Divide. Write your answer as a fraction or mixed number.
$15 \div 2 \frac{6}{7}$
The problem asks us to calculate the value of the division expression \(15 \div 2\frac{6}{7}\). To solve this, we first need to handle the mixed number \(2\frac{6}{7}\) by converting it into an improper fraction. A mixed number of the form \(a\frac{b}{c}\) is equivalent to \(\frac{a \times c + b}{c}\). Applying this to our divisor:
\[2\frac{6}{7} = \frac{2 \times 7 + 6}{7} = \frac{14 + 6}{7} = \frac{20}{7}.\]
Thus, the expression becomes \(15 \div \frac{20}{7}\).
To divide by a fraction, we multiply by its reciprocal. The reciprocal of \(\frac{20}{7}\) is \(\frac{7}{20}\). Therefore, our expression becomes:
\[15 \div \frac{20}{7} = 15 \times \frac{7}{20}.\]
Multiplying the numerator by 15, we get \(\frac{15 \times 7}{20}\). We can simplify this fraction by noting that 15 and 20 both have a common factor of 5:
\[\frac{15 \times 7}{20} = \frac{(3 \times 5) \times 7}{4 \times 5} = \frac{3 \times 7}{4} = \frac{21}{4}.\]
Finally, we convert the improper fraction \(\frac{21}{4}\) back into a mixed number. Dividing 21 by 4 gives a quotient of 5 and a remainder of 1. Thus:
\[\frac{21}{4} = 5\frac{1}{4}.\]
The final answer for the expression \(15 \div 2\frac{6}{7}\) is \(5\frac{1}{4}\).
Understanding how to divide by fractions is essential for tasks like adjusting recipe portions. For example, if a recipe requires 15 cups of flour and you need to scale it by a factor involving a mixed number (like dividing the total amount by a specific bundle size), you use this method to determine exactly how many units are needed. This ensures accurate measurements, preventing waste and ensuring consistency in your cooking.