Let's analyze the statements regarding the product of \(3 \frac{2}{3}\) and \(\frac{10}{9}\).
First, convert \(3 \frac{2}{3}\) to an improper fraction:
\[ 3 \frac{2}{3} = \frac{11}{3} \]
Next, \(\frac{10}{9}\) is already in fractional form. Now we need to determine the product:
\[ \text{Product} = \left(\frac{11}{3}\right) \times \left(\frac{10}{9}\right) = \frac{11 \times 10}{3 \times 9} = \frac{110}{27} \]
Now compare \(\frac{110}{27}\) with \(3 \frac{2}{3}\) (or \(\frac{11}{3}\)).
To make the comparison easier, let's convert both fractions to have the same denominator:
- Convert \(\frac{11}{3}\) to an equivalent fraction with denominator 27:
\[ \frac{11}{3} = \frac{11 \times 9}{3 \times 9} = \frac{99}{27} \]
- Now we compare \(\frac{110}{27}\) and \(\frac{99}{27}\):
Since \(\frac{110}{27} > \frac{99}{27}\), we conclude:
\[ \frac{110}{27} > \frac{11}{3} \]
Thus, the product of \(3 \frac{2}{3}\) and \(\frac{10}{9}\) is greater than \(3 \frac{2}{3}\).
Now, looking at the statements:
- "Because \(10/9\) is greater than 1, the product of \(3 \frac{2}{3}\) and \(10/9\) is less than \(3 \frac{2}{3}\)." (Incorrect)
- "Because \(10/9\) is less than 1, the product of \(3 \frac{2}{3}\) and \(10/9\) is greater than \(3 \frac{2}{3}\)." (Incorrect)
- "Because \(10/9\) is greater than 1, the product of \(3 \frac{2}{3}\) and \(10/9\) is greater than \(3 \frac{2}{3}\)." (Correct)
- "Because \(10/9\) is less than 1, the product of \(3 \frac{2}{3}\) and \(10/9\) is less than \(3 \frac{2}{3}\)." (Incorrect)
The correct statement is:
"Because \(10/9\) is greater than 1, the product of \(3 \frac{2}{3}\) and \(10/9\) is greater than \(3 \frac{2}{3}\)."