Question

Suppose you deposit $1,005.00 into an account 5.00 years from today that earns 10.00%. It will be worth $1,931.00 _____ years from today.

Answer 1

To find out how many years from today the investment of $1,005.00 will grow to $1,931.00, we can use the future value formula:

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

Where:
- \( FV \) is the future value of the investment,
- \( PV \) is the present value or initial deposit,
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of years the money is invested.

In this problem:
- \( FV = 1,931.00 \)
- \( PV = 1,005.00 \)
- \( r = 10\% = 0.10 \)

We can rearrange the formula to solve for \( n \):

\[
n = \frac{\log(FV / PV)}{\log(1 + r)}
\]

Substituting the values:

\[
n = \frac{\log(1,931.00 / 1,005.00)}{\log(1 + 0.10)}
\]

Calculating the fraction inside the logarithm:

\[
\frac{1,931.00}{1,005.00} \approx 1.920
\]

Now calculating the logs:

\[
n = \frac{\log(1.920)}{\log(1.10)}
\]

Using a calculator to find the logarithms:

- \( \log(1.920) \approx 0.283 \)
- \( \log(1.10) \approx 0.0414 \)

Now we compute \( n \):

\[
n \approx \frac{0.283}{0.0414} \approx 6.83
\]

Thus, the time it takes for the account to grow from $1,005.00 to $1,931.00 is approximately 6.83 years.

Since the deposit is made 5.00 years from today, the total time from today will be:

\[
6.83 + 5 = 11.83 \text{ years}
\]

So the answer is:

It will be worth $1,931.00 in approximately **11.83 years from today**.