To simplify the expression \( \frac{2}{3} + \frac{1}{4} \times (1 \frac{1}{2} + \frac{1}{2}) \), we will follow these steps:
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Convert Mixed Number to Improper Fraction: \[ 1 \frac{1}{2} = \frac{3}{2} \]
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Add the Fractions Inside the Parentheses: \[ 1 \frac{1}{2} + \frac{1}{2} = \frac{3}{2} + \frac{1}{2} = \frac{3+1}{2} = \frac{4}{2} = 2 \]
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Substitute Back Into the Expression: \[ \frac{2}{3} + \frac{1}{4} \times 2 \]
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Multiply \(\frac{1}{4}\) by 2: \[ \frac{1}{4} \times 2 = \frac{2}{4} = \frac{1}{2} \]
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Now add \(\frac{2}{3} + \frac{1}{2}\): To do this, we need a common denominator. The least common multiple of 3 and 2 is 6.
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Convert \(\frac{2}{3}\): \[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \]
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Convert \(\frac{1}{2}\): \[ \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \]
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Add the Two Fractions: \[ \frac{4}{6} + \frac{3}{6} = \frac{4 + 3}{6} = \frac{7}{6} \]
Thus, the simplified expression is: \[ \boxed{\frac{7}{6}} \]