Justin contributes $208 each month to a savings account that earns 5% annual interest. Calculate his annuity savings over the course of 25 years if the interest is compounded monthly.
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A.
$9.927.23
B.
$65,520.00
C.
$62,660.00
D.
$123,866.02
First, we need to calculate the monthly interest rate:
r = 5% / 12 = 0.00417
Next, we need to calculate the number of compounding periods:
n = 12 x 25 = 300
Using the annuity formula, we can calculate the total savings:
S = 208 * ((1 + 0.00417)^300 - 1) / (0.00417)
S ≈ $123,866.02
Therefore, the answer is (D) $123,866.02.
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To calculate Justin's annuity savings over 25 years with 5% annual interest compounded monthly, we can use the formula for the future value of an annuity:
S = P * ((1 + r/n)^(nt) - 1) / (r/n)
Where:
S = annuity savings
P = monthly contribution
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
Using the given values:
P = $208 per month
r = 5% or 0.05 (as a decimal)
n = 12 (since interest is compounded monthly)
t = 25 years
Substituting these values into the formula:
S = 208 * ((1 + 0.05/12)^(12*25) - 1) / (0.05/12)
Simplifying this calculation:
S ≈ 208 * (1.00416666667^(300) - 1) / (0.00416666667)
S ≈ 208 * (2.20804065419 - 1) / 0.00416666667
S ≈ 208 * (1.20804065419) / 0.00416666667
S ≈ 24977.1763 / 0.00416666667
S ≈ $5,994,044.08
So, Justin's annuity savings over 25 years would be approximately $5,994,044.08.
None of the given answer choices match this calculation. Please double-check the provided choices or the given information.
To calculate Justin's annuity savings over 25 years with monthly compounding interest, we can use the formula for the future value of an ordinary annuity:
S = P * ((1 + r/n)^(nt) - 1) / (r/n)
Where:
S is the future value of the annuity
P is the monthly contribution amount ($208)
r is the annual interest rate (5% or 0.05)
n is the number of compounding periods per year (12, since it's compounded monthly)
t is the number of years (25)
Substituting the values into the formula:
S = 208 * ((1 + 0.05/12)^(12*25) - 1) / (0.05/12)
Simplifying the equation:
S = 208 * ((1 + 0.00416667)^(300) - 1) / 0.00416667
Using a calculator or spreadsheet software to calculate the expression inside the parentheses (1 + 0.00416667)^(300), we get approximately 12.57827129. Plugging this value back into the formula:
S = 208 * (12.57827129 - 1) / 0.00416667
Simplifying further:
S = 208 * 11.57827129 / 0.00416667
S = 2245.14348287
Hence, Justin will have approximately $2,245.14 in his savings account after 25 years. None of the answer choices provided exactly match this amount, so none of the options (A), (B), (C), or (D) are correct.