Since the information provided in the Venn diagram represents percentages, convert the quantity in the common portion of the diagram from a percentage to a fraction. To convert a percentage to a fraction, divide the percentage by a divisor of \(\displaystyle 100\), then simplify the fraction if possible. 

Common portion is equal to \(\displaystyle 12\%\). Therefore, the solution is:

 

\(\displaystyle 12\div100=\frac{12}{100}=\frac{12\div 2}{100\div 2}=\frac{6}{50}=\frac{6\div 2}{50\div 2}=\frac{3}{25}\)

The common portion of this Venn diagram represents the percentage of respondents that were classified into both groups. In order to calculate the percentage that represents how many students have visted both NYC and Texas, first find the sum of group \(\displaystyle NYC\) and group \(\displaystyle TX\). Then subtract that quantity from \(\displaystyle 100\%\).

The solution is: 

\(\displaystyle 46+32=78\)

\(\displaystyle 100-78=22\)

Since \(\displaystyle 46\%\) of Ms. Dunn's class have visited only NYC, \(\displaystyle 46\) out of every \(\displaystyle 100\) students must have only visited NYC. This can be represented by the ratio \(\displaystyle 46:100\); however, this ratio does not appear as an answer choice, so we must reduce this ratio by dividing each part by their greatest common divisor. 

The solution is:

\(\displaystyle 46:100=46\div2:100\div2=23:50\)

In order to calculate the fraction of Kayla's friends who only play sport, first find the sum of the common portion and video game portion of the Venn diagram. Then subtract that sum from \(\displaystyle 1\) whole. 

The solution is: 

\(\displaystyle \frac{3}{10}+\frac{2}{10}=\frac{5}{10}\)

Note: \(\displaystyle \frac{10}{10}=1\)

Thus, \(\displaystyle \frac{10}{10}-\frac{5}{10}=\frac{5}{10}=\frac{5\div 5}{10\div 5}=\frac{1}{2}\)

This means that half of Kayla's friends only play sports. 

In order to solve this problem, identify that the common portion of this Venn diagram represents Kayla's friends who play sports and video games. Since \(\displaystyle \frac{3}{10}\) of her friends play sports and video games, convert this fraction to a percent.

The solution is:

\(\displaystyle \frac{3}{10}=\frac{3\times 10}{10\times 10}=\frac{30}{100}=30\%\)


Note: the most efficient way to convert this fraction to a percent is to find an equivalent fraction to \(\displaystyle \frac{3}{10}\) with a denominator of \(\displaystyle 100\). 

To find the missing quantity for category \(\displaystyle y\), first calculate the sum from the common portion of the Venn diagram and category \(\displaystyle x\). Then, subtract that sum from \(\displaystyle 100\%\) , because the total percentage of respondents must equal \(\displaystyle 100\%\). 

The algebraic solution is:

\(\displaystyle 39 +12=51\)

\(\displaystyle 100-51=49\)