Prasanna invests certain amounts in three different schemes X, Y and Z at simple interest rates of 10% p.a., 12% p.a. and 15% p.a. respectively. The total interest accrued in one year is Rs. 3,200. The amount invested in Scheme Z is 150% of the amount invested in Scheme X and 240% of the amount invested in Scheme Y. What amount did Prasanna invest in Scheme Y?

Introduction / Context:
This is a typical investment and simple interest problem involving three different schemes with different interest rates. The key challenge is that the amounts invested in the schemes are related to each other by given ratios. We need to convert these relationships into equations, use the total interest information, and solve for the unknown amount invested in one particular scheme.

Given Data / Assumptions:

Interest rate in Scheme X = 10% per annum.
Interest rate in Scheme Y = 12% per annum.
Interest rate in Scheme Z = 15% per annum.
Total simple interest in one year from all schemes = Rs. 3,200.
Amount in Z is 150% of amount in X.
Amount in Z is 240% of amount in Y.
We assume simple interest for one year, so interest = principal * rate.

Concept / Approach:
Let the amounts invested in X, Y and Z be X, Y and Z respectively. We are given two ratio conditions connecting Z with X and Y. Using these, we can express X and Z in terms of Y. Then we write the total interest equation for one year and solve it for Y. This reduces the problem to a single variable equation using percentage and simple interest concepts.

Step-by-Step Solution:
Step 1: Translate the relationships among investments. Z is 150% of X, so Z = 1.5 * X. Z is 240% of Y, so Z = 2.4 * Y. Step 2: Equate the two expressions for Z to relate X and Y. 1.5 * X = 2.4 * Y. So, X = (2.4 / 1.5) * Y = 1.6 * Y. Step 3: Now write all amounts in terms of Y. X = 1.6Y, Z = 2.4Y, and Y = Y. Step 4: Use the total simple interest information. Interest from X in one year = 10% of X = 0.10 * X = 0.10 * 1.6Y = 0.16Y. Interest from Y in one year = 12% of Y = 0.12Y. Interest from Z in one year = 15% of Z = 0.15 * 2.4Y = 0.36Y. Step 5: Add all three interests and equate to 3,200. Total interest = 0.16Y + 0.12Y + 0.36Y = 0.64Y. So, 0.64Y = 3,200. Step 6: Solve for Y. Y = 3,200 / 0.64 = 5,000. Therefore, the amount invested in Scheme Y is Rs. 5,000.

Verification / Alternative check:
Once Y is 5,000, X = 1.6 * 5,000 = 8,000. Z = 2.4 * 5,000 = 12,000. Now compute the interest from each scheme. At 10%, Scheme X yields 800. At 12%, Scheme Y yields 600. At 15%, Scheme Z yields 1,800. Total interest = 800 + 600 + 1,800 = 3,200, which matches the given total. This confirms that the value Y = 5,000 is correct.

Why Other Options Are Wrong:
If Y were 6,000, then the total interest using the same relationships would exceed 3,200.
Values like 4,500 or 7,500 also lead to total interest values that do not equal 3,200 when the given ratios and rates are used.
3,200 is the total interest, not the invested amount in Y, so that option misinterprets the data.

Common Pitfalls:
A common error is to treat 150% and 240% as simple multipliers without converting them properly, or to confuse which amount is bigger. Another mistake is to assign independent variables for X, Y and Z and forget to apply the given percentage relationships, leading to too many unknowns. Some learners also misinterpret the interest formula and multiply by the rate twice. Carefully translating all relationships into a single variable before writing the interest equation makes the problem much easier to manage.

Final Answer:
Prasanna invested Rs. 5,000 in Scheme Y.