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.
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