Answer :
Sure, let's go through the detailed, step-by-step solution for Aaron's RRSP (Registered Retirement Savings Plan) value.
Step 1: Identify the given values
- Periodic payment, \( PMT = \$1200 \)
- Annual interest rate, \( r = 2.8\% \)
- Time in years, \( t = 13.5 \) years
- Compounding frequency: semiannually
Step 2: Calculate the semiannual interest rate
Since the interest is compounded semiannually, we divide the annual interest rate by 2:
[tex]\[ i = \frac{2.8\%}{2} = 1.4\% \][/tex]
Convert this percentage to a decimal for calculations:
[tex]\[ i = 0.014 \][/tex]
Step 3: Determine the number of compounding periods
Semiannual contributions over 13.5 years mean there are \( 2 \times 13.5 \) periods:
[tex]\[ n = 2 \times 13.5 = 27 \][/tex]
Step 4: Use the future value formula for an ordinary annuity
The formula for the future value \( FV \) of an ordinary annuity (where deposits are made at the end of each period) is:
[tex]\[ FV = PMT \left( \frac{(1 + i)^n - 1}{i} \right) \][/tex]
Plug in the values we have:
[tex]\[ FV = 1200 \left( \frac{(1 + 0.014)^{27} - 1}{0.014} \right) \][/tex]
Step 5: Calculate the future value
First, compute \( (1 + i)^n \):
[tex]\[ (1 + 0.014)^{27} = (1.014)^{27} \][/tex]
Next, subtract 1 from the result:
[tex]\[ (1.014)^{27} - 1 \][/tex]
Then, divide by the interest rate \( i = 0.014 \):
[tex]\[ \frac{(1.014)^{27} - 1}{0.014} \][/tex]
Finally, multiply by the periodic payment \( PMT = 1200 \):
[tex]\[ FV = 1200 \left( \frac{(1.014)^{27} - 1}{0.014} \right) \][/tex]
Upon performing these calculations, we find:
[tex]\[ FV = \$39046.47 \][/tex]
Conclusion:
The value of Aaron's RRSP after 13.5 years, with semiannual contributions of \$1200 and an interest rate of 2.8% compounded semiannually, is [tex]\( \boxed{39046.47} \)[/tex].
Step 1: Identify the given values
- Periodic payment, \( PMT = \$1200 \)
- Annual interest rate, \( r = 2.8\% \)
- Time in years, \( t = 13.5 \) years
- Compounding frequency: semiannually
Step 2: Calculate the semiannual interest rate
Since the interest is compounded semiannually, we divide the annual interest rate by 2:
[tex]\[ i = \frac{2.8\%}{2} = 1.4\% \][/tex]
Convert this percentage to a decimal for calculations:
[tex]\[ i = 0.014 \][/tex]
Step 3: Determine the number of compounding periods
Semiannual contributions over 13.5 years mean there are \( 2 \times 13.5 \) periods:
[tex]\[ n = 2 \times 13.5 = 27 \][/tex]
Step 4: Use the future value formula for an ordinary annuity
The formula for the future value \( FV \) of an ordinary annuity (where deposits are made at the end of each period) is:
[tex]\[ FV = PMT \left( \frac{(1 + i)^n - 1}{i} \right) \][/tex]
Plug in the values we have:
[tex]\[ FV = 1200 \left( \frac{(1 + 0.014)^{27} - 1}{0.014} \right) \][/tex]
Step 5: Calculate the future value
First, compute \( (1 + i)^n \):
[tex]\[ (1 + 0.014)^{27} = (1.014)^{27} \][/tex]
Next, subtract 1 from the result:
[tex]\[ (1.014)^{27} - 1 \][/tex]
Then, divide by the interest rate \( i = 0.014 \):
[tex]\[ \frac{(1.014)^{27} - 1}{0.014} \][/tex]
Finally, multiply by the periodic payment \( PMT = 1200 \):
[tex]\[ FV = 1200 \left( \frac{(1.014)^{27} - 1}{0.014} \right) \][/tex]
Upon performing these calculations, we find:
[tex]\[ FV = \$39046.47 \][/tex]
Conclusion:
The value of Aaron's RRSP after 13.5 years, with semiannual contributions of \$1200 and an interest rate of 2.8% compounded semiannually, is [tex]\( \boxed{39046.47} \)[/tex].
