Alright, let's tackle the question step-by-step.
### Part (a): Calculating Interest
#### For Bank A:
Bank A offers a yearly compound interest rate of 6%.
The formula for compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Here:
- [tex]\( P \)[/tex] is the principal amount = Rs. 50,000
- [tex]\( r \)[/tex] is the annual interest rate = 6% = 0.06
- [tex]\( t \)[/tex] is the time the money is invested for = 2 years
- [tex]\( n \)[/tex] is the number of times interest applied per time period, indicating 1 for yearly compounding.
So, substituting the values given:
[tex]\[ A_A = 50000 \left(1 + 0.06\right)^2 \][/tex]
[tex]\[ A_A = 50000 \left(1.06\right)^2 \][/tex]
[tex]\[ A_A = 50000 \times 1.1236 \approx 56180 \][/tex]
The interest earned can be calculated as:
[tex]\[ \text{Interest}_A = A_A - P \][/tex]
[tex]\[ \text{Interest}_A = 56180 - 50000 \][/tex]
[tex]\[ \text{Interest}_A = 6180.00 \][/tex]
#### For Bank B:
Bank B offers a half-yearly compound interest rate of 5%.
The formula for compound interest in this case remains the same, but we need to adjust [tex]\( n \)[/tex] to reflect half-yearly compounding.
Here:
- [tex]\( P \)[/tex] is the principal amount = Rs. 50,000
- [tex]\( r \)[/tex] is the annual interest rate = 5% = 0.05
- [tex]\( t \)[/tex] is the time the money is invested for = 2 years
- [tex]\( n \)[/tex] is the number of times interest applied per time period = 2 (since it's half-yearly).
So, substituting the values given:
[tex]\[ A_B = 50000 \left(1 + \frac{0.05}{2}\right)^{2 \times 2} \][/tex]
[tex]\[ A_B = 50000 \left(1 + 0.025\right)^4 \][/tex]
[tex]\[ A_B = 50000 \left(1.025\right)^4 \][/tex]
[tex]\[ A_B = 50000 \times 1.2155 \approx 55190.64 \][/tex]
The interest earned can be calculated as:
[tex]\[ \text{Interest}_B = A_B - P \][/tex]
[tex]\[ \text{Interest}_B = 55190.64 - 50000 \][/tex]
[tex]\[ \text{Interest}_B = 5190.64 \][/tex]
### Part (b): Which Bank to Save Money In and Why?
Now we need to compare the interests earned from both banks:
- Interest from Bank A: Rs 6180.00
- Interest from Bank B: Rs 5190.64
Clearly, the interest earned from Bank A is higher than that from Bank B.
Conclusion:
You should save your money in Bank A because, after 2 years, it will yield an interest of Rs 6180.00, which is higher than the interest earned from Bank B, which is Rs 5190.64. Therefore, choosing Bank A will provide you with better returns on your investment.
