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Let \( x \) represent the distance that Nicole runs. According to the problem, Nicole's friend runs \( \frac{4}{7} \) of a mile, and this distance is equal to \( \frac{4}{5} \) of Nicole's distance.

So, we can set up the equation:

\[
\frac{4}{5} \cdot x = \frac{4}{7}
\]

To solve for \( x \), divide both sides by \( \frac{4}{5} \), or equivalently, multiply both sides by \( \frac{5}{4} \):

\[
x = \frac{4}{7} \times \frac{5}{4}
\]

Simplify by canceling the 4s:

\[
x = \frac{5}{7}
\]

So, Nicole runs \( \frac{5}{7} \) of a mile.

Would you like a more detailed explanation or have any questions about this solution?

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Here are 5 related questions to expand on this topic:
1. What if Nicole’s friend ran \( \frac{6}{7} \) of a mile and her distance was \( \frac{2}{3} \) of Nicole's distance?
2. How would the solution change if Nicole's friend ran exactly half of Nicole’s distance?
3. What if Nicole and her friend ran a combined distance of 1 mile, with the same ratios?
4. How can you verify the result by substituting back into the original conditions?
5. How would you represent this problem using a proportion instead of an equation?

**Tip:** When solving equations with fractions, multiplying by the reciprocal is a helpful strategy to isolate variables.

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?

Learn how to calculate Nicole's running distance when given that her friend's distance is 4/5 of hers. This problem involves fractions and proportions, ideal for middle school students in Grades 6-8.

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