Approximately how long does it take to obtain $9250 when $5000 is invested for x years at a nominal rate of 8% compounded quarterly

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

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

Where:
- $A$ is the final amount ($9250 in this case),
- $P$ is the principal (initial investment, $5000),
- $r$ is the nominal interest rate (8%, or 0.08 as a decimal),
- $n$ is the number of times the interest is compounded per year (quarterly means $n = 4$),
- $t$ is the number of years (what we are solving for).

### Step-by-step Solution:

1. Start by substituting the known values into the compound interest formula:
   \[
   9250 = 5000 \left(1 + \frac{0.08}{4}\right)^{4t}
   \]
   
2. Simplify the terms inside the parentheses:
   \[
   9250 = 5000 \left(1 + 0.02\right)^{4t}
   \]
   \[
   9250 = 5000 \left(1.02\right)^{4t}
   \]

3. Divide both sides by 5000:
   \[
   1.85 = (1.02)^{4t}
   \]

4. Take the natural logarithm (ln) of both sides to solve for $t$:
   \[
   \ln(1.85) = \ln\left((1.02)^{4t}\right)
   \]
   Using the logarithmic property $\ln(a^b) = b \ln(a)$, we get:
   \[
   \ln(1.85) = 4t \cdot \ln(1.02)
   \]

5. Now, solve for $t$:
   \[
   t = \frac{\ln(1.85)}{4 \ln(1.02)}
   \]

Let me calculate this.The time required to obtain $9250 from a $5000 investment at an 8% nominal rate compounded quarterly is approximately **7.77 years**.

Would you like more details or have any questions?

Here are 5 related questions to further explore:
1. How does increasing the interest rate affect the time required to double an investment?
2. What if the interest were compounded monthly instead of quarterly?
3. How does continuous compounding compare to quarterly compounding for the same investment?
4. What is the difference between nominal and effective interest rates?
5. How long would it take to reach $10,000 under the same conditions?

**Tip:** In compound interest problems, increasing the frequency of compounding reduces the time needed to reach a target amount.

Approximately how long does it take to obtain $9250 when $5000 is invested for x years at a nominal rate of 8% compounded quarterly?

Discover how to calculate the time required to grow an investment of $5000 to $9250 at a nominal rate of 8% compounded quarterly. Learn key mathematical concepts like compound interest, logarithms, and exponential growth with a step-by-step solution. Suitable for high school students in Grades 10-12.

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