Final answer:
To find the common present value of the two annuities, calculate the present values of both Al's and Sal's annuities. The present value of Al's annuity can be found using the formula for the sum of a geometric series. The number of years in Sal's annuity can be calculated by solving for n in the formula for the future value of an annuity.
Step-by-step explanation:
To find the common present value of the two annuities, we need to calculate the present values of both Al's annuity and Sal's annuity at an interest rate of 5%. First, let's calculate the present value of Al's annuity. The payments decrease by 4% annually, so the present value can be calculated as follows:
Present Value of Al's annuity = $100 + $100*(1 - 0.04) + $100*(1 - 0.04)^2 + ... + $100*(1 - 0.04)^14
This is a geometric series, and we can use the formula for the sum of a geometric series to calculate the present value. The formula is as follows:
Sum = a * (1 - r^n) / (1 - r)
Where a is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we get:
Present Value of Al's annuity = $100 * (1 - 0.04^15) / (1 - 0.04)
Next, let's find the value of n, the number of years in Sal's annuity. We know that the accumulated value at the end of n years of Sal's annuity is $1.626,29 when calculated using an interest rate of 5%. This means that the total future amount is $1.626,29, and we need to find n. We can use the formula for the future value of an annuity to solve for n:
Future Value = (Payment / Interest Rate) * ((1 + Interest Rate)^n - 1)
Plugging in the values, we get:
$1.626,29 = ($100 / 0.05) * ((1 + 0.05)^n - 1)
Solving for n using logarithms or trial and error, we find that n is approximately 15.75 (or 16 years rounded up).
Therefore, the common present value of the two annuities is the present value of Al's annuity, which is $100 * (1 - 0.04^15) / (1 - 0.04), and the number of years in Sal's annuity is approximately 16.