Choose all of the expressions that are less than \( 10 \frac{2}{3} \) \( 10 \frac{2}{3} \times \frac{9}{10} \)

To determine if the expression \( 10 \frac{2}{3} \times \frac{9}{10} \) is less than \( 10 \frac{2}{3} \), we can first convert \( 10 \frac{2}{3} \) into an improper fraction. \( 10 \frac{2}{3} = 10 \times 3 + 2 = 30 + 2 = 32 \) with a denominator of 3, so we get: \[ 10 \frac{2}{3} = \frac{32}{3} \] Now, let's evaluate the expression \( 10 \frac{2}{3} \times \frac{9}{10} \): \[ 10 \frac{2}{3} \times \frac{9}{10} = \frac{32}{3} \times \frac{9}{10} = \frac{32 \times 9}{3 \times 10} = \frac{288}{30} = \frac{288 \div 6}{30 \div 6} = \frac{48}{5} \] Next, let's see if \( \frac{48}{5} \) is less than \( \frac{32}{3} \). To compare, we can convert both fractions to have a common denominator. The least common multiple of 5 and 3 is 15: \[ \frac{48}{5} = \frac{48 \times 3}{5 \times 3} = \frac{144}{15} \] \[ \frac{32}{3} = \frac{32 \times 5}{3 \times 5} = \frac{160}{15} \] Now we can see that \( \frac{144}{15} < \frac{160}{15} \). Hence, \[ 10 \frac{2}{3} \times \frac{9}{10} < 10 \frac{2}{3} \] Thus, \( 10 \frac{2}{3} \times \frac{9}{10} \) is indeed less than \( 10 \frac{2}{3} \).