How long will it take $12,000 to grow to $19,000 if the investment earns interest at the rate of 5%/year compounded monthly? (Round your answer to two decimal places.)

yr

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

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

Where:
- $ A $ is the amount of money accumulated after $ n $ years, including interest.
- $ P $ is the principal amount (the initial sum of money).
- $ r $ is the annual interest rate (in decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the number of years the money is invested or borrowed for.

Given:
- $ A = 19000 $ (the future value of the investment)
- $ P = 12000 $ (the initial investment)
- $ r = 0.05 $ (the annual interest rate as a decimal)
- $ n = 12 $ (compounded monthly, so 12 times a year)

We need to find $ t $, the time in years.

Plugging the values into the compound interest formula:

\[
19000 = 12000 \left(1 + \frac{0.05}{12}\right)^{12t}
\]

First, let's simplify the expression inside the parentheses:

\[
1 + \frac{0.05}{12} = 1 + 0.00416667 \approx 1.00416667
\]

Now the equation looks like:

\[
19000 = 12000 \times (1.00416667)^{12t}
\]

To isolate $ t $, we divide both sides by 12000:

\[
\frac{19000}{12000} = (1.00416667)^{12t}
\]

\[
1.5833 = (1.00416667)^{12t}
\]

To solve for $ t $, we take the natural logarithm of both sides:

\[
\ln(1.5833) = \ln\left((1.00416667)^{12t}\right)
\]

\[
\ln(1.5833) = 12t \cdot \ln(1.00416667)
\]

Now, solve for $ t $:

\[
t = \frac{\ln(1.5833)}{12 \cdot \ln(1.00416667)}
\]

Let's calculate the values:

- $ \ln(1.5833) \approx 0.4595 $
- $ \ln(1.00416667) \approx 0.004158 $

Substitute these into the equation:

\[
t = \frac{0.4595}{12 \times 0.004158}
\]

\[
t \approx \frac{0.4595}{0.049896}
\]

\[
t \approx 9.21 \text{ years}
\]

So, it will take approximately **9.21 years** for the $12,000 investment to grow to $19,000 at an interest rate of 5% per year compounded monthly.

Would you like more details on the calculation or have any other questions? Here are some related questions you might find interesting:

1. What would the time be if the interest rate was 4% compounded annually?
2. How much would the investment grow to after 15 years at the same interest rate?
3. How would the time change if the interest was compounded daily instead of monthly?
4. What is the formula for continuous compounding, and how does it differ from monthly compounding?
5. How do different compounding frequencies (e.g., quarterly vs. monthly) affect the total amount of interest earned?

**Tip:** When dealing with compound interest problems, remember that the frequency of compounding significantly affects the amount of interest accumulated. The more frequently interest is compounded, the more interest is earned.

Learn how long it takes for $12,000 to grow to $19,000 with a 5% interest rate compounded monthly. Step-by-step solution using the compound interest formula. Suitable for advanced high school students and anyone interested in financial mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation