Math Snap

STEP 1

1. The interest is calculated using simple interest.
2. The principal amount is $76,982.
3. The future amount after 3 years is $92,273.
4. We need to find the annual simple interest rate.
5. The time period is 3 years.

STEP 2

1. Understand the formula for simple interest.
2. Identify the known values and the unknown variable.
3. Substitute the known values into the formula.
4. Solve for the unknown variable, which is the interest rate.
5. Convert the interest rate to a percentage.

STEP 3

Understand the formula for simple interest. The formula is:
$$A = P(1 + rt)$$ where:
- $A$ is the future amount ($92,273),
- $P$ is the principal amount ($76,982),
- $r$ is the annual interest rate,
- $t$ is the time in years (3 years).

STEP 4

Identify the known values and the unknown variable.
Known values:
- $A = 92,273$
- $P = 76,982$
- $t = 3$
Unknown variable:
- $r$

STEP 5

Substitute the known values into the formula:
$$92,273 = 76,982(1 + 3r)$$

STEP 6

Solve for the unknown variable, $r$.
First, divide both sides by 76,982:
$$\frac{92,273}{76,982} = 1 + 3r$$ Calculate the left side:
$$1.198 = 1 + 3r$$ Subtract 1 from both sides:
$$0.198 = 3r$$ Divide both sides by 3:
$$r = \frac{0.198}{3}$$ $$r = 0.066$$

Convert the interest rate to a percentage:
$$r = 0.066 \times 100\%$$ $$r = 6.60\%$$ The annual simple interest rate Diane Kruger must earn is:
$\boxed{6.60\%}$