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!