2 2/3-3 1/2

To subtract \(3 \frac{1}{2}\) from \(2 \frac{2}{3}\), we first need to convert both mixed numbers into improper fractions.

1. Convert \(2 \frac{2}{3}\): \[ 2 \frac{2}{3} = \left(2 \times 3 + 2\right) / 3 = (6 + 2) / 3 = \frac{8}{3} \]

2. Convert \(3 \frac{1}{2}\): \[ 3 \frac{1}{2} = \left(3 \times 2 + 1\right) / 2 = (6 + 1) / 2 = \frac{7}{2} \]

Now, we need to find a common denominator to perform the subtraction. The least common denominator (LCD) of \(3\) and \(2\) is \(6\).

3. Convert \(\frac{8}{3}\) to sixths: \[ \frac{8}{3} = \frac{8 \times 2}{3 \times 2} = \frac{16}{6} \]

4. Convert \(\frac{7}{2}\) to sixths: \[ \frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6} \]

Now, we can perform the subtraction: \[ \frac{16}{6} - \frac{21}{6} = \frac{16 - 21}{6} = \frac{-5}{6} \]

Thus, the result of \(2 \frac{2}{3} - 3 \frac{1}{2}\) is \(-\frac{5}{6}\).