Provided Information:
Bank A offers a 6% annual interest rate, compounded semi-annually (3% per half-year). Bank C's annual interest rate is \( w \) times that of Bank B.
Raju invests \( P \) in Bank B for \( t \) years. Rupa invests ₹10,000 in Bank C for \( 2t \) years.
Raju's interest from Bank B over \( t \) years equals Bank A's interest over 1 year.
1) Bank A Interest Calculation:
For semi-annual compounding, the amount after 1 year is:
\[ A = P\left(1 + \frac{r}{2}\right)^2 \] where \( r \) is the annual interest rate in decimal form.
Amount after 1 year = \( P \times \left(1 + 0.03\right)^2 = P \times \left(1.03\right)^2 = 1.0609P \)
Bank A's interest for 1 year: \( 1.0609P - P = 0.0609P \)
2) Raju's Interest from Bank B:
Raju's interest from Bank B for \( t \) years is given as 0.0609P. Let Bank B's interest rate be \( R \). The interest is calculated as:
\[ \text{Interest} = P \times R \times t = 0.0609P \]
This simplifies to the equation: \( R \times t = 0.0609 \)
3) Rupa's Interest from Bank C:
Bank C's interest rate is \( wR \). Rupa's interest from Bank C is:
\[ \text{Interest} = 10000 \times wR \times 2t = 20000 \times wR \times t \]
Substituting \( wR \times t = 0.0609 \) from step 2:
\[ \text{Interest} = 20000 \times 0.0609 = ₹1218 \]
Considering Rupa's investment period of \( 2t \) years, the total interest earned is:
\[ 2 \times ₹1218 = ₹2436 \]
Result: The final answer is ₹2436.