Evaluate the following expression

$7\frac{11}{12}$

Explanation

We will translate the words of the problem to numbers and symbols.  Then we will use the order of operations rule to evaluate the resulting expression.  Since the expression involves two operations, addition and subtraction, we need to perform them in the proper order.

The key word difference indicates subtraction.  Since we are to add \(7\frac13\) to the difference, the difference should be written first within parentheses, followed by the addition

• Translate from words to numbers and mathematical symbols.
  • \(\left(\frac56 - \frac14\right) + 7\frac13 \)
• Prepare to subtract the fractions within the parentheses.  Build the fractions so that their denominators are LCD 12. 
  • \(\left(\frac56 - \frac14\right) + 7\frac13 = \left( \frac56 \cdot \frac22 - \frac14 \cdot \frac33 \right) + 7\frac13 \)
• Multiply the numerators.  Multiply the denominators
  • = \( \left( \frac{10}{12} - \frac{3}{12}\right) + 7\frac13 \)
• Subtract the numerators: 10 -3 = 7.  Write the difference over the common denominator 12.
  • = \(\frac{7}{12} + 7\frac13\)
• Prepare to add the fractions.  Build \(\frac13\) so that its denominator is 12: \(\frac13 \cdot \frac44 = \frac{4}{12}\)
  • = \(\frac{7}{12} + 7\frac{4}{12}\)
• Add the numerators of the fractions: 7 + 4 = 11.  Write the sum over the common denominator 12.
  • = \(7\frac{11}{12}\)