Suppose you make an investment of $3,808 at an interest rate of 4.1% compounded quarterly. How long will it take your investment to reach $4,911? round to two decimals

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

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

where:
- $A$ is the final amount (in this case, $4,911),
- $P$ is the initial principal (in this case, $3,808),
- $r$ is the annual interest rate (in this case, 4.1% or 0.041),
- $n$ is the number of times interest is compounded per year (in this case, quarterly, so $n = 4$),
- $t$ is the time in years.

We need to find $t$, the time required for the investment to reach $4,911.

### Step 1: Plug in the known values:

\[
4911 = 3808 \left(1 + \frac{0.041}{4}\right)^{4t}
\]

### Step 2: Solve for $t$:

First, simplify inside the parentheses:

\[
4911 = 3808 \left(1 + \frac{0.041}{4}\right)^{4t} = 3808 \left(1 + 0.01025\right)^{4t} = 3808 \cdot (1.01025)^{4t}
\]

Now, divide both sides by 3808:

\[
\frac{4911}{3808} = (1.01025)^{4t}
\]

Calculate the left side:

\[
1.2899 \approx (1.01025)^{4t}
\]

### Step 3: Apply logarithms to solve for $t$:

Take the natural logarithm (ln) of both sides:

\[
\ln(1.2899) = \ln((1.01025)^{4t})
\]

Use the logarithm power rule, $\ln(a^b) = b \ln(a)$:

\[
\ln(1.2899) = 4t \cdot \ln(1.01025)
\]

Solve for $t$:

\[
t = \frac{\ln(1.2899)}{4 \ln(1.01025)}
\]

Now, let's compute the values to find $t$.It will take approximately **6.24 years** for your investment to grow from $3,808 to $4,911 at an interest rate of 4.1% compounded quarterly.

Would you like more details on this calculation, or do you have any other questions?

Here are some related questions you might find interesting:
1. How would the time change if the interest rate were different?
2. What if the interest were compounded monthly instead of quarterly?
3. How much will the investment be worth in 10 years?
4. How does continuous compounding affect the investment period?
5. What is the effective annual rate (EAR) for an interest rate of 4.1% compounded quarterly?

**Tip:** Understanding how compounding periods affect investment growth can help you make better financial decisions!

Suppose you make an investment of $3,808 at an interest rate of 4.1% compounded quarterly. How long will it take your investment to reach $4,911? Round to two decimals.

Find out how long it will take an investment of $3,808 to grow to $4,911 at an interest rate of 4.1%, compounded quarterly. This detailed guide uses the compound interest formula and logarithmic properties to solve for time. Ideal for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation