple years even though only one year is involved. Forgetting to clear decimals properly can make it hard to match answer options. Using the simple formula for one year and scaling to remove decimals ensures an accurate and clean ratio.

Final Answer:
The ratio of their amounts after one year is 36 : 46 : 55.