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To evaluate the expression \( 5 \frac{1}{5} \div \frac{6}{7} \) and write the answer as a mixed number in its simplest form, let's follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \( 5 \frac{1}{5} \) can be written as an improper fraction by multiplying the whole number (5) by the denominator (5) and adding the numerator (1):

\[
5 \frac{1}{5} = \frac{(5 \times 5) + 1}{5} = \frac{25 + 1}{5} = \frac{26}{5}
\]

### Step 2: Perform the division of fractions
Dividing by a fraction is the same as multiplying by its reciprocal. So:

\[
\frac{26}{5} \div \frac{6}{7} = \frac{26}{5} \times \frac{7}{6}
\]

### Step 3: Multiply the fractions
Now, multiply the numerators and the denominators:

\[
\frac{26}{5} \times \frac{7}{6} = \frac{26 \times 7}{5 \times 6} = \frac{182}{30}
\]

### Step 4: Simplify the fraction
To simplify \( \frac{182}{30} \), we find the greatest common divisor (GCD) of 182 and 30. The GCD is 2.

\[
\frac{182}{30} = \frac{182 \div 2}{30 \div 2} = \frac{91}{15}
\]

### Step 5: Convert to a mixed number
To convert \( \frac{91}{15} \) into a mixed number, divide 91 by 15:

\[
91 \div 15 = 6 \text{ remainder } 1
\]

So, \( \frac{91}{15} \) is equivalent to the mixed number \( 6 \frac{1}{15} \).

### Final Answer:
The answer is \( 6 \frac{1}{15} \).

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Would you like more details on any part of the process? Here are some related questions for further exploration:

1. How do you convert improper fractions to mixed numbers?
2. Why is dividing by a fraction the same as multiplying by its reciprocal?
3. How can we find the greatest common divisor (GCD) of two numbers?
4. What are the steps for simplifying complex fractions?
5. How can you check your final answer for accuracy?

**Tip:** Always simplify your fractions at the end of any division or multiplication process to ensure the final answer is in the simplest form.

Evaluate the expression shown below and write your answer as a mixed number in simplest form: 5 1/5 ÷ 6/7.

Learn how to divide the mixed number 5 1/5 by the fraction 6/7 with a step-by-step solution. This problem involves converting mixed numbers to improper fractions, multiplying by the reciprocal, and simplifying. Suitable for students in Grades 5-7.

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