Evaluate the following expression

Explanation

We will perform any operations within parentheses first. This is the first step of the order of operations rule.

We will begin by performing the subtraction within the first set of parentheses.  The second set of parentheses does not contain an operation to perform.

$\left(\frac78 - \frac14\right) ÷ \left(-2\frac{3}{16}\right)$

• Within the first set of parentheses, prepare to subtract the fractions:  Their LCD is 8, Build \(\frac14\) so that its denominator is 8.
  • = \(\left(\frac78 - \frac14 \cdot \frac22 \right) ÷ \left(-2\frac{3}{16} \right)\)
• Multiply the numerators: 1 • 2 = 2.  Multiply the denominators: 4 • 2 = 8.
  • = \(\left( \frac78 - \frac28 \right) ÷ \left( -2\frac{3}{16}\right)\)
• Subtract the numerators: 7 -2 = 5.  Write the difference over the common denominator 8.
  • = \(\frac58 ÷ \left(-2\frac{3}{16} \right) \)
• Write the mixed number as an improper fraction.
  • = \( \frac58 ÷ \left( -\frac{35}{16}\right)\)
• Use the rule for division of fractions:  Multiply the first fraction by the reciprocal of \(-\frac{35}{16}\)
  • = \(\frac58 \left( -\frac{16}{35}\right)\)
• Multiply the numerators and multiply the denominators.  The product of two fractions with unlike signs is negative.
  • = \(- \frac{5 \cdot 16}{8 \cdot 35}\)
• To simplify, factor 16 as 2 • 8 and factor 35 as 5 • 6.  Remove the common factors of 5 and 8 from the numerator and denominator.
  • = \(\require{cancel} - \frac{\cancel{5}^1 \cdot 2 \cdot \cancel{8}^1}{\cancel{8}_1 \cdot \cancel{5}_1 \cdot 7}\)
• Multiply the remaining factors in the numerator.  Multiply the remaining factors in the denominator.
  • = \(- \frac27\)