Divide. Write your answer as a fraction.
$5 \frac{2}{5} \div 7$
To compute the division \(5 \frac{2}{5} \div 7\), we first convert the mixed number to an improper fraction. A mixed number \(a \frac{b}{c}\) is calculated as \(\frac{a \times c + b}{c}\). Here, \(a = 5\), \(b = 2\), and \(c = 5\). Thus, \(5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5}\).
Next, we set up the division operation with the improper fraction: \(\frac{27}{5} \div 7\). Dividing by a number is equivalent to multiplying by its reciprocal. The reciprocal of \(7\) is \(\frac{1}{7}\). Therefore, we perform the multiplication \(\frac{27}{5} \times \frac{1}{7}\).
Multiply the numerators and the denominators: \(\frac{27 \times 1}{5 \times 7} = \frac{27}{35}\). Checking for common factors, we find that the prime factors of 27 are \(3 \times 3 \times 3\), and the prime factors of 35 are \(5 \times 7\). Since there are no common factors, the fraction \(\frac{27}{35}\) is already in its simplest form.
Dividing fractions is a common task in various real-life scenarios, such as cooking or construction. For instance, if you have a recipe that calls for \(5 \frac{2}{5}\) cups of flour, and you want to scale the recipe down by dividing the ingredients into 7 equal portions, you would need to calculate \(\frac{27}{5} \div 7 = \frac{27}{35}\) cups for each portion. Mastering this skill allows you to accurately adjust measurements for projects, recipes, or resource allocation.