Question

Five years from today, you plan to invest $4,900 for 8 additional years at 7.8 percent compounded annually. How much will you have in your account 13 years from today?

Answer 1

To calculate how much you will have in your account 13 years from today, we can break it down into two parts: the initial 5 years where no additional investment occurs, and the subsequent 8 years of investment.

**Step 1: Calculate the value of the investment after the first 5 years.**

Since you are planning to invest $4,900 today for 5 years at 7.8% compounded annually, we can use the future value formula:

\[
FV = P(1 + r)^n
\]

Where:
- \( FV \) is the future value of the investment.
- \( P \) is the principal amount (initial investment), which is $4,900.
- \( r \) is the annual interest rate (in decimal form), which is 0.078.
- \( n \) is the number of years the money is invested, which is 5.

Plug in the values:

\[
FV = 4900(1 + 0.078)^5
\]

Calculating the expression:

\[
FV = 4900(1.078)^5
\]
\[
FV \approx 4900 \times 1.4636 \quad (\text{using a calculator for } (1.078)^5)
\]
\[
FV \approx 7172.64
\]

So after 5 years, you would have approximately **$7,172.64** in the account.

**Step 2: Now, invest that amount for an additional 8 years at the same interest rate.**

We will take the amount \( 7172.64 \) and calculate the future value after an additional 8 years:

\[
FV = P(1 + r)^n
\]

Where now:
- \( P = 7172.64 \)
- \( r = 0.078 \)
- \( n = 8 \)

Plug in the values:

\[
FV = 7172.64(1 + 0.078)^8
\]
\[
FV = 7172.64(1.078)^8
\]
\[
FV \approx 7172.64 \times 1.8509 \quad (\text{using a calculator for } (1.078)^8)
\]
\[
FV \approx 13256.45
\]

So, after 13 years from today, you will have approximately **$13,256.45** in your account.