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Question

Suppose that 

$3000

 is placed in a savings account at an annual rate of 

4%

, compounded quarterly. Assuming that no withdrawals are made, how long will it take for the account to grow to 

$4335

?

Do not round any intermediate computations, and round your answer to the nearest hundredth.

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 the account.
- $ P $ is the initial principal (starting amount).
- $ r $ is the annual interest rate (in decimal form).
- $ n $ is the number of times the interest is compounded per year.
- $ t $ is the time in years.
- $ A = 4335 $, $ P = 3000 $, $ r = 0.04 $ (since 4% = 0.04), and $ n = 4 $ (quarterly compounding).

We are solving for $ t $, the time it will take for the account to grow to $ 4335 $.

### Step 1: Plug the known values into the formula.

\[
4335 = 3000 \left(1 + \frac{0.04}{4}\right)^{4t}
\]

Simplify:

\[
4335 = 3000 \left(1 + 0.01\right)^{4t}
\]

\[
4335 = 3000 (1.01)^{4t}
\]

### Step 2: Divide both sides by 3000 to isolate the exponential expression.

\[
\frac{4335}{3000} = (1.01)^{4t}
\]

\[
1.445 = (1.01)^{4t}
\]

### Step 3: Take the natural logarithm (ln) of both sides to eliminate the exponent.

\[
\ln(1.445) = \ln\left((1.01)^{4t}\right)
\]

\[
\ln(1.445) = 4t \cdot \ln(1.01)
\]

### Step 4: Solve for $ t $.

First, calculate the logarithms:

\[
\ln(1.445) \approx 0.367
\]
\[
\ln(1.01) \approx 0.00995
\]

Now, plug these values into the equation:

\[
0.367 = 4t \cdot 0.00995
\]

\[
t = \frac{0.367}{4 \cdot 0.00995}
\]

\[
t \approx \frac{0.367}{0.0398}
\]

\[
t \approx 9.22 \text{ years}
\]

So, it will take approximately **9.22 years** for the account to grow to $4335.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the frequency of compounding affect the time needed for growth?
2. What happens if the interest rate is compounded monthly instead of quarterly?
3. Can you solve the same problem using a different interest formula?
4. How would a change in the principal amount affect the time to reach $4335?
5. How does continuous compounding compare to quarterly compounding in this scenario?

**Tip:** Always convert the interest rate and time into consistent units (e.g., years, months) for accurate compound interest calculations.

Suppose that $3000 is placed in a savings account at an annual rate of 4%, compounded quarterly. Assuming that no withdrawals are made, how long will it take for the account to grow to $4335? Do not round any intermediate computations, and round your answer to the nearest hundredth.

Learn how to solve a compound interest problem where $3000 is placed in a savings account at 4% annual interest, compounded quarterly. Discover how long it takes for the account to grow to $4335 using exponential growth and logarithmic functions. Suitable for Grades 11-12 students.

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