A person invests 9500 dollars in a bank. The bank pays 4.5% interest compounded quarterly. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 22100 dollars?


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

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

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

Given:
- $ A = 22100 $ dollars
- $ P = 9500 $ dollars
- $ r = 4.5\% = 0.045 $
- $ n = 4 $ (since interest is compounded quarterly)

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

### Step 1: Plug values into the formula

\[
22100 = 9500 \left(1 + \frac{0.045}{4}\right)^{4t}
\]

### Step 2: Simplify the equation

\[
22100 = 9500 \left(1 + 0.01125\right)^{4t}
\]

\[
22100 = 9500 \left(1.01125\right)^{4t}
\]

### Step 3: Divide both sides by 9500

\[
\frac{22100}{9500} = \left(1.01125\right)^{4t}
\]

\[
2.3263 = \left(1.01125\right)^{4t}
\]

### Step 4: Take the natural logarithm (ln) of both sides

\[
\ln(2.3263) = \ln\left(1.01125^{4t}\right)
\]

\[
\ln(2.3263) = 4t \cdot \ln(1.01125)
\]

### Step 5: Solve for $ t $

\[
t = \frac{\ln(2.3263)}{4 \cdot \ln(1.01125)}
\]

Using a calculator:

\[
t \approx \frac{0.8434}{4 \cdot 0.01119}
\]

\[
t \approx \frac{0.8434}{0.04476}
\]

\[
t \approx 18.8 \text{ years}
\]

### Final Answer:
The person must leave the money in the bank for approximately **18.8 years** to reach $22,100.

Would you like further clarification on any part of this solution?

Here are 5 related questions for more practice:
1. What happens if the interest rate changes to 5%? How would that affect the time required?
2. What would the final amount be if the person only leaves the money in the bank for 10 years?
3. How does the number of compounding periods per year (e.g., monthly vs. quarterly) affect the result?
4. How long would it take for the investment to double at the same interest rate?
5. What is the continuous compound interest formula, and how would it apply here?

**Tip**: When dealing with compound interest problems, always check if the interest is compounded annually, semi-annually, quarterly, or monthly, as this significantly affects the results.

A person invests 9500 dollars in a bank. The bank pays 4.5% interest compounded quarterly. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 22100 dollars?

Learn how to calculate the time needed to grow $9500 to $22100 at a 4.5% interest rate, compounded quarterly. This step-by-step guide covers the compound interest formula and logarithmic methods, making it suitable for students in Grades 9-12.

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