We want to find out how long it will take for $39,000 to grow to $230,000 at an annual interest rate of 4.6%.



To find out how long it will take for an initial investment of $39,000 to grow to $230,000 at an annual interest rate of 4.6%, we can use the formula for compound interest:

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

Where:
- $A$ is the final amount ($230,000),
- $P$ is the principal or initial amount ($39,000),
- $r$ is the annual interest rate (4.6% or 0.046),
- $n$ is the number of times the interest is compounded per year,
- $t$ is the time in years.

Assuming the interest is compounded annually ($n = 1$), the formula simplifies to:

\[
A = P (1 + r)^t
\]

We need to solve for $t$, the number of years. Substituting the given values:

\[
230,000 = 39,000 (1 + 0.046)^t
\]

\[
\frac{230,000}{39,000} = (1.046)^t
\]

\[
5.8974 = (1.046)^t
\]

Now, take the natural logarithm (ln) of both sides to solve for $t$:

\[
\ln(5.8974) = t \cdot \ln(1.046)
\]

\[
t = \frac{\ln(5.8974)}{\ln(1.046)}
\]

Let me calculate that for you.It will take approximately 39.46 years for $39,000 to grow to $230,000 at an annual interest rate of 4.6%.

Would you like further details on any part of the calculation or have any questions?

Here are 5 related questions:
1. How does compounding frequency affect the growth of the investment?
2. What is the difference between simple interest and compound interest?
3. How much would the investment grow to in 20 years at the same interest rate?
4. What is the continuous compounding formula, and how would it affect the result?
5. How can inflation impact the real value of this investment over time?

**Tip:** The more frequently interest is compounded, the faster the investment grows!

We want to find out how long it will take for $39,000 to grow to $230,000 at an annual interest rate of 4.6%.

Learn how to calculate the time required for $39,000 to grow to $230,000 with an annual interest rate of 4.6%. This detailed guide explains the use of the compound interest formula and logarithms to solve the problem. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation