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 of
Jacob’s account?

At the beginning of year 4 ---> at the end of year 3

Pick Year 6 as your "focal point" on a time graph.
Amount = 25000(1.05)^12 + 40000(1.05)^6
= 98500.23

To find the balance of Jacob's account at the end of six years, we need to calculate the interest for each of the deposits and add them up.

Let's break down the problem step-by-step:

1. Calculate the interest on the initial deposit of $25,000 for six years compounded semiannually at a 10% interest rate.

First, we need to determine the number of compounding periods. Since the interest is compounded semiannually, and there are six years, we have a total of 12 compounding periods (6 years multiplied by 2 compounding periods per year).

Next, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:
- A is the final amount.
- P is the principal amount (initial deposit).
- r is the annual interest rate (as a decimal).
- n is the number of times that interest is compounded per year.
- t is the number of years.

For the initial deposit:
P = $25,000
r = 10% = 0.10 (as a decimal)
n = 2 (semiannual compounding)
t = 6

A = $25,000(1 + 0.10/2)^(2*6)
A = $25,000(1 + 0.05)^12
A = $25,000(1.05)^12
A ≈ $44,133.35

2. Calculate the interest on the additional deposit of $40,000 made at the beginning of Year 4.

Since this deposit is made at the beginning of Year 4 (after three years), we need to calculate the interest for three years and then add it to the balance.

Using the same compound interest formula:

For the additional deposit:
P = $40,000
r = 10% = 0.10 (as a decimal)
n = 2 (semiannual compounding)
t = 3

A = $40,000(1 + 0.10/2)^(2*3)
A = $40,000(1 + 0.05)^6
A = $40,000(1.05)^6
A ≈ $53,651.03

3. Calculate the final balance of Jacob's account by adding the two amounts calculated in steps 1 and 2.

Final balance = $44,133.35 + $53,651.03
Final balance ≈ $97,784.38

Therefore, the balance of Jacob's account at the end of six years is approximately $97,784.38.