To determine the interest rate Sarah needs to achieve her investment goal, we can use the compound interest formula:

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

Where:

• $A$ is the amount of money accumulated after interest (future value),
• $P$ is the principal amount (initial investment),
• $r$ is the annual interest rate (as a decimal),
• $n$ is the number of times interest is compounded per year,
• $t$ is the time the money is invested for in years.

Given:

• $A = 121,000$,
• $P = 79,000$,
• $n = 4$ (since interest is compounded quarterly),
• $t = 7$.

We need to find the annual interest rate $r$.

Step 1: Substitute the known values into the formula

$121,000 = 79,000 \left(1 + \frac{r}{4}\right)^{4 \times 7}$

Simplifying:

$121,000 = 79,000 \left(1 + \frac{r}{4}\right)^{28}$

Step 2: Divide both sides by 79,000

$\frac{121,000}{79,000} = \left(1 + \frac{r}{4}\right)^{28}$

$1.53165 = \left(1 + \frac{r}{4}\right)^{28}$

Step 3: Take the 28th root of both sides

$(1.53165)^{\frac{1}{28}} = 1 + \frac{r}{4}$

$1.0161 = 1 + \frac{r}{4}$

Step 4: Solve for $r$

$\frac{r}{4} = 1.0161 - 1$

$\frac{r}{4} = 0.0161$

$r = 0.0161 \times 4 = 0.0644$

Step 5: Convert $r$ to a percentage

$r = 0.0644 \times 100 = 6.44\%$

So, the required interest rate is 6.4% (rounded to the nearest tenth of a percent).

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Would you like further clarification on this calculation? Or do you have any other related questions?

Related Questions:

1. How would the result change if the interest were compounded monthly instead of quarterly?
2. What would happen if Sarah invested for 10 years instead of 7 years?
3. How do you calculate the compound interest earned after a certain period?
4. How does the number of compounding periods per year affect the interest rate needed?
5. What if Sarah wanted to earn $130,000 instead of $121,000 in the same time frame?

Tip:

When solving compound interest problems, be sure to convert the annual interest rate to a decimal before plugging it into the formula. Also, ensure the number of compounding periods matches the given information (quarterly, monthly, etc.).