how long will it take $3000 to grow to $14000 if it is invested at 5% compounded monthly

To solve this problem, we can use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:

- $ A $ is the final amount of money after interest,
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times interest is compounded per year,
- $ t $ is the time the money is invested for in years.

Given values:

- $ P = 3000 $ (initial investment),
- $ A = 14000 $ (final amount),
- $ r = 0.05 $ (5% annual interest rate),
- $ n = 12 $ (compounded monthly).

We need to find $ t $, the number of years it will take for the investment to grow from $3000 to $14000.

### Step 1: Plug the values into the compound interest formula.

\[
14000 = 3000 \left(1 + \frac{0.05}{12}\right)^{12t}
\]

### Step 2: Simplify the equation.

\[
14000 = 3000 \left(1 + \frac{0.05}{12}\right)^{12t}
\]

\[
\frac{14000}{3000} = \left(1 + \frac{0.05}{12}\right)^{12t}
\]

\[
\frac{14000}{3000} = \left(1.0041667\right)^{12t}
\]

\[
\frac{14000}{3000} = 4.6667
\]

### Step 3: Take the natural logarithm of both sides to solve for $ t $.

\[
\ln(4.6667) = \ln\left(\left(1.0041667\right)^{12t}\right)
\]

\[
\ln(4.6667) = 12t \cdot \ln(1.0041667)
\]

\[
t = \frac{\ln(4.6667)}{12 \cdot \ln(1.0041667)}
\]

### Step 4: Calculate $ t $.

\[
t = \frac{\ln(4.6667)}{12 \cdot \ln(1.0041667)}
\]

\[
t \approx \frac{1.541}{12 \cdot 0.004158}
\]

\[
t \approx \frac{1.541}{0.049896}
\]

\[
t \approx 30.88
\]

So, it will take approximately **30.88 years** for $3000 to grow to $14000 at a 5% annual interest rate compounded monthly.

Would you like more details or have any questions?

### Related Questions

1. How much would the investment grow to if the interest rate were 6% compounded monthly?
2. What would be the final amount if $3000 is invested at 5% compounded annually for 30.88 years?
3. How long will it take for the investment to double at 5% compounded monthly?
4. What would be the difference in growth if the interest were compounded quarterly instead of monthly?
5. If $5000 were invested instead of $3000, how long would it take to reach $14000 at 5% compounded monthly?

### Tip

Understanding how different compounding frequencies (monthly, quarterly, annually) affect the growth of investments can help you make better financial decisions.

Discover how to calculate the time it takes for a $3000 investment to grow to $14000 at a 5% interest rate compounded monthly. This problem involves compound interest calculations, exponential growth, and logarithms. Suitable for high school students in Grades 10-12.

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