Sets Practice Questions

For an overlapping sets problem, it is best to use a double-set matrix to organize the information and solve. Fill in the information in the order in which it is given.

Of the films Empty Set Studios released last year, \(60\%\) were comedies and the rest were horror films.

$$\begin{array}{|c|c|c|c|} \hline & \text{Comedies} & \text{Horror Films} & \text{Total} \\ \hline \text{Profitable} & & & \\ \hline \text{Unprofitable} & & & \\ \hline \text{Total} & 0.6x & 0.4x & x \\ \hline \end{array}$$

\(75\%\) of the comedies were profitable, but \(75\%\) of the horror films were unprofitable.

$$\begin{array}{|c|c|c|c|} \hline & \text{Comedies} & \text{Horror Films} & \text{Total} \\ \hline \text{Profitable} & 0.75(0.6x) & & \\ \hline \text{Unprofitable} & & 0.75(0.4x) & \\ \hline \text{Total} & 0.6x & 0.4x & x \\ \hline \end{array}$$

If the studio made a total of \(40\) films...

$$\begin{array}{|c|c|c|c|} \hline & \text{Comedies} & \text{Horror Films} & \text{Total} \\ \hline \text{Profitable} & 0.75(24) = 18 & & \\ \hline \text{Unprofitable} & & 0.75(16) = 12 & \\ \hline \text{Total} & 0.6(40) = 24 & 0.4(40) = 16 & x = 40 \\ \hline \end{array}$$

Since each row and each column must sum to the total value, we can fill in the remaining boxes.

$$\begin{array}{|c|c|c|c|} \hline & \text{Comedies} & \text{Horror Films} & \text{Total} \\ \hline \text{Profitable} & 18 & 4 & 22 \\ \hline \text{Unprofitable} & 6 & 12 & 18 \\ \hline \text{Total} & 24 & 16 & 40 \\ \hline \end{array}$$ The problem seeks the total number of profitable films, which is \(22\).
The correct answer is E.

For an overlapping-sets problem we can use a double-set matrix to organize our information and solve. Because the values are in percents, we can assign a value of 100 for the total number of interns at the hospital. Then, carefully fill in the matrix based on the information provided in the problem. The matrix below details this information. Notice that the variable \( x \) is used to detail the number of interns who receive 6 or more hours of sleep, \( 70\% \) of whom reported no feelings of tiredness. $$ \begin{array}{|l|c|c|c|} \hline & \text{Tired} & \text{Not Tired} & \text{TOTAL} \\ \hline \text{6 or more hours} & .3x & .7x & x \\ \hline \text{Fewer than 6 hours} & 75 & & 80 \\ \hline \text{TOTAL} & & & 100 \\ \hline \end{array} $$ In a double-set matrix, the sum of the first two rows equals the third and the sum of the first two columns equals the third. Thus, the boldfaced entries below were derived using the above matrix. $$ \begin{array}{|l|c|c|c|} \hline & \text{Tired} & \text{Not Tired} & \text{TOTAL} \\ \hline \text{6 or more hours} & \mathbf{6} & \mathbf{14} & \mathbf{20} \\ \hline \text{Fewer than 6 hours} & 75 & \mathbf{5} & 80 \\ \hline \text{TOTAL} & \mathbf{81} & \mathbf{19} & 100 \\ \hline \end{array} $$ We were asked to find the percentage of interns who reported no feelings of tiredness, or \( 19\% \) of the interns. The correct answer is C.

For an overlapping set problem we can use a double-set matrix to organize our information and solve. Because the values here are percents, we can assign a value of 100 to the total number of lights at Hotel California. The information given to us in the question is shown in the matrix in boldface. An x was assigned to the lights that were "Supposed To Be Off" since the values given in the problem reference that amount. The other values were filled in using the fact that in a double-set matrix the sum of the first two rows equals the third and the sum of the first two columns equals the third.

$$\begin{array}{|l|c|c|c|}\hline & \text{Supposed To Be On} & \text{Supposed To Be Off} & \text{TOTAL} \\\hline \text{Actually on} & & 0.4x & 80 \\\hline \text{Actually off} & 0.1(100 - x) & 0.6x & 20 \\\hline \text{TOTAL} & 100 - x & x & 100 \\\hline \end{array}$$

Using the relationships inherent in the matrix, we see that:
0.1(100 − x) + 0.6x = 20
10 − 0.1x + 0.6x = 20
0.5x = 10 so x = 20

We can now fill in the matrix with values:

$$\begin{array}{|l|c|c|c|}\hline & \text{Supposed To Be On} & \text{Supposed To Be Off} & \text{TOTAL} \\\hline \text{Actually on} & 72 & 8 & 80 \\\hline \text{Actually off} & 8 & 12 & 20 \\\hline \text{TOTAL} & 80 & 20 & 100 \\\hline \end{array}$$

Of the 80 lights that are actually on, 8, or \( 10\% \), are supposed to be off.

The correct answer is D.