Answer :
To determine the exact value of [tex]\( i \)[/tex] in the context of a present value ordinary annuity formula where the account is increasing at a rate of [tex]\( 3.3\% \)[/tex] compounded semiannually, we need to convert the given annual percentage rate to a corresponding semiannual rate.
### Step-by-Step Solution:
1. Understanding the Annual Rate:
- The given annual interest rate is [tex]\( 3.3\% \)[/tex].
2. Conversion to a Semiannual Rate:
- Since the interest is compounded semiannually, we need to find the effective rate per semiannual period.
- There are two semiannual periods in a year, so we divide the annual rate by 2 to find the semiannual rate.
3. Calculation:
- The annual interest rate [tex]\( 3.3\% \)[/tex] can be expressed as a decimal: [tex]\( 3.3\% = 0.033 \)[/tex].
- The semiannual interest rate [tex]\( i \)[/tex] is obtained by dividing this annual rate by 2:
[tex]\[ i = \frac{0.033}{2} = 0.0165 \][/tex]
4. Final Answer:
- The value of [tex]\( i \)[/tex], which represents the semiannual interest rate in decimal form, is [tex]\( 0.0165 \)[/tex].
Thus, the exact value of [tex]\( i \)[/tex] for the present value ordinary annuity formula is [tex]\( 0.0165 \)[/tex], which corresponds to the processed answer from the original compound interest adjustments.
Answer explanation for the provided choices:
- (a) [tex]\( 3.3 \)[/tex]: Incorrect, as this is the annual rate in percentage.
- (b) [tex]\( \frac{0.033}{100} \)[/tex]: Incorrect, this would convert [tex]\( 0.033 \)[/tex] to a percentage form ([tex]\( 0.00033\)[/tex]).
- (c) [tex]\( \frac{0.033}{2} \)[/tex]: Correct, as it simplifies to [tex]\( 0.0165 \)[/tex].
- (d) [tex]\( \frac{0.33}{2} \)[/tex]: Incorrect, as this yields [tex]\( 0.165 \)[/tex], a much higher rate.
Therefore, the correct choice is:
[tex]\( c. \frac{0.033}{2} \)[/tex]
### Step-by-Step Solution:
1. Understanding the Annual Rate:
- The given annual interest rate is [tex]\( 3.3\% \)[/tex].
2. Conversion to a Semiannual Rate:
- Since the interest is compounded semiannually, we need to find the effective rate per semiannual period.
- There are two semiannual periods in a year, so we divide the annual rate by 2 to find the semiannual rate.
3. Calculation:
- The annual interest rate [tex]\( 3.3\% \)[/tex] can be expressed as a decimal: [tex]\( 3.3\% = 0.033 \)[/tex].
- The semiannual interest rate [tex]\( i \)[/tex] is obtained by dividing this annual rate by 2:
[tex]\[ i = \frac{0.033}{2} = 0.0165 \][/tex]
4. Final Answer:
- The value of [tex]\( i \)[/tex], which represents the semiannual interest rate in decimal form, is [tex]\( 0.0165 \)[/tex].
Thus, the exact value of [tex]\( i \)[/tex] for the present value ordinary annuity formula is [tex]\( 0.0165 \)[/tex], which corresponds to the processed answer from the original compound interest adjustments.
Answer explanation for the provided choices:
- (a) [tex]\( 3.3 \)[/tex]: Incorrect, as this is the annual rate in percentage.
- (b) [tex]\( \frac{0.033}{100} \)[/tex]: Incorrect, this would convert [tex]\( 0.033 \)[/tex] to a percentage form ([tex]\( 0.00033\)[/tex]).
- (c) [tex]\( \frac{0.033}{2} \)[/tex]: Correct, as it simplifies to [tex]\( 0.0165 \)[/tex].
- (d) [tex]\( \frac{0.33}{2} \)[/tex]: Incorrect, as this yields [tex]\( 0.165 \)[/tex], a much higher rate.
Therefore, the correct choice is:
[tex]\( c. \frac{0.033}{2} \)[/tex]
