To evaluate the expression \( 4 \frac{1}{2} - 2 - 5 \frac{2}{3} \), we first convert the mixed numbers to improper fractions.
-
Convert \( 4 \frac{1}{2} \):
\[
4 \frac{1}{2} = 4 \times 2 + 1 = 8 + 1 = \frac{9}{2}
\]
-
Convert \( 5 \frac{2}{3} \):
\[
5 \frac{2}{3} = 5 \times 3 + 2 = 15 + 2 = \frac{17}{3}
\]
Now, rewrite the expression using these improper fractions:
\[
\frac{9}{2} - 2 - \frac{17}{3}
\]
Next, we convert \(2\) to a fraction with the denominator of 1:
\[
2 = \frac{2}{1}
\]
Now, let's find a common denominator for \( \frac{9}{2} \), \( \frac{2}{1} \), and \( \frac{17}{3} \). The least common multiple (LCM) of the denominators 2, 1, and 3 is 6.
Convert each fraction:
-
Convert \( \frac{9}{2} \) to a common denominator of 6:
\[
\frac{9}{2} = \frac{9 \times 3}{2 \times 3} = \frac{27}{6}
\]
-
Convert \( \frac{2}{1} \) to a common denominator of 6:
\[
\frac{2}{1} = \frac{2 \times 6}{1 \times 6} = \frac{12}{6}
\]
-
Convert \( \frac{17}{3} \) to a common denominator of 6:
\[
\frac{17}{3} = \frac{17 \times 2}{3 \times 2} = \frac{34}{6}
\]
Now we can rewrite the expression:
\[
\frac{27}{6} - \frac{12}{6} - \frac{34}{6}
\]
Since all fractions have the same denominator, subtract them:
\[
\frac{27 - 12 - 34}{6} = \frac{27 - 12 = 15}{6} - \frac{34}{6} = \frac{15 - 34}{6} = \frac{-19}{6}
\]
Now convert \(\frac{-19}{6}\) to a mixed number:
- Divide \(19\) by \(6\):
- \(19 \div 6 = 3\) (whole number)
- Remainder: \(19 - (6 \times 3) = 19 - 18 = 1\)
So, \(\frac{-19}{6}\) is:
\[
-3 \frac{1}{6}
\]
Thus, the evaluation of \( 4 \frac{1}{2} - 2 - 5 \frac{2}{3} \) gives us:
\[
\text{Final Answer: } -3 \frac{1}{6}
\]
The correct response is:
A: \(-3 \frac{1}{6}\)
