To evaluate the expression \( 19 - 3 \div 2 + 20 \), we need to follow the order of operations, which is parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
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First, we perform the division: \[ 3 \div 2 = 1.5 \]
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Next, substitute this result back into the expression: \[ 19 - 1.5 + 20 \]
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Now perform the subtraction: \[ 19 - 1.5 = 17.5 \]
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Finally, add 20: \[ 17.5 + 20 = 37.5 \]
Thus, the value of the expression is \( \boxed{37.5} \).