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

0.4x
80
Actually off	0.1(100 – x)	0.6x	20
TOTAL	100 – x	x	100

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:

	Supposed To Be On	Supposed To Be Off	TOTAL
Actually on	72	8	80
Actually off	8	12	20
TOTAL	80	20	100

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

The correct answer is D.

6.
This question involves overlapping sets so we can employ a double-set matrix to help us. The two sets are speckled/rainbow and male/female. We can fill in 645 for the total number of total speckled trout based on the first sentence. Also, we can assign a variable, x, for female speckled trout and the expression 2x + 45 for male speckled trout, also based on the first sentence.

	Male	Female	Total
Speckled	2x + 45	x	645
Rainbow
Total

We can solve for x with the following equation: 3x + 45 = 645. Therefore, x = 200.

	Male	Female	Total
Speckled	445	200	645
Rainbow
Total

If the ratio of female speckled trout to male rainbow trout is 4:3, then there must be 150 male rainbow trout. We can easily solve for this with the below proportion where y represents male rainbow trout:

4 3

=

200 y

Therefore, y = 150. Also, if the ratio of male rainbow trout to all trout is 3:20, then there must be 1000 total trout using the below proportion, where z represents all trout:

3 20

=

150 z

	Male	Female	Total
Speckled	445	200	645
Rainbow	150
Total			1000

Now we can just fill in the empty boxes to get the number of female rainbow trout.

	Male	Female	Total
Speckled	445	200	645
Rainbow	150	205	355
Total			1000

The correct answer is D.

7.
Begin by constructing a double-set matrix and filling in the information given in the problem. Assume there are 100 major airline companies in total since this is an easy number to work with when dealing with percent problems.

	Wireless	No Wireless	TOTAL
Snacks	? MAX ?		70
NO Snacks			30
TOTAL	30	70	100

Notice that we are trying to maximize the cell where wireless intersects with snacks. What is the maximum possible value we could put in this cell. Since the total of the snacks row is 70 and the total of the wireless column is 30, it is clear that 30 is the limiting number. The maximum value we can put in the wireless-snacks cell is therefore 30. We can put 30 in this cell and then complete the rest of the matrix to ensure that all the sums will work correctly.

	Wireless	No Wireless	TOTAL
Snacks	30	40	70
NO Snacks	0	30	30
TOTAL	30	70	100

The correct answer is B.

8.
For an overlapping set problem we can use a double-set matrix to organize our information and solve. Because the given values are all percentages, we can assign a value of 100 to the total number of people in country Z. The matrix is filled out below based on the information provided in the question. 

The first sentence tells us that 10% of all of the people do have their job of choice but do not have a diploma, so we can enter a 10 into the relevant box, below. The second sentence tells us that 25% of those who do not have their job of choice have a diploma. We don't know how many people do not have their job of choice, so we enter a variable (in this case, x) into that box. Now we can enter 25% of those people, or 0.25x, into the relevant box, below. Finally, we're told that 40% of all of the people have their job of choice.

	University Diploma	NO University Diploma	TOTAL
Job of Choice		10	40
NOT Job of Choice	0.25x		x
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 relationships (for example: 40 + x = 100, therefore x = 60. 0.25 × 60 = 15. And so on.). 

	University Diploma	NO University Diploma	TOTAL
Job of Choice	30	10	40
NOT Job of Choice	15	45	60
TOTAL	45	55	100

We were asked to find the percent of the people who have a university diploma, or 45%.

The correct answer is B.

9.
This is a problem that involves two overlapping sets so it can be solved using a double-set matrix. The problem tells us that there are 800 total students of whom 70% or 560 are male. This means that 240 are female and we can begin filling in the matrix as follows:

	Male	Female	TOTAL
Sport
No Sport			maximize
TOTAL	560	240	800

The question asks us to MAXIMIZE the total number of students who do NOT participate in a sport. In order to maximize this total, we will need to maximize the number of females who do NOT participate in and the number of males who do NOT participate in a sport.
The problem states that at least 10% of the female students, or 24 female students, participate in a sport. This leaves 216 female students who may or may not participate in a sport. Since we want to maximize the number of female students who do NOT participate in a sport, we will assume that all 216 of these remaining female students do not participate in a sport.
The problem states that fewer than 30% of the male students do NOT participate in a sport. Thus, fewer than 168 male students (30% of 560) do NOT participate in a sport. Thus anywhere from 0 to 167 male students do NOT participate in a sport. Since we want to maximize the number of male students who do NOT participate in a sport, we will assume that 167 male students do NOT participate in a sport. This leaves 393 male students who do participate in a sport.
Thus, our matrix can now be completed as follows:

	Male	Female	TOTAL
Sport	393	24	417
No Sport	167	216	383
TOTAL	560	240	800

Therefore, the maximum possible number of students in School T who do not participate in a sport is 383.

The correct answer is B.

10.
This is an overlapping sets problem, which can be solved most efficiently by using a double set matrix. Our first step in using the double set matrix is to fill in the information given in the question. Because there are no real values given in the question, the problem can be solved more easily using 'smart numbers'; in this case, we can assume the total number of rooms to be 100 since we are dealing with percentages. With this assumption, we can fill the following information into our matrix:

There are 100 rooms total at the Stagecoach Inn.

Of those 100 rooms, 75 have a queen-sized bed, while 25 have a king-sized bed.

Of the non-smoking rooms (let's call this unknown n), 60% or .6n have queen-sized beds.

10 rooms are non-smoking with king-sized beds.

Let's fill this information into the double set matrix, including the variable n for the value we need to solve the problem:

	SMOKING	NON-SMOKING	TOTALS
KING BED		10	25
QUEEN BED		.6n	75
TOTALS		n	100

In a double-set matrix, the first two rows sum to the third, and the first two columns sum to the third. We can therefore solve for n using basic algebra:

10 + .6n = n
10 = .4n
n = 25
We could solve for the remaining empty fields, but this is unnecessary work. Observe that the total number of smoking rooms equals 100 – n = 100 – 25 = 75. Recall that we are working with smart numbers that represent percentages, so 75% of the rooms at the Stagecoach Inn permit smoking.

The correct answer is E.

11.
For an overlapping set problem we can use a double-set matrix to organize our information and solve. The boldfaced values were given in the question. The non-boldfaced values were derived 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. The variable p was used for the total number of pink roses, so that the total number of pink and red roses could be solved using the additional information given in the question.

	Red	Pink	White	TOTAL
Long-stemmed			0	80
Short-stemmed	5