Questions: Given the rate per compounding period, find r, the annual rate. 4.95% per half-year r= % (Round to three decimal places as needed.)

Solution Steps

To find the annual rate given the rate per compounding period, we need to use the formula for compound interest. Since the rate is given per half-year, we need to compound it twice to get the annual rate. The formula to convert the rate per period to the annual rate is:

\[ (1 + \text{rate per period})^n - 1 \]

where \( n \) is the number of compounding periods per year. In this case, \( n = 2 \) because the rate is given per half-year.

We are given the rate per compounding period, which is \( 4.95\% \) per half-year. We need to find the annual rate \( r \).

First, we convert the percentage to a decimal for calculations: \[ \text{rate per half-year} = \frac{4.95}{100} = 0.0495 \]

Since the rate is given for half a year, the number of compounding periods per year \( n \) is: \[ n = 2 \]

Using the formula for the annual rate: \[ r = (1 + \text{rate per period})^n - 1 \] we substitute the values: \[ r = (1 + 0.0495)^2 - 1 \] Calculating this gives: \[ r \approx 0.1014502500000003 \]

To express the annual rate as a percentage, we multiply by 100: \[ r \approx 0.1014502500000003 \times 100 \approx 10.145 \]

Final Answer