Divide. Write your answer as a fraction.
$5 \frac{2}{5} \div 7$
The problem asks us to divide the mixed fraction \(5 \frac{2}{5}\) by the integer \(7\). First, we convert the mixed fraction into an improper fraction. A mixed fraction \(a \frac{b}{c}\) is equivalent to the improper fraction \(\frac{a \times c + b}{c}\). Here, \(a=5, b=2,\) and \(c=5\). Thus, \(5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5}\).
Now we need to compute \(\frac{27}{5} \div 7\). Dividing by a number is equivalent to multiplying by its reciprocal. The reciprocal of \(7\) is \(\frac{1}{7}\). So, we perform the calculation:
\[ \frac{27}{5} \div 7 = \frac{27}{5} \times \frac{1}{7} \]
Multiplying the numerators together and the denominators together, we get:
\[ \frac{27 \times 1}{5 \times 7} = \frac{27}{35} \]
Finally, we check if the fraction \(\frac{27}{35}\) can be simplified. We identify the factors of both numbers:
The factors of \(27\) are \(1, 3, 9, 27\).
The factors of \(35\) are \(1, 5, 7, 35\).
Since the only common factor is \(1\), the fraction \(\frac{27}{35}\) is already in its simplest form. The result of the division is \(\boxed{\frac{27}{35}}\).
Dividing fractions is very useful in everyday life, such as when baking. For example, if a recipe calls for \(5 \frac{2}{5}\) cups of flour for a large batch of cookies and you want to split this into 7 smaller, equal portions, you need to divide the amount by 7. Calculating \(\frac{27}{5} \div 7 = \frac{27}{35}\) tells you that each portion will require \(\frac{27}{35}\) cups of flour. This ensures that you distribute the ingredients accurately for consistent results.