Sets 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

SETS

1.
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.

	Total__Profitable'>Comedies	Horror Films	Total
Profitable
Unprofitable
Total	0.6x	0.4x	x

75% of the comedies were profitable, but 75% of the horror moves were unprofitable. 

	Comedies	Horror Films	Total
Profitable	0.75(0.6x)
Unprofitable		0.75(0.4x)
Total	0.6x	0.4x	x

If the studio made a total of 40 films...

	Comedies	Horror Films	Total
Profitable	0.75(24) = 18
Unprofitable		0.75(16) = 12
Total	0.6(40) = 24	0.4(40) = 16	x = 40

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

	Comedies	Horror Films	Total
Profitable	18	4	22
Unprofitable	6	12	18
Total	24	16	40

The problem seeks the total number of profitable films, which is 22.

The correct answer is E.

2.
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.

	Tired	Not Tired	TOTAL
6 or more hours	.3x	.7x	x
Fewer than 6 hours	75		80
TOTAL			100

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.

	Tired	Not Tired	TOTAL
6 or more hours	6	14	20
Fewer than 6 hours	75	5	80
TOTAL	81	19	100

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.

3.
This is an overlapping sets problem concerning two groups (students in either band or orchestra) and the overlap between them (students in both band and orchestra).

If the problem gave information about the students only in terms of percents, then a smart number to use for the total number of students would be 100. However, this problem gives an actual number of students (“there are 119 students in the band”) in addition to the percentages given. Therefore, we cannot assume that the total number of students is 100.

Instead, first do the problem in terms of percents. There are three types of students: those in band, those in orchestra, and those in both. 80% of the students are in only one group. Thus, 20% of the students are in both groups. 50% of the students are in the band only. We can use those two figures to determine the percentage of students left over: 100% - 20% - 50% = 30% of the students are in the orchestra only.

Great - so 30% of the students are in the orchestra only. But although 30 is an answer choice, watch out! The question doesn't ask for the percentage of students in the orchestra only, it asks for the number of students in the orchestra only. We must figure out how many students are in Music High School altogether.

The question tells us that 119 students are in the band. We know that 70% of the students are in the band: 50% in band only, plus 20% in both band and orchestra. If we let x be the total number of students, then 119 students are 70% of x, or 119 = .7x. Therefore, x = 119 / .7 = 170 students total.

The number of students in the orchestra only is 30% of 170, or .3 × 170 = 51.

The correct answer is B.

4.

For an overlapping set problem we can use a double-set matrix to organize our information and solve. Let's call P the number of people at the convention. The boldface entries in the matrix below were given in the question. For example, we are told that one sixth of the attendees are female students, so we put a value of P/6 in the female students cell.

	FEMALE	NOT FEMALE	TOTALS
STUDENTS	P/6	P/6	P/3
NON STUDENTS	P/2