Rithika has Rs 8000. She invests some of this amount in savings scheme P for 3 years and the remaining amount in savings scheme Q for 2 years. Scheme P offers simple interest at 20% per annum, while scheme Q offers compound interest at 30% per annum compounded annually. The interest from scheme P is Rs 930 more than the interest from scheme Q. If Rs 1500 is later debited from the amount invested in scheme P, what is the difference between the amounts invested in P and Q?

Introduction / Context:
This question combines simple interest and compound interest concepts with an additional condition about withdrawing part of the investment later. Rithika splits her money between two schemes with different interest types and durations. We know the difference between the interests earned from the two schemes and must determine the original investments in each scheme, then adjust the investment in one scheme by debiting an amount and finally find the difference between the adjusted amounts.

Given Data / Assumptions:
Total money available to Rithika is Rs 8000. She invests x rupees in scheme P for 3 years at simple interest 20% per annum. She invests 8000 - x rupees in scheme Q for 2 years at compound interest 30% per annum, compounded annually. Interest from scheme P is Rs 930 more than interest from scheme Q. Later, Rs 1500 is debited from the amount invested in scheme P. We must find the difference between the amounts invested in P and Q after this debit.

Concept / Approach:
We first express the interest from each scheme in terms of x. For simple interest, interest = principal * rate * time. For compound interest, we compute the amount after 2 years and subtract the principal to get interest. Using the condition that interest from P exceeds interest from Q by Rs 930, we form an equation in x and solve it. Once we know x, we identify the investments in P and Q. Then we reduce the investment in P by Rs 1500 and compare the adjusted amounts to get the required difference.

Step-by-Step Solution:
Step 1: Let amount invested in P be x rupees, so amount in Q is 8000 - x rupees. Step 2: Simple interest from P for 3 years at 20% per annum is I_P = x * 20 / 100 * 3 = 0.6x. Step 3: For Q, principal is 8000 - x, rate is 30% per annum, time is 2 years and interest is compound. Step 4: Amount in Q after 2 years is A_Q = (8000 - x) * (1 + 30 / 100)^2 = (8000 - x) * 1.3^2 = (8000 - x) * 1.69. Step 5: Compound interest from Q is I_Q = A_Q - (8000 - x) = (8000 - x) * (1.69 - 1) = (8000 - x) * 0.69. Step 6: We are told that interest from P is Rs 930 more than interest from Q, so 0.6x = 0.69 * (8000 - x) + 930. Step 7: Expand the right side: 0.69 * (8000 - x) = 0.69 * 8000 - 0.69x = 5520 - 0.69x. Step 8: So the equation becomes 0.6x = 5520 - 0.69x + 930 = 6450 - 0.69x. Step 9: Bring x terms together: 0.6x + 0.69x = 6450, so 1.29x = 6450. Step 10: Therefore x = 6450 / 1.29 = 5000. Step 11: So original investment in P is Rs 5000 and in Q is 8000 - 5000 = Rs 3000. Step 12: Now Rs 1500 is debited from the amount in P, so adjusted investment in P is 5000 - 1500 = Rs 3500. Step 13: Adjusted investment in Q remains Rs 3000. Step 14: Difference between amounts in P and Q after debit = 3500 - 3000 = Rs 500.

Verification / Alternative check:
With x = 5000, interest from P is I_P = 0.6 * 5000 = Rs 3000. For Q, principal is 3000. Amount after 2 years at 30% compound interest is 3000 * 1.3^2 = 3000 * 1.69 = 5070, so interest I_Q = 5070 - 3000 = Rs 2070. The difference I_P - I_Q = 3000 - 2070 = Rs 930, which matches the condition in the question. This confirms that x = 5000 is correct and that the later debit of Rs 1500 from P leading to a difference of Rs 500 between P and Q is consistent.

Why Other Options Are Wrong:
If the final difference between investments in P and Q were Rs 200, Rs 300 or Rs 400, it would imply different initial investments and would break the interest difference relation of Rs 930. Only the initial split 5000 and 3000 satisfies the interest condition, and after debiting Rs 1500 from P the resulting difference is exactly Rs 500. Therefore the other numerical options cannot match all the constraints of the problem.

Common Pitfalls:
A common mistake is to treat scheme Q as simple interest rather than compound interest, leading to a different interest equation. Another pitfall is to forget that Q runs for only 2 years while P runs for 3 years. Some learners may also misread the phrase about debiting Rs 1500 and incorrectly apply this change before computing the interest relation. Carefully distinguishing original investments, interest calculations and later adjustments helps to avoid these errors.

Final Answer:
After debiting Rs 1500 from the investment in scheme P, the difference between the amounts invested in P and Q is Rs 500.