MATH DAT practice Q

How many ways can you choose 7 people from a group of 10? A. 70 ways B. 120 ways C. 140 ways D. 240 ways E. 300 ways

B. 120 ways 10! / 7! x 3!

Let x,y, and z be distinct non zero integers Quantiy A: (x/y) / z Quantity B: x / (y/z) A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

D. The relationship cannot be determined from the information given.

Ellen used a coupon at a restaurant for a certain percent discount of the meal. What percent discount did she receive? 1) Ellen paid $14 after the coupon.2) Had Ellen spent $5 more, she would have saved an additional $2 from the coupon. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.D. EACH statement ALONE is sufficient.E. Statements (1) and (2) TOGETHER are NOT sufficient.

When we look at Statement (1), we are told Ellen paid $14 dollars after using the coupon. However, with only this information, we do not know how much Ellen would have paid without the coupon, and so Statement (1) clearly is insufficient on its own. Statement (2) tells us nothing about how much Ellen spent. It does tell us though that had she spent $5 more, she would have saved an extra $2. In effect, this is telling us the percent discount for the coupon: save $2 for every $5 you spend. No information from Statement (1) was necessary for this either, and so our answer must be [B] Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

x,y,z> 0 x - y > 0 x - z < 0 quantity A: x/y quantity B: y/z A. Quantity A is greater .B. Quantity B is greater. C. The two quantities are equal .D. The relationship cannot be determined from the information given.

A. Quantity A is greater We start out being told all of our numbers, x, y,and z, are all greater than 0, which certainly simplifies matters, as negatives would lead to extra circumstances we'd need to check. The next two statements tell us how y and z relate to x, namely: x > y x < z Additionally, because of the transitive property, which tells us if A is greater than B, and B is greater than C, then A is also greater than C, we can construct a relation between y and z: y < z This relation is crucial to solving our problem. We can tell that Quantity A must be greater than 1, since "x > y", and when the numerator is greater than the denominator, the fraction is greater than 1. For the same reason, we can tell that Quantity B is less than 1, from our last inequality. Since Quantity A must be greater than 1, and Quantity B must be less than 1, we know our answer must be [A] Quantity A is greater.

x > 0 y = x^2 + 6x + 5 z = 2x + 2 Quantity A: (y - z) / (y + z) Quantity B: 3/7 A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

A. Quantity A is greater.

The number of jackets a college student owns is normally distributed with a mean of 4 and a standard deviation of 1. What is the probability that a student has 6 or less jackets? A. 5% B. 50% C. 84% D. 95% E. 97.5%

E. 97.5%

A bathtub can hold a maximum of 50 gallons of water. Water can be drained out of the tub at a rate of 2 gallons per minute. If the tub is initially completely filled by a faucet at a rate of 1 gallon per minute, how long will it take to drain the full tub if the drain is opened but the faucet is kept on? A. 25 minutes B. 50 minutes C. 100 minutes D. 120 minutes E. 150 minutes

B. 50 minutes

If the complement of an angle and the supplement of the same angle summed together equals 100 degrees, find the angle. A. 20 degrees B. 38 degrees C. 50 degrees D. 75 degrees E. 85 degrees

E. 85 degrees To solve this problem, we can set up an equation using the definitions of a complementary angle (the angle you add to make 90 degrees), and a supplementary angle (the angle you add to make 180 degrees): 100 = (180 - x) + (90 - x) solve for x: 100 = 270 - 2x -170 = -2x x = 85

According to a survey, the number of patients in a given dental office in a given month is normally distributed with a mean of 1,100 patients and a standard deviation of 100 patients. If a dental office is chosen at random, what is the probability that more than 1,400 patients visit this dental office? A. 0.15% B. 0.3% C. 5% D. 68% E. 99.7%

A. 0.15% The key to solving this problem involves understanding a Gaussian distribution. With this unique distribution, 68% of the data is contained within one standard deviation from the mean, 95% of the data lies within 2 standard deviations, and 99.7% of the data lies within 3 standard deviations from the mean. In this case, we can find that 1,400 patients is 3 standard deviations away from the mean. This means that 99.7% of the dental offices have between 800 and 1,400 patients in a given month. This also implies that 0.3% of dental offices lie outside of this range, and have either less than 800 patients per month or more than 1,400 patients per month. However, the question specifically asks for the probability that a randomly chosen dental office has more than 1,400 patients. Due to the symmetry of a Gaussian distribution, we can say that 0.15% (0.3 ÷ 2) have more than 1,400 patients in a month. Therefore, answer choice [A] is correct.

The chart below shows the snow total in inches for four different ski resorts from Monday through Friday. If one resort saw more snow than any other on Saturday, which resort was that? 1) From Monday through Saturday, each resort had the most snow, or tied for the most snow, at least one day. 2) From Monday through Saturday, no resort had the most snow, or tied for the most snow, every day. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement 1 tells us that each of the four resorts spends at least one day from Monday through Saturday with the most snowfall. From the graph, we can see every ski resort at least ties for that distinction from Monday through Friday other than "Chute and Sweet". Since the question also tells us that one resort saw more than any other on Saturday, with the benefit of Statement 1, we know that resort must be "Chute and Sweet", and so Statement 1 is sufficient for answering the given question. Statement 2 tells us that no resort had the most snow every day. However, this is already satisfied Monday through Friday; in other words, the statement doesn't provide us with any information not already found in the graph, and so it must be insufficient, since the graph contains no information about Saturday. As Statement 1 sufficiently answered the given question, but Statement 2 did not, our answer must be [A] Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

How many ways can 4 identical footballs and 3 identical basketballs be ordered on a shelf? A. 12 ways B. 35 ways C. 64 ways D. 125 ways E. 210 ways

B. 35 ways 7! / 3! x 4! = 35

Four dice are rolled. What is the probability of getting exactly three "4's" and one "2" in any order? A. 1/1,296 B. 4/1,296 C. 1/396 D. 1/216 E. 1/200

B. 4/1,296

y is a positive, whole number. Is y a prime number? 1) y2 has exactly 3 positive factors. 2) y is an odd number. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. a prime number has only 2 factors, 1 and itself. statement 1 tells use that y^2 has exactly 3 positive factors. So y can be 3 because 3^2 is 9 and has 3 factors

There are 8 marbles in a bag: 4 are blue, 3 are green, and 1 is red. What is the probability that of two randomly chosen marbles without replacement, 1 is blue and 1 is red? A. 1/7 B. 1/6 C. 2/5 D. 1/2 E. 3/4

A. 1/7 The probability of pulling a blue marble on the first pull would be 4/8, or 1/2. Since there is no replacement, the probability of getting a red marble on the second pull would be 1/7 because one blue marble was removed from the bag. Therefore, the total probability of both events occurring would be 1/2 * 1/7 = 1/14. The probability of pulling a red marble on the first pull would be 1/8. Since there is no replacement, the probability of then getting a blue marble on the second pull would be 4/7 because one red marble was already removed. Therefore, the total probability of both events occurring would be 4/7 * 1/8 = 1/14. 1/14 + 1/14 = 1/14 or *1/7*

Tank A is made up of a 50% solution of salt water. Tank B is made up of a 90% solution of salt water. Part of each tank is mixed together to make a 75% solution of salt water. If 4 liters were taken from tank A to make the solution, how much total solution was made? A. 32/3 liters B. 12 liters C. 16 liters D. 18 liters E. 94/5 liters

A. 32/3 liters Let's start by writing the following expression, since we know that the amount of salt in the two parts of solution will be the same as the amount of salt in the final solution. x represents the amount of solution taken from Tank B. (0.5 x 4 liters) + (0.9 x (x liters)) = 0.75 x (4 + x)liters Simplify the expression: 2 + 0.9x = 3 + 0.75x 0.15x = 1 x = 6.66 liters Therefore, 4 (taken from Tank A) + 6.66 (taken from Tank B) = 10.66 liters, or 32/3 liters of total solution were made. Answer choice [A] is correct.

|y| < x^2 - 81 x>0 Quantity A: (x-9) Quantity B: y/(x+9) A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

A. Quantity A is greater. We can expand the difference of 2 squares in our problem statement to equal: |y| < (x+9)(x-9) In order to remove the absolute value sign, we state that y must also be greater than the negative of the term on the right: -(x+9)(x-9)<|y|<(x+9)(x-9) We can then divide the entire inequality by "x + 9", giving: -(x-9)< y/(x+9) < (x - 9) Because x was already restricted to being greater than 0, we do not need to worry about dividing our inequality by a negative number, which would require the signs to flip. We can now clearly see that our term in Quantity B must be less than our term in Quantity A, and so our answer is [A] Quantity A is greater.

The chart below demonstrates energy usage over one month. Do Joe and Stef combined use, on average, 100 kWh less per day than Bob and Ashley for one month? Pie chart stats: 66% bob and ashley 21% joe 13% stef 1) The month was April (30 days long). 2) Everyone combined to use 10,000 kWh of electricity. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. The question is asking about total numbers, not percents. The pie chart above only tells us percents, and so to make any determination on a total number, we will need information about total numbers to be provided in the statements. As Statement 2 is the only statement to give us this information, let's begin with that one. Statement 2 tells us "everyone combined to use 10,000 kWh of electricity". This will allow us to directly convert from percents to total numbers, as 66% of 10,000 = 6,600, 21% of 10,000 = 2,100, and 13% of 10,000 = 1,300. We can than see that Bob and Ashley used 3,200 kWh more electricity than Joe and Stef did. Even in a 31 day month, which is the greatest possible days in any month, this would mean Joe and Stef used, on average, more than 100 kWh less than their parents, as 3,200 ÷ 31 is larger than 100 (3,200 ÷ 32 = 100), and so Statement 2 is sufficient in answering our question. Statement 1 tells us the month was April, meaning there are only 30 days. This is even better for Joe and Stef with respect to Statement 2, but on its own, it provides no information about total numbers, and is insufficient on its own. As Statement 2 was sufficient on its own, but Statement 1 was not, our answer must be [B] Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

A bag of chips normally costs $2. If there is a sale to get 12 bags of chips for the price of 6 bags, what is the price per bag of chip, taking into account a 10% sales tax? A. $1.00 B. $1.10 C. $1.25 D. $1.30 E. $1.42

B. $1.10 Let's start by calculating the cost of 12 bags of chips before tax. Since we are getting 12 bags of chips for the normal price of six ($2 each), we are getting 12 bags for $12. Now, we can calculate the price including sales tax on the total purchase. Sales tax is 10% so by increasing the price of the 12 bags by 10%, we can find the total price of the bags with the sales tax added on. To do this, we multiply the price of the bags by 110% or 1.1: $1.2 x 1.1 = $13.20 Divide the total price by 12 to get the price per bag: $13.20/ 12 = 1.10 per bag

How many distinct ways can 2 people sit in 5 chairs? A. 10 ways B. 20 ways C. 50 ways D. 60 ways E. 80 ways

B. 20 ways 5! / (5 - 2)! 5! / 3! = 20

In a group of people, 30 of them wear glasses and 17 of them wear earrings. If 12 people wear both glasses and earrings, how many people wear only earrings or only glasses? A. 17 people B. 23 people C. 30 people D. 35 people E. 42 people

B. 23 people 30 wear glasses and 17 wear earrings 12 people wear both 30 - 12 = 18 17 - 12 = 5 18 + 5 = 23

A regular hexagon has a perimeter of 12. What is the area of the hexagon? A. 6 B. 6√3 C. 8 D. 8√3 E. 12√3

B. 6√3 Let's recognize that a hexagon can be divided into six identical equilateral triangles: Since the perimeter of the hexagon is 12 and there are 6 equal sides, the length of each triangle's sides is 2. To find the area of the whole hexagon, we can start by finding the area of one of the equilateral triangles. We can drop a perpendicular line to bisect an equilateral triangle into two 30-60-90 triangles: Since the length of the hypotenuse is 2 and the length of the shorter leg is 1, the length of the longer leg must be √3. We can now use this information to find the area of the equilateral triangle: Area = (1/2)bh area = (1/2) x 2 x squrt3 = sqrt 3 Since there are six equilateral triangles in the hexagon, the total area of the hexagon must be 6√3, or answer choice You can also memorize the equation for the area of a regular hexagon! "s" = length of one side: Area of a regular hexagon = (3sqrt3 / 2) x s^2

Quantity A: -5x^2 -1 Quantity B: -(5x^2-1) A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

B. Quantity B is greater. To solve this we simply distribute the (-) on the outside of the parenthesis on Quantity B. We then get -5x^2+1 Its similar to Quantity A only that it has a +1 instead of a -1 so Quantity B will always be larger than Quantity A

What is the larger angle in the right triangle shown below, A or B? 1) sin(A) = cos(B) 2) tan(A) > tan(B) A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient.E. Statements (1) and (2) TOGETHER are NOT sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. In a right triangle, the sine of one acute angle will always equal the cosine of the other, and vice versa. As such, Statement 1 gives us no actual information, eliminating [A], [C], and [D] as possible answers. The tangent function monotonically increases as its argument increases. Therefore, Statement 2 is essentially stating A > B. As such, our answer must be [B] Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

There are 12 pens in a pencil case: 4 are blue, 6 are black, and 2 is red. What is the probability that of three randomly chosen pens with replacement, 2 are black and 1 is red? A. 1/24 B. 1/12 C. 1/8 D. 1/4 E. 1/3

C. 1/8 The 3 different posibilities are BBR, BRB, and RBB. BBR = 1/2 x 1/2 x 1/6 = 1/24 all the possibilities are going to equal 1/24 so we just add all possibilities: 1/24 + 1/24 + 1/24 = 3/24 or *1/8*

Nate and Izzy ran a five mile race, starting and finishing at the same point, and beginning next to one another. Who ran faster? 1) Nate passed Izzy once during the race. 2) Izzy passed Nate after he'd passed her once. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. The question asks us who ran faster, though considering they ran the same distance, and started at the same time, we can also interpret this as who finished first, as whoever finished first ran the race faster. Statement 1 tells us "Nate passed Izzy once during the race". If we knew Izzy never caught back up to Nate, then this would tell us Nate ran faster. But we do not have that information, so Statement 1 on its own is insufficient. Statement 2 tells us "Izzy passed Nate after he'd passed her once". This tell us at some point Nate passed Izzy, and then later, she passed him. However, Nate could've passed her again, and without knowing this, Statement 2 on its own is insufficient. However, if we use Statement 1, which tells us Nate passed Izzy once, when looking at Statement 2, we know he couldn't have passed her again, since that would mean he passed her more than once, and so we can definitively say that Izzy finished first and ran faster.

A population of rabbits grows by 10% every month. If the population begins with 1,000 rabbits, how many rabbits will there be in 6 months? A. 1,234 rabbits B. 1,451 rabbits C. 1,772 rabbits D. 2,003 rabbits E. 2,304 rabbits

C. 1,772 rabbits use the formula for compound interest: A = P(1 + (r/n))^nt A= final population P = starting population r = interest rate n = number of times rabbit population is compounded t = number of time periods A = 1000(1+ (0.1/1))^1 x 6 A = 1000(1.1)^6 A = 1771.6 rabbits

A standard deck of playing cards consists of 52 cards, half of which are black and half of which are red. If three cards are picked at random from the deck with replacement, what is the probability that either all three are black or all three are red? A. 1/16 B. 1/8 C. 1/4 D. 3/8 E. 1/2

C. 1/4 the desired outcomes are BBB and RRR. to get BBB the probability would be: 1/2 x 1/2 x 1/2 = 1/8 To get RRR the probability would be: 1/2 x 1/2 x 1/2 = 1/8 Therefore, the total probability of either event occurring would be: 1/8 + 1/8 = 2/8 = 1/4

A bird launches into flight from a point on the ground. The bird can fly 8 feet per second horizontally and 6 feet per second vertically. After 10 seconds, how far has the bird flown relative to its initial spot on the ground? A. 10 feet B. 50 feet C. 100 feet D. 500 feet E. 1,000 feet

C. 100 feet We need to calculate the horizontal and vertical distances that the bird has traveled in order to find the distance the bird has flown relative to its starting point: Horizontally: 8 ft per sec x 10 seconds = 80 ft vertically: 6 ft per sec x 10 sec = 60 ft Therefore, we can use these distances to calculate the "hypotenuse" of the triangle: A^2 + B^2 = C^2 60^2 + 80^2 = c^2 3600 + 6400 = c^2 10000 = c^2 c = 100 ft

A soda bottle can be filled completely by a faucet in 10 seconds. The bottle also has a hole that can drain the full soda bottle to empty in 30 seconds. If the bottle is initially empty when the faucet is turned on, how long will it take for the bottle to be fully filled? A. 10 seconds B. 12 seconds C. 15 seconds D. 20 seconds E. 30 seconds

C. 15 seconds This problem is similar to classic rate problems; however, it is not as straightforward because the bottle is being simultaneously filled and emptied. We can approach this problem in the same way and then modify the equation to account for this complication. Using the framework: *(1 / A time) + (1 / B time) = ( 1 / A and B working together time)* We can set up and solve the equation: (1/ 10 seconds) - (1 / 30 seconds) = 1/x ( 3 / 30) - (1 / 30) = 1/x (2 / 30) = 1/x 2x = 30 x = 15 seconds

In Gainesville, one in four people have blonde hair. What is the chance that if four people are chosen at random from Gainesville, at least one person has blonde hair? A. 1/256 B. 81/256 C. 175/256 D. 255/256 E. 1

C. 175/256 The fastest approach to this problem is to realize that the probability of choosing at least one blonde person equals one minus the probability of choosing zero blonde people. You can use the following equation: P(at least one blonde person chosen) = 1 - P(zero blonde people chosen) The probability of choosing zero blonde people is: P(zero blonde people chosen) (3/4) x (3/4) x (3/4) x (3/4) = 81/256 P(at least one blonde person chosen) = 1 - P(zero blonde people chosen) = 1 - (81/256) = (256/256) - (81/256) = *(175/256)*

Pizza Hat sells 3 diameter sizes of pizza: 8", 10" and 12". Determine the area ratio of the 12" to the 8" pizza. A. 0.44 B. 1.5 C. 2.25 D. 3.66 E. 4.25

C. 2.25 Recall the formula for the area of a circle: A = πr2 Now, we can set up the ratio of the areas: Ration = (πR2 / πr2) Ratio = π(6)^2 / π(4)^2 Ratio = 36 / 16 *Ratio = 2.25*

Two normal dice are rolled separately. What is the probability of getting a sum of 7 or 11? A. 1/36 B. 1/9 C. 2/9 D. 1/2 E. 3/4

C. 2/9 Sum of 7: 1,6 or 6,1 or 2,5 or 5,2 or 3,4 or 4,3 Sum of 11: 5,6 or 6,5 8/36 = 2/9

3 liters of 50% salt solution is mixed with 7 liters of 13% salt solution. What is the percent of salt in the mixture? A. 13.6% B. 17.4% C. 24.1% D. 33.2% E. 50.1%

C. 24.1% 3 x .5 = 1.5 7 x .13 = .91 1.5 + .91 = 2.41 2.41 / 10 = .241 or 24.1%

In a local kickball league, the ratio of college graduates with a graduate degree to non-college graduates is 1:8, and the ratio of college graduates without a graduate degree to non-college graduates is 2:3. If one random college graduate is picked, what is the probability that they hold a graduate degree? A. 1/24 B. 1/8 C. 3/19 D. 3/16 E. 1/2

C. 3/19 With the given ratios, we are asked to find a probability. Let's start by setting some variables: X = college graduates with a graduate degree Y = college graduates without a graduate degree Z = non-college graduates The given ratios then become: X:Z = 1:8 Y:Z = 2:3 Let's make the common term the same so that we can combine the ratios. 3(X:Z) = 3(1:8) = 3:24 8(Y:Z) = 8(2:3) = 16:24 X:Y:Z = 3:16:24 Since the question only asks for the probability of choosing from college graduates, we can ignore the non-college graduates (Z), and end up with the ratio X:Y, or 3:16. To translate this to a probability, we add the parts of the ratio together (3 + 16 = 19), and since 3 parts of the 19 are college graduates with graduate degrees, the final probability is 3/19, or answer choice [C].

Ben is 6 feet tall. At a certain time of day, Ben's shadow appears on the ground to be 6 feet long. What is the angle of elevation of the sun at this time? A. 25 degrees B. 30 degrees C. 45 degrees D. 60 degrees E. 75 degrees

C. 45 degrees To start the problem draw the situation, ben is 6 ft so the height of the triangle will be 6 ft, the shadow is 6ft, so the base will also be 6ft. THis makes a 90° angle so the shadow should be 45° We can immediately recognize that this is one of the "special" triangles of trigonometry - the 45-45-90 isosceles right triangle:

Bill can mow a lawn in one hour and twenty minutes. If Bill and Dave can mow the lawn together in a half hour, how long does it take for Dave to mow the lawn alone? A. 30 minutes B. 36 minutes C. 48 minutes D. 50 minutes E. 90 minutes

C. 48 minutes We can use a classic rate equation here with three variables: the amount of time Person A needs to finish the job individually, the amount of time Person B needs to finish the job individually, and the amount of time needed to complete the job if Persons A and B work together. Sample equation: (1 / Person A time) + (1 / Person B time) = (1 / Person A and B working together time) In this problem, Bill can mow the lawn in 80 minutes and together, Bill and Dave can mow the lawn in 30 minutes. The equation we set up is as follows: 1/80 + 1/x = 1/30 1/x = 1/30 - 1/80 1/x = 8/240 - 3/240 1/x = 5/240 1/x = 1/48 x = 48 minutes

Half of the population of Cool Town owns a bicycle, and 25% of the population owns a car. If 10% of the population owns both a car and a bicycle, what is the probability that a person chosen at random from Cool Town owns either a car or a bicycle or both? A. 10% B. 50% C. 65% D. 75% E. 100%

C. 65% 50% owns a bike 25% own a car 10% own both To solve: 50%-10% = 40% 25%-10% = 15% 40% + 15% = 55% + 10% = 65%

The price of wheat increases 10% in one week, then decreases 30% the next week, then increases 20% the following week. What is the net change in the price of wheat? A. 0% B. 5% C. 7.6% D. 8.2% E. 10.8%

C. 7.6%

How many different ways (orders) can you arrange 6 different playing cards? A. 6 ways B. 36 ways C. 720 ways D. 1,296 ways E. 46,656 ways

C. 720 ways answer is simply 6!

Sarah finds herself with 9 different cheeses and 9 different wines. In how many ways can she pair one cheese to one wine? A. 18 ways B. 72 ways C. 81 ways D. 9!/2! ways E. 99 ways

C. 81 ways 9 x 9 This problem can be solved with some careful thought and multiplication. Let's say you selected one certain type of cheese already. How many different bottles of wine do you have to go with that? For each type of cheese, there are 9 different combinations. Since there are 9 types of cheeses, the total number of combinations would be 9 * 9, or 81 combinations.

For which value of q would the following system of linear equations have no solution? 2x + y = 4 qx + 3y = 9 A. -6 B. -3 C. 3 D. 6 E. 8

D. 6 A quick way to solve this problem would be to multiply the second equation (2x + y) by 3 on both sides: (2x + y) * 3 = 4 * 3 6x + 3y = 12 This allows us to more directly compare this equation to the first equation: qx + 3y = 9 vs. 6x + 3y = 12 We can see that if we set q = 6, then: 6x + 3y = 9 vs. 6x + 3y = 12 Only one of these statements can be true at a time, so there is no solution that satisfies both equations when q = 6. Answer choice [D] is correct.

Solve for x: 1) y = sqrt x 2) y = sqrt -x A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. The square root function is only defined when the radicand (√) is greater than or equal to zero. Knowing this, let's assess the statements: Statement 1 → x must be a non-negative (zero or a positive) number Statement 2 → x must be a non-positive (zero or a negative) number Only one number, zero, is both non-negative and non-positive, and so our answer must be [C] BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Consider a parallelogram with perimeter 36. What is its area? 1) One side is twice as long as one other 2) One angle is 80º A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. We are given the perimeter of the parallelogram which is 36. Equation for area of parallelogram is: Base x Height = area or L1 x L2 x sinø If we look at statement 1 alone, it tells us that one side is twice as long as the other one. Lets says L1 is the left and right sides and L2 is our base and top. The equation for the perimeter now would be 2L1 + 2L2 = 36 Since it says L2= 2 x L1 we can plug that in and solve for L1 2L1 + 2(2 x L1) = 36 6L1 = 36 *L1 = 6 so L2 must equal 12* *statement 1 alone does not give us enough info to solve for area Looking at statement 2 alone, it also doesnt give us enough info to solve for area, it only gives us ø. When we combine statements 1 and 2, we have enough info to solve for area of the parallelogram using L1 x L2 x sinø = area*

Is x a prime number? 1) x is a factor of 56. 2) x is a two-digit number. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. We are trying to determine whether x is a prime number. Statement (1) tells us it is a factor of 56. That gives us the following possibilities for what x could be: 1,2,3,4,8,14,28,56 Some of these numbers are primes and some are not, so Statement (1) is insufficient. Statement (2) on its own is also insufficient, as there are plenty of two-digit numbers which are and are not primes. However, only three of the factors of 56 are two-digit: 14, 28, and 56. All three of these numbers are not prime, and so we can determine that x is not a prime number. Since it took both statements to reach this conclusion, our answer must be [C] BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Set X contains all numbers divisible by 10 between 1 and 100. Set Y contains all numbers divisible by 12 between 1 and 100. Quantity A: the intersection of the two sets Quantity B: 60 A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

C. The two quantities are equal. The intersection of the two sets is 60 because Set X, 60 is divisible by 10, and in Set Y, 60 is divisible by 12. So Quantity A is equal to Quantity B

JIm has $4 With $4, jim can buy exactly 16 apples 4 apples cost $0.20 more than 5 oranges Quantity A: the number of oranges Jim can buy Quantity B: 25 A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

C. The two quantities are equal. To solve this problem, we can first determine the cost of an apple: $4 = 16x x = (4/16) = 0.25 Each apple costs at least $0.25, so 4 apples would cost $1.00. Now, we can find the cost of each orange: 4 apples = $1.00 = 5 oranges + $0.20 5 oranges = $0.80 orange = $0.16 Since each orange costs $0.16, Jim can buy $4 / $0.16 per orange = 25 oranges

y = x^2 - 4x + 4 Quantity A: the root of the equation Quantity B: 2 A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

C. The two quantities are equal. To solve this problem, we should first factor the given equation. *Remember that the root of an equation is a solution to the equation when f(x) = 0. So you set y equal to 0 to solve:* y = x^2 - 4x + 4 y = (x - 2)(x - 2) *x = 2) Therefore, Quantity A is equal to Quantity B and answer choice [C] is correct.

The average of a large data set is 60. Which of the following must be greater than or equal to 60? A. mode B. median C. mean D. median and mean E. mode and mean

C. mean For example, consider the following small data set: {1, 1, 1, 1, 1, 355} The median of the data set is 1 because it is the middle number. the mean will still be 60 The mean of this data set is 60, but the median is 1. We could come up with a number of data sets that have averages of 60 with medians that are less than, equal to, or greater than 60; therefore, since this question asks which MUST be greater than or equal to 60, the median is not included. Answer [D] is incorrect. The mode is the value that occurs most often in the data set. If there are not any numbers in the data set that are repeated, there is NO mode. We can use the example above to determine if the mode is a correct answer for this problem. The mode of that data set is clearly 1, which is less than 60. The mode has no direct correlation upon the mean or median, so answer [E] is incorrect. We are left with only answer choice [C] as the correct response. The mean is equal to the average!

In a right triangle, one acute angle is 22 degrees less than 3 times the other acute angle. What is the measure of the larger acute angle? A. 18 degrees B. 46 degrees C. 56 degrees D. 62 degrees E. 70 degrees

D. 62 degrees Let's start by writing an expression to represent the question stem. We will use x to represent one of the acute angles: x + (3x - 22) = 90 We can set this equation equal to 90 because one of the angles in the triangle is a right angle, leaving 90 degrees total for the other two angles. Now, we can solve for x: x + (3x - 22) = 90 4x = 112 x = 28 The measure of the other acute angle will be: 3x - 22 (3 * 28) - 22 84 - 22 = 62 degrees Answer choice [D] is correct.

Is a even or odd? 1) a = 2x, for some positive whole number x. 2) a = 3y, for some positive whole number y. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D. EACH statement ALONE is sufficient. Statement 1 tells us "a = 2x, for some positive whole number x". This means one of the factors of a is 2, and any number with 2 as a factor must be even, so Statement 1 is sufficient for telling a is even. Statement 2 tells us "a = 3y, for some positive whole number y". This means none of the factors of a are 2, as 2 is not a factor of 3, and any number without 2 as a factor must be odd, so Statement 2 is sufficient for telling a is odd. As either statement could sufficiently tell us whether a is even or odd, our answer must be [D] EACH statement ALONE is sufficient.

If "m + 1 = n", what is the value of "m * n"? 1) n/m = 2 2) 2m * 2n = 8 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D. EACH statement ALONE is sufficient. Statement 1 tells us "n/m = 2". There are many numbers that would satisfy this relation, but only the numbers 1 and 2, for m and n respectively, would also satisfy "m + 1 = n". And so, we know the value of "m * n" equals 2, and we can say Statement 1 is sufficient. Statement 2 tells us "2m * 2n = 8". We can rearrange this to read as "2 * 2 * m * n = 8", or "4 * m * n = 8". Dividing both sides by 4 then gives "m * n = 2", and so we can say Statement 2 is also sufficient. As either statement was sufficient in answering the given question, our answer must be [D] EACH statement ALONE is sufficient.

Three dice are rolled. What is the probability that all three dice come up with the same number? A. 1/216 B. 1/108 C. 1/72 D. 1/36 E. 1/2

D. 1/36 If you roll 3 dice, the probability of the first die being any number 1 through 6 is 1/1 = 1. Then, the probability of getting that same number on die #2 is 1/6 and getting that same number on die #3 is also 1/6. ( 1/1)(1/6)(1/6) = 1/36

What is the probability of pulling two cards from the same suit in a row, with each card coming from a separate 52 card deck? A. 1/32 B. 1/16 C. 1/8 D. 1/4 E. 4/13

D. 1/4 The probability of drawing the first card is 52/52, since it does not matter which suit is drawn. Once a card is pulled and a suit is selected, the probability of drawing the same suit from a separate deck of cards is 13/52. There are 13 cards in a suit, and since it is a separate deck of cards, all 13 cards from the suit are present. Therefore, the probability of pulling two cards from the same suit in a row is: 52/52 * 13/52 = 13/52 = 1/4

If you have 2 distinct black playing cards and 2 distinct red playing cards, how many ways can you arrange the four cards given that the red cards can never be next to each other? A. 2 ways B. 4 ways C. 8 ways D. 12 ways E. 24 ways

D. 12 ways We need to determine how many permutations have the red cards NOT next to each other. Let's think about the different scenarios in which the red cards are not next to each other. RBBR, RBRB, BRBR Since the black and red playing cards are all unique, there are 4 possible arrangements for the 3 scenarios (RBBR, RBRB, BRBR). To visualize this, let's write out one the possible arrangements of one of the scenarios (RBBR): R1B1B2R2 R1B2B1R2 R2B1B2R1 R2B2B1R1 These 4 possible permutations apply for all 3 scenarios, so the total number of arrangements with the red cards never next to each other is 4 * 3 = 12 ways, or answer choice [D].

How many ways can the letters in APPALOOSA be arranged? A. 2,405 ways B. 2,505 ways C. 9,050 ways D. 13,530 ways E. 15,120 ways

D. 13,530 ways This problem is a little tricky because of the duplicate letters in the word APPALOOSA. If this word did not have any repeating letters, the answer would simply be 9!. However, if we label all the A's in APPALOOSA as A1, A2, and A3 it is easy to see that A1PPA2LOOSA3 is the exact same as A1PPA3LOOSA2. Therefore, duplications prevent 9! from being the final answer. To help figure out the exact number of outcomes, it may help to rearrange the word so that all the repeating letters are together, AAAPPOOLS. In this orientation, we see that there are 3 A's, 2 P's and 2 O's. The general formula to correct repeating elements in a set is: Number of total possibilites = (maximum number of combinations without repeats / A1!, A2!, A3! .....) Where a1...ai represent the number of repeated entries for each item that repeats. Using this formula we get: Number of possibilites = (9! / 3! x 2! x 2!) = 15,120 possibilities 3! is used to correct for the 3 A's, while 2! is used to correct for the multiple P's and O's.

What is the measure of an internal angle of a regular 18-gon? A. 30 degrees B. 120 degrees C. 150 degrees D. 160 degrees E. 175 degrees

D. 160 degrees To solve this problem, we will use the formula for the sum of internal angles of a polygon: *Sum of internal angles = 180(n-2)* We can simply plug in the number of sides: 180(18 - 2) = 2880° Divide this sum by the number of angles to find the measure of each interior angle: *angle = 2880/18 = 160°*

The sum of two numbers is 168. If the larger number is divided by the smaller number, the quotient is 6 with a remainder of 7. What is the smaller number? A. 19 B. 21 C. 22 D. 23 E. 27

D. 23 Let's start by setting up the following equation. We will represent the smaller number with the variable x, and the larger number with "6x + 7" since it is divisible by the smaller number 6 times with a remainder of 7: x + (6x + 7) = 168 Simplify the expression and solve for x: 7x = 161 x = 23 Answer choice [D] is correct.

Julia took 4 courses at her college this past semester. She earned an A in Chemistry (3 credits), a B in Chemistry Lab (1 credit), an A in Spanish (4 credits), and a C in Fruit Science (1 credit). If grades are weighted such that an A is four grade points, a B is three grades points, and a C is two grade points, what is Julia's grade point average from this past semester? A. 2.89 B. 3.00 C. 3.50 D. 3.67 E. 4.00

D. 3.67 GPA = total grade points / # of credits taken total grade points = (grade points x credits) (3 x 4) + (3 x 1) + (4 x 4) + (2 x 1) / (3 + 1 + 4 + 1) (12) + (3) + (16) + (2) / (9) 33/9 = *3.67*

Train A leaves from City A traveling towards City B. Train B leaves from City B traveling the same path as Train A towards City A. City A and B are 500 miles apart. Train A travels at 100 miles per hour, while Train B travels at 150 miles per hour. When the two trains cross paths, how far has Train B traveled? A. 100 miles B. 200 miles C. 250 miles D. 300 miles E. 500 miles

D. 300 miles

The number of ants in an anthill is found to be normally distributed, with a mean of 1,000 ants and a standard deviation of 35 ants. Approximately what percent of anthills have a population between 965 and 1,035 ants? A. 5% B. 34% C. 50% D. 68% E. 95%

D. 68% This problem gives you both the mean number of ants, μ = 1,000 ants, and the standard deviation, σ = 35 ants, and asks you to determine the percent of anthills that contain between 965 and 1,035 ants. 965 is one standard deviation below the mean and 1,035 is one standard deviation above the mean, so the percent of anthills that contain between 965 and 1,035 ants is 68%, [D]. To answer this problem, you must know the 68-95-99.7 rule of normal distributions. This is explained in the picture below. In a normally distributed data set, 68% of the data lies between 1 standard deviation above and below the mean (μ ± σ), 95% of the data lies between 2 standard deviations above and below the mean (μ ± 2σ), and 99.7% of the data lies between 3 standard deviations above and below the mean (μ ± 3σ).

13 business men all have to shake each others' hands. How many handshakes will take place in total? Assume that each businessman only shakes hands with everyone once. A. 13 handshakes B. 39 handshakes C. 52 handshakes D. 78 handshakes E. 156 handshakes

D. 78 handshakes 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 78 handshakes

constanats are any letters that are not A,E,I,O,U Constantine picks two letters at random from the word CONSTANTINOPLE with replacement. What is the probability that both letters picked are consonants? A. 14/196 B. 25/196 C. 5/14 D. 81/196 E. 9/14

D. 81/196

Grace is twice as old as her sister. How old is Grace? 1) Grace was three times as old as her sister two years ago. 2) The sum of Grace's age and her sister's age equals 12. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D. EACH statement ALONE is sufficient. Our initial problem statement gives us the following equation: G = 2S where G is Grace's age and S is her sister's age. Statement 1 gives us a second equation: G - 2 = 3(S - 2) Distributing the 3(S-2) term and adding 2 to both sides gives: G = 3s - 4 We needn't solve our system of equations, but simply realize that we have two non-identical equations and 2 unknowns: our problem is solvable, so Statement 1 is clearly sufficient. Statement 2 tells us: G + S = 12 Again, this equation is non-identical to our initial equation, and so again we have 2 equations and 2 unknowns and our problem is solvable, meaning that Statement 2 is also sufficient. Since both equations give us a second, non-identical equation, which makes our problem solvable, our answer must be [D] EACH statement ALONE is sufficient.

x, y, and z are prime numbers. What values of x, y, and z fulfill x + y + z = 15? 1)x * y * z = 44 2)x * y * z = 105 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D. EACH statement ALONE is sufficient. Statements 1 and 2 both call for prime factorizations. Since every number has a unique prime factorization and luckily these numbers only have 3 prime factors, [A], [B], and [E] can be disregarded. Examining the prime factorization of each statement yields: 44 = 2 x 22 = 2 x 2 x 11 105 = 5 x 21 = 5 x 3 x 7 Even though x, y, and z are different in Statements 1 and 2, their sum up to 15 in both cases, so the correct answer is [D] EACH statement ALONE is sufficient.

x > 0 Quantity A: - 17 Quantity B: squrt(x) A. Quantity A is greater B. Quantity B is greater C. The two quantities are equal D. The relationship cannot be determined from the information given

D. The relationship cannot be determined from the information given Since we know that x is greater than zero (a positive number), we do not need to worry about taking the square root of a negative number. When we take the square root of a positive number, we get two answers: a positive number and a negative number. For example, when we take the square root of 9, we get +3 and -3. Because of this rule, Quantity B can be positive or negative. Although Quantity B can be positive or negative, we are given no other information about the magnitude of x. Because of this uncertainty, our answer is [D] The relationship cannot be determined from the information given.

-2 ≤ x ≤ 2 Quantity A: x^3 - 4x Quantity B: 0 A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

D. The relationship cannot be determined from the information given. Quantity A contains a cubic polynomial, whereas Quantity B is simply the number 0, which we can treat as a horizontal line along the x-axis. Immediately we can eliminate [C] as we know over a range of values for x a cubic function will not remain equal to a straight line. To gain better insight into the relation of the two quantities, we can attempt to factor Quantity A. x3 - 4x = x(x2 - 4) x(x2 - 4) = x(x + 2)(x - 2) Now we have the three roots of the equation: -2, 0, 2. We have a root at "x = 0", which is within our range of values for x. Since there are three total roots, we know that the sign must change on either side of "x = 0", meaning that within our range of values for x, Quantity A may be either negative or positive. Since we do not know for certain whether Quantity A will be greater or less than 0, our answer must be [D] The relationship cannot be determined from the information given.

Mr. Andrews' test consist of ten questions worth 10 points each. He also gives an extra credit question with two options for its value. The first test option is for it to be equal to the student's average individual question score. The second test option is for it to be equal to the student's median individual question score. A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

D. The relationship cannot be determined from the information given. The crux of this question concerns the relationship between a median and an average. Quantity A would be larger if the average is larger than the median, and Quantity B would be larger if the median is larger than the average. However, the average and median values depend on the distribution. Thus, we need more information and answer choice [D] is correct.

A z score of 1.78 corresponds to a p value of 0.0375. In testing the null hypothesis that μ ≤ 12 at a confidence level of 90%, the sample data yields a z score of 1.78. What is the difference between the test p value and the significance level, and should the null hypothesis be rejected or does it fail to be rejected? A. 0.025; fail to be rejected B. 0.025; reject C. 0.0375; fail to be rejected D. 0.0625; fail to be rejected E. 0.0625; reject

E. 0.0625; reject Our alternate hypothesis is: μ > 12 and its a one tailed test becasue the null hypothesis is μ≤12. It says our test p value is 0.0375 To find out significance level, we simply subtract the confidence level from 100% so it would be 100 - 90 = 10% Now to find the difference between the test p value and the significance level we just subtract: 0.10 - 0.0375 = 0.0625 *Since our p value is less than our significance level, we reject the null hypothesis*

Jeff can paint a wall in 3 hours, while Will can paint the same size wall in 5 hours. If they work together and charge $40 an hour to paint walls, how much should they charge to paint 3 walls? A. $80 B. $160 C. $175 D. $205 E. $225

E. $225 First lets find how fast they can work together 1/ 3 hours + 1/ 5 hours = 1/x 5/15 + 3/15 = 1/x 8/15 = 1/x 8x = 15 *x = 15/8 or 1.875 hours for 1 wall* Next we need to determine how much they should charge: (1.875 hours/ wall) x (40$ / hour) = 75$ / wall *75$ x 3 walls = 225$*

One liter of water contains 1,000 cubic centimeters (cc) of water. How many cubic millimeters (mm3) of water make up one liter? A. 100 mm3 B. 1,000 mm3 C. 10,000 mm3 D. 100,000 mm3 E. 1,000,000 mm3

E. 1,000,000 mm3 10 x 10 = 100 100 x 10 = 1000 1000 x 1000 = 1000000

How many distinct amounts of money can be made from any combination of 1 quarter, 3 dimes, and 1 nickel? A. 8 amounts B. 9 amounts C. 10 amounts D. 11 amounts E. 12 amounts

E. 12 amounts

McDowell's sells x burgers at a price of $10. How much would 2 burgers cost at McDowell's? A. x/20 B. x/5 C. 5/x D. 10/x E. 20/x

E. 20/x (10 dollars / x burgers) x 2 burgers = 20 dollars / x burgers

If the area of square ABCD with side length 8 is equal to the area of triangle XYZ with height 5, then how long is line ZX in the triangle? The figures are not drawn to scale. ZX is the base of the triangle A. 5.0 B. 11.8 C. 12.8 D. 14.6 E. 25.6

E. 25.6

A farmer redistributes his cows among three pastures. The first has 10 cows less than half of the total, the second has 8 cows less than one third the total, and the third has 6 less than one fourth the total. How many cows does he have? A. 58 cows B. 80 cows C. 166 cows D. 254 cows E. 288 cows

E. 288 cows Let's start by writing an equation based on the question stem. We will set the total number of cows as x: ((x/2) - 10) + ((x/3) - 8) + ((x/4) - 6) = x Simplify the expression: (x/2) + (x/3) + (x/4) - 10 - 8 - 6 = x (x/2) + (x/3) + (x/4) - 24 = x (x/2) + (x/3) + (x/4) - x = 24 Now, we can combine the fractions by finding a common denominator: (6x/12) + (4x/12) + (3x/12) - (12x/12) = 24 (13x/12) - (12x/12) = 24 x/12 = 24 Solve for x: x = 24 x 12 x = 288

An ellipse is defined by the equation x2/9 + y2/16 = 1. In the diagram, point A is at the intersection of the minor axis and the ellipse, and point B is at the intersection of the major axis and the ellipse. What is the length of the line segment AB? A. 2 B. 3 C. 3√2 D. 4 E. 5

E. 5 To find the length of line AB, we should construct the following right triangle: ( its oval shaped and the A axis is the longer axis (height), the B axis is the shorter axis (width)) We can see that the longer leg of this triangle will be equal to half the length of the major axis (a), and the shorter leg of the triangle will be equal to half the length of the minor axis (b). We can look at the standard formula for a vertical ellipse and compare to the given equation: (y^2 / a^2) + (x^2 / b^2) = 1 And see that a will be 4 and b will be 3. Therefore, the right triangle is one of the "special" right triangles (the 3-4-5 right triangle), and the length of AB will be 5, or answer choice [E].

A tin can is a closed cylinder of radius 3 centimeters and height 9 centimeters. If the top of the can was removed to get at the contents inside, what fraction of the surface area of the can is left? A. 1/8 B. 1/2 C. 5/8 D. 3/4 E. 7/8

E. 7/8 To solve this problem, we can start by finding the surface area of the intact can using the formula for the surface area of a cylinder: *A= 2πrh + 2πr^2* A = 2π(3)(9) + 2π(3)^2 A = 2π(27)+ 2π(9) A = 54π + 18π *A = 72π* Now, we can find what fraction of the surface area of the can was lost when the top was removed. The surface area of the top is: *A = πr^2* A = π(3)^2 A = 9π Therefore, 63π (72π - 9π) out of the original 72π still remains. This fraction would be: *63π/72π which equals 7/8*

For which of the following could set {-100, -50, -25, 25, 50, 100} satisfy G ∪ F? A. F = {x ≥ 100}, G = {x ≤ -100} B. F = {|100|}, G = {-50, -25} C. F = {x < 100}, G = {x < -100} D. F = {x < 25}, G = {x < -25} E. F = {x ≥ 14}, G = {x ≤ -14}

E. F = {x ≥ 14}, G = {x ≤ -14} Its asking for what values of F and G union will include all the numbers in the data set. Only E includes all the data set because it includes everything greater than 14 and less than -14

Is "ax2 + bx + c" > 0? 1) ab > 0 2) bc > 0 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient. For our parabola to be concave up and above the x-axis, the coefficient (a) of our quadratic term (x2) must be positive and the coefficient (c) must be greater than or equal to zero respectively. Statement 1 does not assure us that "a" is positive and fails to mention "c". The product of "a" and "b" can be greater than zero if both values are negative. Statement 2 does not assure us that "c" is positive with an identical reasoning and fails to mention "a". Since we cannot tell the sign of our "a" nor our "c" coefficients from our question statements, we cannot determine whether our parabola lies above the x-axis (from using the statements), so our answer must be [E] Statements (1) and (2) TOGETHER are NOT sufficient.

There are 20 people in a room under 40 years old. Is their median age over or under 30 years old? 1) Three people are under 25 years old. 2) Nine people are 31 years old. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient. If we consider both statements together, there are 8 people whose ages we know nothing about. If they are all over 31, then our median is 31. If they are all under 31, then we do not know our median. If they are between 25 and 30, then our median is under 30. Since our median could be over or under 30, and the statements do not clarify which is the definite answer, our answer must be [E] Statements (1) and (2) TOGETHER are NOT sufficient.

Let x and y be the coordinates of a point on the perimeter of a circle centered at the origin. What is the value of x + y? 1) x/y = 1 2) x2 + y2 = 4 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient. Since we are on the perimeter of a circle, any point must satisfy the relation: x^2 + y^2 = r^2 So, from statement 2 we know the radius of the circle (radius = 2). Statement 1 tells us that x and y have the same value. This may appear sufficient, but in fact there are two points on a circle with the same value of x and y, one in the first quadrant and the other in the third. As such, all we know is: 2x^2 = 4 x^2 = 2 x and y = squrt 2 or -squrt 2

If K represents the set of negative integers greater than or equal to -5, and L represents the set of integers less than -2, which of the following represents the set K ∩ L? A. {..., -5, -4, -3} B. {-2, -1, 0, ...} C. {-5, -4, -3, -2, -1} D. {-4, -3, -2} E. {-5, -4, -3}

E. {-5, -4, -3}