To simplify the expression \((1 \frac{1}{2} + \frac{5}{6}) - (2 \frac{3}{4} - \frac{8}{9})\), let's first convert the mixed numbers to improper fractions.
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Convert \(1 \frac{1}{2}\) to an improper fraction: \[ 1 \frac{1}{2} = \frac{3}{2} \]
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Convert \(2 \frac{3}{4}\) to an improper fraction: \[ 2 \frac{3}{4} = \frac{11}{4} \]
Now, let's rewrite the expression with these improper fractions: \[ \left( \frac{3}{2} + \frac{5}{6} \right) - \left( \frac{11}{4} - \frac{8}{9} \right) \]
Next, we will simplify each part inside the parentheses.
Part 1: \( \frac{3}{2} + \frac{5}{6} \)
To add these fractions, we need a common denominator. The least common multiple of 2 and 6 is 6.
Convert \( \frac{3}{2} \): \[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]
Now add: \[ \frac{9}{6} + \frac{5}{6} = \frac{9 + 5}{6} = \frac{14}{6} \]
Simplifying \( \frac{14}{6} \): \[ \frac{14}{6} = \frac{7}{3} \]
Part 2: \( \frac{11}{4} - \frac{8}{9} \)
For this subtraction, we need a common denominator. The least common multiple of 4 and 9 is 36.
Convert \( \frac{11}{4} \): \[ \frac{11}{4} = \frac{11 \times 9}{4 \times 9} = \frac{99}{36} \]
Convert \( \frac{8}{9} \): \[ \frac{8}{9} = \frac{8 \times 4}{9 \times 4} = \frac{32}{36} \]
Now subtract: \[ \frac{99}{36} - \frac{32}{36} = \frac{99 - 32}{36} = \frac{67}{36} \]
Now combine the results:
We have: \[ \frac{7}{3} - \frac{67}{36} \]
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 36 is 36.
Convert \( \frac{7}{3} \): \[ \frac{7}{3} = \frac{7 \times 12}{3 \times 12} = \frac{84}{36} \]
Now subtract: \[ \frac{84}{36} - \frac{67}{36} = \frac{84 - 67}{36} = \frac{17}{36} \]
Thus, the simplified expression is: \[ \boxed{\frac{17}{36}} \]