Let's simplify the expression step by step:
-
Convert mixed numbers to improper fractions:
-
For \(1 \frac{1}{2}\): \[ 1 \frac{1}{2} = \frac{3}{2} \]
-
For \(2 \frac{3}{4}\): \[ 2 \frac{3}{4} = \frac{11}{4} \]
-
-
Now rewrite the expression: \[ \left(\frac{3}{2} + \frac{5}{6}\right) - \left(\frac{11}{4} - 8\right) \]
-
First, simplify \(\frac{3}{2} + \frac{5}{6}\):
- Find a common denominator (which is 6): \[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]
Now add: \[ \frac{9}{6} + \frac{5}{6} = \frac{14}{6} = \frac{7}{3} \]
-
Next, simplify \(\frac{11}{4} - 8\):
- Convert 8 to a fraction with a denominator of 4: \[ 8 = \frac{32}{4} \]
Now subtract: \[ \frac{11}{4} - \frac{32}{4} = \frac{11 - 32}{4} = \frac{-21}{4} \]
-
Now substitute back into the expression: \[ \frac{7}{3} - \left(\frac{-21}{4}\right) = \frac{7}{3} + \frac{21}{4} \]
-
Find a common denominator for \(\frac{7}{3}\) and \(\frac{21}{4}\) (which is 12):
-
Convert \(\frac{7}{3}\): \[ \frac{7}{3} = \frac{7 \times 4}{3 \times 4} = \frac{28}{12} \]
-
Convert \(\frac{21}{4}\): \[ \frac{21}{4} = \frac{21 \times 3}{4 \times 3} = \frac{63}{12} \]
-
-
Add the two fractions: \[ \frac{28}{12} + \frac{63}{12} = \frac{28 + 63}{12} = \frac{91}{12} \]
Thus, the simplified expression is: \[ \frac{91}{12} \]