Certainly! Let's solve the given problem step by step.
We need to calculate the product of a whole number and a mixed number:
[tex]\[
11 \times 2 \frac{7}{11}
\][/tex]
First, let's convert the mixed number [tex]\(2 \frac{7}{11}\)[/tex] into an improper fraction. A mixed number has the form [tex]\( \text{whole part} + \frac{\text{numerator}}{\text{denominator}}\)[/tex]. So:
[tex]\[
2 \frac{7}{11} = 2 + \frac{7}{11}
\][/tex]
To add these, we need a common denominator. The simplest way is to convert the whole number 2 into a fraction with the same denominator:
[tex]\[
2 = \frac{2 \times 11}{11} = \frac{22}{11}
\][/tex]
Now, add this fraction to [tex]\(\frac{7}{11}\)[/tex]:
[tex]\[
\frac{22}{11} + \frac{7}{11} = \frac{22 + 7}{11} = \frac{29}{11}
\][/tex]
With this, the mixed number [tex]\(2 \frac{7}{11}\)[/tex] becomes the improper fraction [tex]\(\frac{29}{11}\)[/tex].
Next, we multiply the whole number 11 by this improper fraction:
[tex]\[
11 \times \frac{29}{11}
\][/tex]
When multiplying by a fraction, if the denominator of the fraction and the whole number are the same, they cancel out:
[tex]\[
11 \times \frac{29}{11} = 29
\][/tex]
Therefore, the product is 29.
Finally, we need to select the closest value to 29 from the given choices:
[tex]\[
\begin{array}{lr}
33 & 29 \\
31 & 35
\end{array}
\][/tex]
Clearly, the closest value is 29.
To summarize, the product of [tex]\(11 \times 2 \frac{7}{11}\)[/tex] is 29, and among the choices provided, the correct answer is:
[tex]\[
\boxed{29}
\][/tex]
