To find the annual interest rate $r$ when interest is compounded annually, we can use the compound interest formula:

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

Where:

• $A$ is the final amount in the account,
• $P$ is the initial principal (deposit),
• $r$ is the annual interest rate (as a decimal),
• $n$ is the number of times the interest is compounded per year,
• $t$ is the time (in years).

Since interest is compounded annually, $n = 1$, and we can simplify the formula to:

$A = P(1 + r)^t$

We are given:

• $A = 20,000$,
• $P = 14,000$,
• $t = 8$ years.

Now, plug in the values and solve for $r$:

$20,000 = 14,000 (1 + r)^8$

Step-by-step Solution:

1. Divide both sides by 14,000:

$\frac{20,000}{14,000} = (1 + r)^8$

$1.42857 = (1 + r)^8$

2. Take the 8th root of both sides to solve for $1 + r$:

$(1 + r) = \left( 1.42857 \right)^{\frac{1}{8}}$

3. Subtract 1 from both sides to find $r$:

$r = \left( 1.42857 \right)^{\frac{1}{8}} - 1$

Let me calculate this for the precise value.The approximate annual interest rate is $4.56\%$, rounded to the nearest hundredth.

Would you like more details on any part of the process or another explanation?

Here are 5 related questions to explore:

1. What is the formula for compound interest if compounded quarterly?
2. How does the frequency of compounding affect the interest rate?
3. What would the final amount be with a different initial deposit but the same interest rate?
4. How do you calculate simple interest compared to compound interest?
5. How would the interest rate change if the time period was doubled?

Tip: Always convert percentages back to decimals when using them in equations.