Question

Anindita invests a total of 1 lakh rupees distributed across three schemes A, B and C for a period of two years. These schemes offer an interest rate of 10%, 8% and 12% per annum, respectively, each compounded annually. If the initial investment amount in scheme A is 30000 rupees and the total interest earned from all the three schemes during the first year is 10600 rupees, then the total interest earned, in rupees, from all the three schemes for the second year is ________

• A. 10308
• B. 19708
• C. 11748
• D. 22348

Hint:

An alternative way to calculate the second year's interest is to realize it's simply the first year's interest, plus interest on the first year's interest. Total 2nd year interest = (Total 1st year interest) * (1 + blended rate). A simpler method is as shown above: calculate the new principal for each scheme and then find the interest for the next year.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves compound interest and solving a system of linear equations. First, we need to determine the initial investment in each scheme using the given information. Then, we calculate the interest for the second year based on the compounded principal.
Step 2: Key Formula or Approach:
1. Set up equations for the total investment and total first-year interest. 2. Solve the system of equations to find the investment in schemes B and C. 3. For compound interest, the interest for the second year is calculated on the principal of the first year plus the interest earned in the first year. Interest for 2nd year = (Principal + Interest for 1st year) \( \times \) rate.
Step 3: Detailed Explanation:
Let the investments in schemes A, B, and C be \(P_A, P_B, P_C\).
Total Investment: \(P_A + P_B + P_C = 100000\).
Given \(P_A = 30000\), so \(30000 + P_B + P_C = 100000 \implies P_B + P_C = 70000\) (Eq. 1).
Total interest in the first year is 10600.
Interest = \( P_A \times 0.10 + P_B \times 0.08 + P_C \times 0.12 \).
\( 10600 = 30000 \times 0.10 + 0.08 P_B + 0.12 P_C \)
\( 10600 = 3000 + 0.08 P_B + 0.12 P_C \)
\( 7600 = 0.08 P_B + 0.12 P_C \) (Eq. 2).
Now, solve the system of equations:
From Eq. 1, \( P_C = 70000 - P_B \). Substitute this into Eq. 2:
\( 7600 = 0.08 P_B + 0.12 (70000 - P_B) \)
\( 7600 = 0.08 P_B + 8400 - 0.12 P_B \)
\( 7600 - 8400 = -0.04 P_B \)
\( -800 = -0.04 P_B \)
\( P_B = \frac{800}{0.04} = \frac{80000}{4} = 20000 \).
So, \( P_C = 70000 - 20000 = 50000 \).
The initial investments are: \(P_A = 30000, P_B = 20000, P_C = 50000\).
Now, calculate the interest for the second year. The principal for the second year is the initial principal plus the first year's interest.

Scheme A: Principal for 2nd year = \( 30000 + (30000 \times 0.10) = 30000 + 3000 = 33000 \). Interest for 2nd year = \( 33000 \times 0.10 = 3300 \).
Scheme B: Principal for 2nd year = \( 20000 + (20000 \times 0.08) = 20000 + 1600 = 21600 \). Interest for 2nd year = \( 21600 \times 0.08 = 1728 \).
Scheme C: Principal for 2nd year = \( 50000 + (50000 \times 0.12) = 50000 + 6000 = 56000 \). Interest for 2nd year = \( 56000 \times 0.12 = 6720 \).
Total interest earned in the second year = \( 3300 + 1728 + 6720 = 11748 \).
Step 4: Final Answer:
The total interest earned for the second year is 11748 rupees.