https://file.aiquickdraw.com/mathbot/file/1730193201167aibiq1eh.jpg

To solve this problem, let's define some variables and set up an equation:

Let \( x \) represent the distance that Nicole runs in miles.

According to the problem:
1. Nicole's friend runs \( \frac{4}{7} \) of a mile.
2. Her friend’s distance (\( \frac{4}{7} \) miles) is only \( \frac{4}{5} \) of Nicole's distance.

Thus, we can set up the equation:
\[
\frac{4}{5} \cdot x = \frac{4}{7}
\]

Now, to find \( x \), we'll solve for it by isolating \( x \) on one side of the equation.

### Step 1: Divide both sides by \( \frac{4}{5} \)
Dividing by \( \frac{4}{5} \) is the same as multiplying by \( \frac{5}{4} \):
\[
x = \frac{4}{7} \cdot \frac{5}{4}
\]

### Step 2: Simplify the equation
Multiply the fractions:
\[
x = \frac{4 \times 5}{7 \times 4} = \frac{20}{28}
\]

### Step 3: Simplify \( \frac{20}{28} \) further
\[
x = \frac{5}{7}
\]

### Conclusion
Nicole runs \( \frac{5}{7} \) miles.

### Answer
The correct answer is:
**a. \( \frac{5}{7} \) miles**

Would you like further details or have any questions?

Here are some related questions to expand on this topic:
1. How would the solution change if Nicole's friend ran a different fraction of Nicole's distance?
2. Can you represent the relationship between Nicole's distance and her friend's distance as a proportion?
3. What if Nicole's friend ran an additional half-mile? How would that change Nicole's distance?
4. How would the answer change if Nicole’s friend’s distance were increased by a fixed distance, not as a fraction?
5. Can you solve a similar problem if both distances were unknown but their ratio was given?

**Tip:** When dealing with fractions, try converting multiplication or division into simpler terms by canceling out common factors.

Nicole’s friend runs 4/7 of a mile, but her distance is only 4/5 of Nicole’s distance. How far does Nicole run?

Solve for Nicole's running distance given that her friend's distance is 4/7 of a mile and is only 4/5 of Nicole's distance. This problem involves setting up a proportion and solving with fraction multiplication, ideal for middle school students in Grades 6-8.

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