To solve the expression \(1 + 1 + 4 \times 1 \frac{1}{2} \times 3 \times 1 \frac{3}{4} + 2\), we will follow the order of operations (also known as PEMDAS/BODMAS rules: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
Let's solve it step by step:
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Convert the mixed numbers to improper fractions:
- \(1 \frac{1}{2} = \frac{3}{2}\)
- \(1 \frac{3}{4} = \frac{7}{4}\)
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Now, rewrite the expression: \[ 1 + 1 + 4 \times \frac{3}{2} \times 3 \times \frac{7}{4} + 2 \]
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Calculate \(4 \times \frac{3}{2}\): \[ 4 \times \frac{3}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6 \]
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Now, substitute back into the expression: \[ 1 + 1 + 6 \times 3 \times \frac{7}{4} + 2 \]
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Calculate \(6 \times 3\): \[ 6 \times 3 = 18 \]
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Substitute back: \[ 1 + 1 + 18 \times \frac{7}{4} + 2 \]
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Now calculate \(18 \times \frac{7}{4}\): \[ 18 \times \frac{7}{4} = \frac{18 \times 7}{4} = \frac{126}{4} = 31.5 \]
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Substitute back into the expression: \[ 1 + 1 + 31.5 + 2 \]
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Now, add the numbers together: \[ 1 + 1 = 2 \] \[ 2 + 31.5 = 33.5 \] \[ 33.5 + 2 = 35.5 \]
Thus, the value of the expression is \(\boxed{35.5}\).