Jacob Fonda deposited $25,000 in a savings account at 10% interest compounded semiannually. At the beginning of Year 4, Jacob deposits an additional $40,000 at 10% interest compounded semiannually. At the end of six years, what is the balance in Jacob’s account?
To find the balance in Jacob's account at the end of six years, we can break down the problem into two parts: the first four years and the remaining two years.
1. First, let's calculate the balance at the end of the first four years. Jacob initially deposited $25,000 in the savings account at 10% interest compounded semiannually. This means that interest is accrued twice a year.
To find the balance after four years, we need to calculate the value of the initial deposit plus the interest earned over the four-year period. The formula to calculate compound interest is:
A = P(1 + r/n)^(n*t)
where:
A = the final amount
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case, P = $25,000, r = 10% or 0.10, n = 2 (compounded semiannually), and t = 4.
Plugging in the values, we have:
A = $25,000 * (1 + 0.10/2)^(2*4)
Simplifying the exponents:
A = $25,000 * (1 + 0.05)^8
Calculating the value inside the parentheses:
A = $25,000 * (1.05)^8
Using a calculator, we find:
A ≈ $38,698.50 (rounded to the nearest cent)
2. Now, let's calculate the balance at the end of the remaining two years. At the beginning of year 4, Jacob deposits an additional $40,000 at 10% interest compounded semiannually.
Using the same formula as before with the new principal amount, we have:
P = $38,698.50 (the balance at the end of four years) + $40,000 (additional deposit)
A = $38,698.50 + $40,000 * (1 + 0.10/2)^(2*2)
Simplifying:
A = $38,698.50 + $40,000 * (1.05)^4
Using a calculator, we find:
A ≈ $95,998.58 (rounded to the nearest cent)
Therefore, at the end of six years, the balance in Jacob's account is approximately $95,998.58.