Answer :
To determine the annual interest rate for an account that compounds monthly at a rate of 2.5%, we can use the concept of compound interest. Here’s the step-by-step solution:
1. Understand the Input Data:
- Monthly interest rate: 2.5% (which is 0.025 as a decimal).
- Compounding frequency: Monthly means there are 12 compounding periods in a year.
2. Convert Monthly Rate to Decimal:
- Convert the interest rate from percentage to a decimal by dividing by 100:
[tex]\[ \text{monthly rate} = \frac{2.5}{100} = 0.025 \][/tex]
3. Calculate the Annual Rate:
- The formula to convert a monthly interest rate to an annual rate while accounting for compounding is:
[tex]\[ \text{Annual Rate} = \left(1 + \text{monthly rate}\right)^{12} - 1 \][/tex]
- Plugging in the values:
[tex]\[ \text{Annual Rate} = \left(1 + 0.025\right)^{12} - 1 \][/tex]
4. Convert Annual Rate to Percentage:
- Convert the annual rate back to a percentage by multiplying by 100:
[tex]\[ \text{Annual Rate Percentage} = \left(\left(1 + 0.025\right)^{12} - 1\right) \times 100 \][/tex]
From the given information and calculation, the annual interest rate for an account that compounds monthly at a rate of 2.5% is approximately 34.4889%.
1. Understand the Input Data:
- Monthly interest rate: 2.5% (which is 0.025 as a decimal).
- Compounding frequency: Monthly means there are 12 compounding periods in a year.
2. Convert Monthly Rate to Decimal:
- Convert the interest rate from percentage to a decimal by dividing by 100:
[tex]\[ \text{monthly rate} = \frac{2.5}{100} = 0.025 \][/tex]
3. Calculate the Annual Rate:
- The formula to convert a monthly interest rate to an annual rate while accounting for compounding is:
[tex]\[ \text{Annual Rate} = \left(1 + \text{monthly rate}\right)^{12} - 1 \][/tex]
- Plugging in the values:
[tex]\[ \text{Annual Rate} = \left(1 + 0.025\right)^{12} - 1 \][/tex]
4. Convert Annual Rate to Percentage:
- Convert the annual rate back to a percentage by multiplying by 100:
[tex]\[ \text{Annual Rate Percentage} = \left(\left(1 + 0.025\right)^{12} - 1\right) \times 100 \][/tex]
From the given information and calculation, the annual interest rate for an account that compounds monthly at a rate of 2.5% is approximately 34.4889%.
