Solve Andrea's Running Equation: How Many 3 km Runs Equal 56 km?

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation using the weighted average formula.
• Step 2: Solve the equation to find the number of 3 km runs.

Now, let's work through each step:
Step 1: The average distance is given by $5.92 = \frac{\text{Total Distance}}{\text{Total Runs}}$.
Let $x$ be the number of times Andrea runs 3 km. The total number of runs is $7 + x$, because she runs 8 km 7 times.
The total distance she runs is $8 \times 7 + 3 \times x = 56 + 3x$.
Using the weighted average formula, we have:
$\frac{56 + 3x}{7 + x} = 5.92$

Step 2: Solve the equation for $x$:
Multiply both sides by $7 + x$ to clear the fraction:
$56 + 3x = 5.92(7 + x)$
Expand the right side:
$56 + 3x = 41.44 + 5.92x$
Rearranging gives:
$56 - 41.44 = 5.92x - 3x$
$14.56 = 2.92x$
Divide both sides by 2.92 to solve for $x$:
$x \approx \frac{14.56}{2.92} \approx 5$

Therefore, Andrea runs a distance of $3$ km $5$ times.