Dylan is going to invest $360 and leave it in an account for 5 years. Assuming the interest is compounded quarterly, what interest rate, *to the nearest hundredth of a percent*, would be required in order for Dylan to end up with $420?

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

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

Where:
- $ A $ is the amount of money after interest (in this case, $420),
- $ P $ is the principal amount (initial investment, $360),
- $ r $ is the annual interest rate (which we need to find),
- $ n $ is the number of times interest is compounded per year (quarterly means $ n = 4 $),
- $ t $ is the time the money is invested for in years (5 years).

Let's plug in the known values:

\[
420 = 360 \left(1 + \frac{r}{4}\right)^{4 \times 5}
\]

Simplifying:

\[
420 = 360 \left(1 + \frac{r}{4}\right)^{20}
\]

Now, we solve for $ r $.

1. Divide both sides by 360:

\[
\frac{420}{360} = \left(1 + \frac{r}{4}\right)^{20}
\]

\[
1.1667 = \left(1 + \frac{r}{4}\right)^{20}
\]

2. Take the 20th root of both sides:

\[
\left(1.1667\right)^{\frac{1}{20}} = 1 + \frac{r}{4}
\]

3. Calculate $ 1.1667^{\frac{1}{20}} $:

\[
1.00765 \approx 1 + \frac{r}{4}
\]

4. Subtract 1 from both sides:

\[
0.00765 = \frac{r}{4}
\]

5. Multiply both sides by 4:

\[
r = 0.0306
\]

Convert to a percentage by multiplying by 100:

\[
r \approx 3.06\%
\]

So, the interest rate required is approximately **3.06%**.

Would you like further details or clarification on any step?

### Related Questions:
1. How would the result change if the interest were compounded annually instead of quarterly?
2. What would the interest rate be if Dylan wanted $450 instead of $420 after 5 years?
3. How much interest would be earned if the rate were exactly 3.06%?
4. What would the final amount be if Dylan invested $500 at 3.06% compounded quarterly for 5 years?
5. How long would it take to reach $420 if the interest rate were fixed at 3% compounded quarterly?

### Tip:
The more frequently interest is compounded, the more interest is earned over the same time period!

Dylan is going to invest $360 and leave it in an account for 5 years. Assuming the interest is compounded quarterly, what interest rate, to the nearest hundredth of a percent, would be required in order for Dylan to end up with $420?

This guide explains how to calculate the interest rate required for a $360 investment compounded quarterly to grow to $420 in 5 years. Using the compound interest formula, we show step-by-step how to determine the rate, rounding to the nearest hundredth of a percent. Ideal for high school students and anyone interested in finance math.

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