missed quantitative problems

Samuel borrowed $5,000 to help get his business started. He took out a 9-month loan with simple interest, based on an annual interest rate of 6 percent. What was the total amount Sam was required to pay back at the end of the loan period?

$5,225

Which of the following is equivalent to ((6+3)^2(11−2)^2)^1/4?

(5 + 4)(15 - 6)^1/2

Which of the following are equal to (1/4)^4 (1/2)^2 (1/8)^2 (1/16)^2 8^-2 4^-4 (1/2)^8

(C), (E), and (F)

Which of the following does 1/5 of 0.03 percent equal?

0.00006 One percent means 1/100. So 0.03 percent means 0.03 times 1/100. Since the answer choices are decimals, let's work with decimals. Then one percent means 0.01. The decimal equivalent of 1/5 is 0.2. So 1/5 of 0.03 percent is 0.2 × 0.03 × 0.01. Let's begin by multiplying out 2 × 3 × 1. We have that 2 × 3 × 1 = 6 × 1 = 6. There is one place to the right of the decimal point in 0.2, there are two places to the right of the decimal point in 0.03, and there are two places to the right of the decimal point in 0.01. So there will be 1 + 2 + 2 = 5 places to the right of the decimal point in 0.2 × 0.03 × 0.01. So 0.2 × 0.03 × 0.01 = 0.00006. Choice (B) is correct.

Partha is accessorizing for a costume party. She wants to wear 5 different bangles and 2 different necklaces. If she has 9 different bangles and 6 different necklaces to choose from, how many different combinations of 5 bangles and 2 necklaces are possible?

1,890 combinations problem

Jim can paint a room in 3 hours, Steve can paint an identical room in 6 hours, and Jackson can paint such a room in x hours. Working together, Jim, Steve, and Jackson can paint 4 such rooms in 3 hours. What is the value of x ?

1.2 If Jim can paint a room in 3 hours, his painting rate can be expressed as 1room/3hrs Similarly, if Steve can paint the room in 6 hours, his painting rate can be expressed as 1 room / 6 hours And, if Jackson can paint the room in x hours, his painting rate can be expressed as 1 room / x hrs If all three people can paint 4 identical rooms in 3 hours when working together, then, as a group, they paint 4 rooms / 3 hours 1/3+1/6+1/x=4/3 To solve for x in your equation above, start by clearing both sides of the fractions by multiplying by a common multiple of all of the denominators, such as 6x ----> 1.2

In the figure above, ABE is an equilateral triangle, BCDE is a square, and the area of the circle is 36π. What is the area of polygon ABCDE ?

108+27 (3^1/2)

Geometry question. In the figure above, what is the value of y ?

110 Because AB = BC, the angles opposite these equal sides are also equal. So, ∠BAC = ∠BCA. Since ∠BCA is supplementary to the 125° angle, its measure is 180°- 125° = 55°; ∠BAC also equals 55°. Because the sum of the interior angles of a triangle is 180°, ∠ABC = 180°- 55°- 55° = 70°. Finally, y is supplemental to ∠ABC, so y = 180°- 70° = 110°, which is (D). Alternatively, you could have used the rule that every external angle of a triangle equals the sum of the two nonadjacent internal angles to find that y = 55 + 55 = 110, (D). Confirm your answer Check your logic and calculations and be certain that you used the correct values. You should have sketched the figure on your scratch paper to keep track of values as you calculated them. If you drew it to scale, starting with a 125-degree exterior angle on the left, you can eyeball the triangle to see that y is somewhat greater than 90°, which comports with the 110° value. TAKEAWAY: Knowing all the rules for the properties of triangles can pay dividends on Test Day.

In a class of 30 female and male students, the ratio of males to females is 3 to 2. What is the number of females in the class?

12

If the area of sector ABC is 3π, what is the length of the diameter of the circle?

12 In any circle, the ratio of a central angle's measure in degrees to the circle's total number of degrees is equivalent to the ratio of the area of that circle's corresponding sector to the area of the entire circle. Set up that proportion of part-to-whole ratios, substituting the given values to solve for the value of the circle's radius. Finally, double the radius's value to obtain the diameter's. measure of central angle/360°=area of corresponding sector/area of circle 30°/360°=3π/πr2 1/12=3/r^2 r^2=36 r=6 d=2r=(2)(6)=12

Every student in a room is either a junior or a senior. There is at least one junior and at least one senior in the room. If 1/4 of the juniors is equal to 3/5 of the seniors, what fraction of the students in the room are juniors?

12/17

How many different committees of 7 people can be formed from a group of 10 people?

120

Factories A, B, and C produced a total of x widgets. Of these x widgets, 15 were produced by factory A, 47 were produced by factory B, and the remaining 400 were produced by factory C. The fraction of the widgets produced by factory A that were defective was 170, and the fraction of the widgets produced by factory B that were defective was 140. If a widget is selected at random from the x widgets, the probability that it is defective is 135. How many of the widgets produced by factory C were defective?

20

A phone card charges a flat fee for the first minute of a phone call, plus an additional charge per each extra minute of the call. For a caller using the phone card, an 11-minute call costs 55 cents, and a 23-minute call costs 97 cents. How much does the phone card charge for the first minute of a call?

20 cents translate the given information into two equations. Call the cost of the first minute F and the cost per minute after the first minute r. Since an 11-minute call costs 55 cents, the first equation will be this: 55 = F + 10r The coefficient for r is 10 instead of 11, since the rate r only applies to the 10 minutes after the first minute. Similarly, the 23-minute, 97-cent call can be represented by this equation: 97 = F + 22r Using combination, subtract the first equation from the second to get 42 = 12r, or r = 3.5 cents. If r = 3.5, then plug this back into the first equation to get 55 = F + 10(3.5) 55 = F + 35 20 = F Alternatively, you could solve by sidestepping the equations and reasoning this way. The extra 12 minutes between the two calls cost an extra 42 cents, so each extra minute costs 3.5 cents. Therefore, 10 minutes on the first call cost 35 cents, so the charge for the initial minute must be the total cost of 55 cents - 35 cents = 20 cents.

If x= 1/y =2, what is the ratio of x4 to y4 ?

256/1 x^4 = 2^4 = 16 y^4 = (1/2)^4 = 1/ 16

If x is divided by 7, the remainder is 1. If y is divided by 7, the remainder is 2. What is the remainder if x + y is divided by 7?

3 Identify the task Find the remainder when the sum of the two unknowns is divided by 7. Approach strategically To solve this question quickly, pick numbers for x and y that satisfy the description. x divided by 7 has a remainder of 1. That means x may be any number that is 1 greater than a multiple of 7. The simplest choice is to let x = 8. Likewise, when y is divided by 7, the remainder is 2. That means y may be any number 2 greater than a multiple of 7. The simplest choice for y is 9. 8 + 9 is 17, and dividing by 7 gives 2 with a remainder of 3. Remember that when picking numbers to test number properties, only one set of numbers need be picked; if it didn't work for all possible values of x and y, the question couldn't be asked. Confirm your answer The correct answer is 3. TAKEAWAY: Pick numbers to make number properties questions more concrete.

The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. If 4 disks are to be selected one after the other, with each disk selected at random and without replacement, what is the probability that the range of the numbers on the disks selected is 7 ?

3/14 The probability formula is Probability=Number of desired outcomes/Number of possible outcomes The range of a group of numbers is the greatest number minus the smallest number. If the range is to be 7, then we want 2 of the numbers to differ by 7 and the other 2 numbers to both be between the 2 numbers that differ by 7. Thus, if we select 1 and 8, the other 2 numbers can be 2 different numbers among 2, 3, 4, 5, 6, and 7. If we select 2 and 9, the other 2 numbers can be 2 different numbers among 3, 4, 5, 6, 7, and 8. If we select 3 and 10, the other 2 numbers can be 2 different numbers among 4, 5, 6, 7, 8, and 9. nCk is used to denote the number of different subgroups of k different objects that can be selected from a group of n different objects, where n is a positive integer, k is a nonnegative integer, and 0 ≤ k ≤ n. The combinations formula is Ck n=n!/ k!(n−k)!

At Deb's Deli, a customer may choose either a sandwich and a salad or a sandwich and a soup for the lunch special. There are 5 choices of sandwich, 4 choices of salad, and 3 choices of soup. How many possible lunch special combinations can be ordered?

35

In the rectangular solid shown, PQ = 5, QR = 12, and the volume is 480. What is the total surface area of the solid, in square units?

392 rectangular solid volume = LxWxH surface area = 2(WxL)+ 2(WxH) + 2(LxH)

Jasmine drives the first 150 miles of her trip at an average speed of 50 miles per hour. If she drives the remaining 80 miles of her trip at an average of 40 miles per hour, what is her average speed, in miles per hour, for the entire trip?

46 The average speed formula is Average speed=Total distanceTotal time. The total distance can be calculated from , so you'll need to determine the total time by finding the time for each part of the trip: for the first part, 150 miles/50 mph=3 hours and for the second, 80 miles/40 mph=2 hours. So her average speed overall was 230 miles/5 hours=46 mph. That's (C).

How many different lists containing each of the numbers 1, 4, 5, 8, 17, and 21 exactly once, and nothing else, are there in which every odd integer appears before any even integer?

48 There are 4 odd integers in the list (1, 5, 17, and 21) and 2 even integers (4 and 8). An acceptable list must have the 4 odd integers appearing before the 2 even integers. The odd integers will be arranged in the first four positions, in any order. The number of ways to arrange four items is 4! = 4 × 3 × 2 × 1 = 24. For each of those arrangements, the even numbers will be arranged in either of 2 ways (4, 8 or 8, 4). So, the total number of arrangements is 24 × 2 = 48. That makes (C) correct.

The cities of Hickory and Deerlick are connected by two straight parallel train tracks. The cities are 150 miles apart. Train A leaves Hickory for Deerlick at 12:00 p.m., and Train B leaves Deerlick for Hickory on the same day at 12:40 p.m. Both trains move at a constant speed for the duration of the journey. Train A travels at a constant speed of 45 miles per hour, and the trains pass each other at 2:10 p.m. At approximately what time will Train B arrive at Hickory?

4:57 p.m. In order to find when Train B arrives at its destination, you'll need to know its speed. Its starting time is known, as well as the time it passes Train A. In order to find its speed, you'd need the distance between those two points, which is not given. It can, however, be determined from what's known about Train A.

If x > 0 and x2 + 6x + 6 = 22, what is the value of x + 3?

5 The equation is a quadratic, so set the equation equal to zero and see if it can be factored. x2 + 6x + 6 = 22 x2 + 6x + 6 - 22 = 22 - 22 x2 + 6x - 16 = 0 Look for two numbers whose product is -16 and whose sum is +6. (x + 8)(x - 2) = 0 When the product of two factors is zero, at least one of the factors must be zero. Set each of these factors equal to zero in turn to find that the roots of the equation are -8 and 2. Since the question says that x > 0, you want the positive root, 2. If x = 2, then x + 3 = 5. Confirm your answer Make sure you answer with the value of x + 3, not x itself. TAKEAWAY: In a numeric entry question, it's particularly important to make sure you're answering the right question, since you don't have the answer choices to guide you.

There are 35 distinct numbers in set M, there are 28 distinct numbers in set N, and there are 12 distinct numbers that are in both sets M and N. Set H is the set containing the elements that are in at least one of sets M and N. How many elements are in set H ?

51 Total number of elements in two sets = (Number of elements in one set) + (Number of elements in the other set) - (Number of elements in both sets) Here, the number of elements in set M is 35, the number of elements in set N is 28, and the number of elements in both sets M and N is 12. Then the number of elements in set H is 35 + 28 - 12 = 63 - 12 = 51.

If in a sequence the term an+1=(2^an)+1 and a1=1, what is the value of a4 ?

513

If x2 + 25y2 = 274 and y=21/x, then what is the value of (x - 5y)2 ?

64 You might begin by expanding (x - 5y)2 to make it look more like the equation given in the question stem: (x - 5y)2 = x2- 10xy + 25y2. Multiply both side of y=21/x by x to eliminate the fraction and find that xy = 21. This value can be substituted into the expression for xy, so it becomes x2- 10(21) + 25y2 = x2- 210 + 25y2. Rearrange this as x2 + 25y2- 210. The equation in the question stem says that x2 + 25y2 = 274, so substitute 274 for x2 + 25y2 in the expression to get 274 - 210 = 64. (C) is correct.

Triangle ABC is a right triangle in which angle ABC measures 90 degrees. D is a point on AC and E is a point on BC such that DE is parallel to AB. If EC = 2 and BE = 4, what is ratio of the area of triangle ABC to the area of triangle DEC?

9:1 Since DE is parallel to AB, angle DEC corresponds to angle ABC, so DEC is also a right angle. Angle BAC corresponds to angle EDC, so those angles are also equal. Finally, triangle ABC and triangle DEC share angle C. Therefore, all the angles are equal, so the triangles are similar. In similar triangles, the lengths of corresponding sides are all have the same ratio to each other. BC, the base of triangle ABC, is 6, while EC, the base of triangle DEC, is 2. Thus, all the side lengths in triangle ABC are 3 times as long as the corresponding sides in triangle DEC. Even though you cannot determine the area of triangle ABC or DEC, you can determine the ratio of the areas.The formula for the area of a triangle is A=1/2bh The lengths of the bases are given as 2 and 6; if the height of the smaller triangle is h, then the height of the larger one is 3h. Now you can set up the ratio of the two areas: (1/2(6)(3h)) / (1/2(2)(h)) = > 9 /1 thus d is correct Alternatively, you could pick a number for AB. If AB = 9, then DE = 3. Now you can calculate the areas of triangles ABC and DEC and compare them. Triangle ABC has an area of 27, and triangle DEC has an area of 3, which simplifies to a 9:1 ratio.

A standard deck of 52 cards is shuffled, and 2 cards are drawn, one after another without replacement. A standard deck has 4 aces. It also has 12 face cards: 4 kings, 4 queens, and 4 jacks. For the first draw all of the aces are removed. For the second draw the aces are returned to the deck and all of the face cards are removed. What is the probability that the first draw is a face card and the second draw is an ace?

Analyze the question There is a deck of 52 cards with 4 aces and 12 face cards. There were no aces in the deck for the first draw, and no face cards for the second draw. Cards are not replaced when drawn. The answer choices are probabilities expressed as fractions. Identify the task Determine the likelihood that the first draw was a face card, and the second draw was an ace. approach strategically To figure out the likelihood of two events occurring, determine the probability for each, then multiply those probabilities together. Keep in mind that with each card drawn, the number of cards in the deck goes down by one. The probability formula is probability =number of favorable outcomes / number of all possible outcomes. Because the 4 aces were removed before the first draw, the total number of cards is 52 − 4 = 48. The desired outcome is a face card, of which there are 12. Therefore, the probability of drawing a face card on the first draw is 12/48=1/4. Now there are 47 cards left in the deck. Before the second draw, the 4 aces are returned and the 11 face cards are removed, which means there are 47 + 4 − 11 = 40 cards total in the deck for the second draw. The desired outcome is an ace, of which there are 4. Therefore, the probability of drawing an ace on the second draw is 4/40=1/10, since on the second draw, there was one fewer card in the deck. (Note that the question stem says that an ace wasn't drawn first, so there's no need to worry about there possibly being one fewer ace in the deck.) Multiply the two probabilities: 1/4×1/10=1/40. (C) is the correct answer.

The grass in a city park can be mowed by 5 gardeners in 6 hours. Working at the same rate, how many hours would it take 8 gardeners to mow that same grass?

Analyze the question This is a combined work problem. The question gives the rate at which 5 gardeners can mow the grass in a park. Identify the task Find out how long it would take 8 gardeners to mow the grass at the same rate. Approach strategically Since 5 gardeners complete the task in 6 hours, it takes a total of 30 gardener-hours to cut the grass, which means it would take 1 gardener 30 hours to mow the lawn. Divide 30 by 8 to determine how long it will take 8 gardeners to complete the task: 30/8=3 & 3/4. So (B) is correct. TAKEAWAY: When a combined work problem tells you how much time some number of people takes to do a job, a good first step is often to find the rate for one person. Then use that rate to find the time for a different number of people.

The value of Ms. Smith's house is $10,000 more than the value of Mr. Jackson's house and $40,000 more than the value of Mrs. Cooper's house. Quantity A The average (arithmetic mean) of the values of the houses Quantity B The median of the values of the houses

B This problem can seem a bit confusing, as it doesn't provide any actual values for the houses, so let's solve it by Picking Numbers. If the value of Ms. Smith's house is $100,000, then the value of Mr. Jackson's house is $100,000 - $10,000 = $90,000, and the value of Mrs. Cooper's house is $100,000 - $40,000 = $60,000. The median value of the 3 houses is the middle number, or $90,000. If this median were equidistant from $60,000 and $100,000, it would equal the average. However, the median is much closer to $100,000 than it is to $60,000, so the median is greater than the average. The correct answer is choice (B).

Samantha saved $100 over 5 weeks. Each week she saved $6 more than she did during the previous week. How much money did she save in the first week?

Backsolve. Choice (D) seems a little high, since even if she didn't increase her saving from week to week, $16 per week adds up to $80 total, which is already pretty close to $100. (B) is a better choice to try. If $8 is saved in week 1, the other amounts saved are $14, $20, $26, and $32. If we add these amounts together, we get $8 + $14 + $20 + $26 + $32 = $100. The correct answer is (B).

If y is an integer, which of the following must be an odd integer? y2 + 4y + 4 y2 + 3y + 8 y2 - 7y + 3 y2 - 11y - 10 y2 + 8y - 3

C Because there are variables in the answer choices, we can use Picking Numbers to determine the correct answer. Let's keep in mind that when we use the method of Picking Numbers, all 4 incorrect answer choices must be eliminated because sometimes one or more incorrect answer choices will work for the particular value that we choose. First, let's substitute 2 for y in all the expressions in the answer choices. If the resulting value is even, then we can eliminate that answer choice: (A) gives us 16. Eliminate. (B) gives us 18. Eliminate. (C) gives us -7. Leave. (D) gives us -28. Eliminate. (E) gives us 17. Leave. Next, we should use an odd number to eliminate (C) or (E). Because 1 is the smallest positive odd number, it will be the easiest to work with. (E) gives us 6. Eliminate. Choice (C) is the only remaining answer. Therefore, choice (C) must be correct.

If §n = n - 4 and ♦n = 2n, which of the following is the value of ♦(§(♦4))?

Each symbol represents a mathematical operation. To find the value of ♦(§(♦4)), apply the operation described in the question stem each time the symbol appears. In any expression with multiple functions, you must begin with the innermost set of parentheses. Since ♦n = 2n, ♦4 = 2 × 4 = 8. Next calculate (§8). Since §n = n - 4, §8 = 8 - 4 = 4. All that is left is to calculate ♦4, which we already determined equals 2 × 4 = 8. Choice (D) is correct.

In a class of 30 female and male students, the ratio of males to females is 3 to 2. What is the number of females in the class?

It is given that there are 3 male students (m) for every 2 female students (f), so there are these numbers of students for every 5 students (3 + 2) in the class (T); thus, m : f : T = 3 : 2 : 5. Since it is also given that there are 30 total students, divide that number by its corresponding piece in the ratio: 30/5=6. You can now use this multiplier of 6 to obtain any other number of students from the original ratio. Since the stem asks for the number of female students, multiply 6 by 2 (f = 2 in the ratio) to get 12, which is the correct answer.

How many positive integers that are less than 741 are NOT multiples of both 4 and 7 ?

Let's first find the number of positive integers less than 741 that are multiples of both 4 and 7, and then let's subtract that number from the number of positive integers that are less than 741. Since the number of positive integers less than 741 is 740, we will subtract from 740 the number of positive integers less than 741 that are both multiples of 4 and 7. Now 4 and 7 have no common factors greater than 1. So the least common multiple of 4 and 7 is 4 × 7 = 28. Now let's divide 740 by 28 to find how many positive multiples of 28 there are that are less than or equal to 740. Now 740 divided by 28 is 26 with a remainder of 12. The positive multiples of 28 that are less than 741 are 1 × 28, 2 × 28, 3 × 28, 4 × 28, ..., 26 × 28. There are 26 positive integers less than 741 that are both multiples of 4 and 7. Let's note that 26 × 28 = 728 is the greatest multiple of 28 that is less than 741. Since the number of positive integers that are less than 741 and are multiples of 4 and 7 is 26, and the number of positive integers that are less than 741 is 740, the number of positive integers that are less than 741 and are not multiples of 4 and 7 is 740 − 26 = 714. Choice (D) is correct.

This is a general rule in triangles: the length of a side opposite a greater angle is greater than the length of a side opposite a smaller angle.

Now side AB is opposite angle ACB whose measure is 46°, while side AC is opposite angle ABC whose measure is 44°. Since angle ACB is greater than angle ABC, AB is greater than AC

If a and b are prime numbers such that a > b, which of the following CANNOT be true? a + b is prime. ab is odd. a(a - b) is odd. a - b is prime. ab is even.

Picking numbers is often a good approach to questions involving prime numbers. Seek to make each answer true by picking permissible numbers. Evaluate each choice in turn. Start with a = 3 and b = 2: (A): 3 + 2 = 5. This is prime, so eliminate (A). (B) 3 × 2 = 6. This is not odd, so keep this. (C) 3(3 - 2) = 3 × 1 = 3. This is odd, so eliminate (C). (D) 3 - 2 = 1. This is not prime, so keep this. (E) 32 = 9. This is not even, so keep this. With three choices remaining, you'll need to pick another set of numbers. Try a = 5 and b = 3 in the remaining choices: (B) 5 × 3 = 15. This is odd, so eliminate (B). (D) 5 - 3 = 2. This is prime, so eliminate (D). (E) 53 = 125. Again, this is not even, and it's the only answer you couldn't eliminate, so the correct answer is (E). TAKEAWAY: Questions that ask about number properties but present only variables are well-suited to picking numbers.

Quantity A (3^−1 + 3^−2)^−1 Quantity B (2^3)^2 / 8^2

Quantity A is greater

The function f(x) = x^2 + 3x− 4, and the function g(x)=1/2(x−2)^2−2. Both functions are graphed on the xy-plane. Quantity A The positive difference between the values of x at the x-intercepts of f(x) Quantity B The positive difference between the values of x at the x-intercepts of g(x)

Quantity A is greater.

x > 0, y > 0, and z > 0 Quantity A ( x+7 )/ y+ y /z Quantity B xz+y^2+5z / yz

Quantity A is greater.

geometry question Quantity A Length of the arc AB/Ab+CD+EF Quantity B 1/2

Quantity A is greater. Evaluate Quantity A in order to compare that value to 12. Because AB is a diameter, arc AB is a semicircle. The circumference of a circle is 2πr, so the length of the arc AB is πr. Line segments AB, CD, and EF all pass through the center of the circle, so they are all diameters with a length of 2r. So,Length of the arc AB/AB+CD+EF => πr / 6r= > π / 6≈ 3.14 / 6. thus A is slightly greater and correct

x/y = z/4 x, y, and z are positive. Quantity A 6x Quantity B 2yz

Quantity B is greater from cross multiplying the centered information, you know that 4x = yz. Multiply both sides of the equation by 2, and you have 8x = 2yz. Therefore, Quantity B, 2yz, is equal to 8x. Quantity A is only 6x, and because all the variables are positive, Quantity A must be less than Quantity B. The correct choice is (B).

geometry problem Quantity A a + c Quantity B 230

Quantity B is greater. The centered information gives us a triangle, with an interior angle at the top measuring b degrees and two exterior angles measuring a degrees and c degrees, respectively. The angle marked a° and the interior angle of the triangle adjacent to the angle marked a° make up a straight line. So the interior angle of the triangle adjacent to the angle marked a° has a measure of (180 - a)°. The angle marked c° and the interior angle of the triangle adjacent to the angle marked c° make up a straight line. So the interior angle of the triangle adjacent to the angle marked c° has a measure of (180 - c)°. The sum of the interior angles of any triangle is 180°. The interior angles of this triangle have measures of (180 - a)°, b°, and (180 - c)°. Therefore, (180 - a) + b + (180 - c) = 180. Let's work with the equation (180 - a) + b + (180 - c) = 180. (180−a)+b+(180−c) = 180 180−a+b+180−c = 180 360−a+b−c = 180 180−a+b−c = 0 180+b = a + c Thus, a + c = 180 + b. Now the centered information says that b < 47. Since a + c = 180 + b, a + c < 180 + 47, and then a + c < 227. Quantity A, which is a + c, is less than 227, while Quantity B is 230. Quantity B is greater and choice (B) is correct.

The sum of a list of positive numbers equals 4 times the average (arithmetic mean) of the numbers. How many numbers are in the list?

Recall the average formula: Average=Sum of terms / Number of terms The question asks for the number of terms, which can be called n. Let A be the average. The sum of the terms is four times that, or 4A. Plug these values in and solve for n. A=4A/n nA=4A n=4 So, there are 4 numbers in the list. Picking Numbers can be helpful. Say the list had an average of 5. The sum would be four times that, or 20. Plug those numbers in and solve for the number of terms: 5=20/n 5n=20 n=4 TAKEAWAY: You don't need to know the specific items in a list to apply the average formula.

A sandwich shop has a list of 8 different ingredients to choose from when making a sandwich. How many different types of sandwiches can be made if each sandwich has 3 different ingredients?

So, 56 different sandwiches can be made with 3 different ingredients from a choice of 8 different ingredients. The correct answer is 56. combination problem / formula

geometry question

The degree measure of the angle at point Z could be less than 60 degrees, equal to 60 degrees, or larger than 60 degrees. So side XY might be shorter than side YZ, equal to side YZ, or larger than side YZ. Since more than one relationship between the quantities is possible, choice (D) is correct.

A fair six-sided die with sides labeled 1 through 6 is rolled. Quantity A The probability that the number rolled is closer to 4 than it is to 2 Quantity B The probability that the number rolled is closer to 5 than it is to 3

The formula for the probability of an event in which all the possible outcomes have the same probability is Probability = Number of desired outcomes / Number of possible outcomes. In this case, the possible outcomes when rolling a die once are 1, 2, 3, 4, 5, and 6. So the Number of possible outcomes when rolling a die once is 6. Among the integers 1 through 6, the integers closer to 4 than to 2 are 4, 5, and 6. So the Number of desired outcomes is 3. Quantity A, which is probability that the number rolled is closer to 4 than it is to 2, is 3/6=1/2 Among the integers 1 through 6, the integers closer to 5 than to 3 are 5 and 6. So the Number of desired outcomes is 2. Quantity B, which is the probability that the number rolled is closer to 5 than it is to 3, is 2/6=1/3. Quantity A is 1/2 and Quantity B is 1/3. Quantity A is greater and choice (A) is correct.

If 7x − 2y = 8 and 3y − 6x = −5, which of the following is/are greater than x + y? Choose all that apply. −7 −1 4 8 15

The question asks which answers are greater than the value of x + y, but it does not ask for the value of x or y individually. So before we do the complex work of solving for x and y, we should first consider rearranging the equations and combining them to solve: 7x-2y=8 +-6x+3y=5 --------- -> x+y = 3 This combination just happens to give us the needed value of x + y, so no further algebra is necessary. Since x + y = 3, any answer choice greater than 3 will be correct. The correct answers are (C), (D), and (E)

(34x)(98x) > (274x)(3280) Quantity A x Quantity B 37

The relationship cannot be determined from the information given.

At noon of a certain day, when 5 pens and 3 pencils were placed in a drawer, the ratio of the number of pens to the number of pencils in that drawer became 47 to 17. Quantity A The ratio of the number of pens to the number of pencils in the drawer immediately before noon of that day Quantity B 3/1

The relationship cannot be determined from the information given.

x < 0 < y + z z≠ 0 Quantity A (y+z)/x Quantity B y/z

The relationship cannot be determined from the information given.

a > 0 and a ≠ 1. Quantity A a2−√a Quantity B 0

The relationship cannot be determined from the information given. Watch out for traps. It seems pretty obvious that a2−a√>0, after all isn't a2 always greater than a√? No! This is a classic trap, one that occurs on almost every GRE. If you only think about positive integers, you'll get the question wrong. Try thinking about what would happen if the variable was a fraction between 0 and 1, say a=14. In that case, a2=(14)2=14×14=116 and a√=14‾‾√=12. So in this case a2−a√=116−12=−716. But if a is a positive integer, say a = 4, then a2−a√=42−4‾√=16−2=14. Since Quantity A can be less than Quantity B or greater than Quantity B, the relationship between the quantities cannot be determined.

geometry question

The two quantities are equal.

|x + 3| = 4x Quantity A x Quantity B 1

The two quantities are equal.

Quantity A Area of square I Quantity B Area of square II + Area of square III

The two quantities are equal. Analyze the centered information and quantities The centered information shows three squares labeled I, II, and III. They are arranged in such a way that a right triangle is created in the space between them. The whole figure is therefore a visual representation of the Pythagorean theorem a2 + b2 = c2. Quantity A is the area of square I, while Quantity B is the sum of the areas of squares II and III. Approach strategically No values are given for the side lengths. Nonetheless, because a right triangle is involved and right triangles have defined side relationships, it may be possible to find a definite relationship between the quantities. Since the triangle in the center is a right triangle, an effective way to pick numbers would be to use the Pythagorean triple 3:4:5. So the length of a side of square I (the hypotenuse of the triangle) is 5, the length of a side of square II is 4, and the length of a side of square III is 3. In Quantity A, then, the area of square I is 5 × 5 = 25. In Quantity B, the area of square II is 4 × 4 = 16, and the area of square III is 3 × 3 = 9; the sum of the two areas is 9 + 16 = 25. The two quantities are equal, making (C) correct. TAKEAWAY: You could have picked any numbers for the two legs of the triangle and calculated to find the hypotenuse, but using a small Pythagorean triple like 3:4:5 saves you that step, and with it valuable time.

x > 1; y > 1 Quantity A (y/x) / (x^3 / y)^1/2 Quantity B (xy)^1/2

The two quantities are equal. square both sides to end up with xy and xy TAKEAWAY: When both quantities involve positive values, you can square both quantities without changing the relationship.

1/x > 1/x^2 , x doesnt equal zero Quantity A x Quantity B 1/4

The value of x2 will be positive, so it is safe to multiply both sides of the inequality by x2 and retain the inequality sign. When you multiply the inequality by x2, you get x > 1. So, the answer is choice (A). Quantity A is greater.

a, b, and c are the lengths of the sides of a triangle Quantity A 2b Quantity B a + b + c

The variable b appears in both quantities, so subtract b from both to simplify the comparison. Quantity A becomes 2b - b = b, and Quantity B becomes a + b + c - b = a + c. In all triangles, the sum of any two side lengths is greater than the third side length. So, a + c is greater than b. This can be seen by sketching a triangle; the only way to make b the same length as a + c would be to flatten out sides a and c until they formed a straight line, but then the shape would no longer be a triangle. Quantity B is greater, so (B) is correct. TAKEAWAY: For all triangles, each side length is less than the sum of the other two side lengths.

Can A is a right circular cylinder with a height of h and a diameter of d. Can B is a right circular cylinder with a height of 2h and a diameter of d. Quantity A The volume of can B as a percent of the volume of can A Quantity B 200%

The volume of a cylinder with a radius r and a height h is πr^2h. Each can has a radius of d/2. Let's call that radius r. Can A has a volume of πr^2h and can B has a volume of πr^2(2h) = 2πr^2h. The volume of can B as a percent of the volume of can A is Volume of can B/ Volume of can A×100% = 2πr2h / πr2h ×1 00% =2×100% =200%. The quantities are equal. Choice (C) is correct.

At which points does the graph of the equation x^2 - 10 = y^2 - 1 intersect the x-axis? Indicate all possible correct answers.

The y-coordinate of any point on the x-axis is 0. Knowing this allows you to eliminate answer choices (C), (D), (E), and (F) because they do not have a y coordinate of 0. Now let's substitute 0 for y into the equation x2 - 10 = y2 - 1. Then x2 - 10 = 02 - 1, x2 - 10 = 0 - 1, x2 - 10 = -1, and x2 = 9. If x2 = 9, then x = 3 or x = -3. The points at which the graph of the equation x2 - 10 = y2 - 1 intersect the x-axis are (3, 0) and (-3, 0). Choices (B) and (G) are correct.

Amanda is choosing photos to display in 2 frames. Each frame holds 4 photos. She is choosing from a number of family photos to arrange in the first frame and a number of vacation photos to arrange in the second frame. Which numbers of family photos and vacation photos would result in more than 500,000 ways to arrange the photos in the frames? Indicate all that apply.

b & c Consider choice (A) first. There are 5 different photos Amanda could put into the first picture slot of the first frame. Once she commits a photo to the first slot, there are 4 photos left, any of which could go into the second slot. That leaves 3 photos for the third slot and 2 for the last. Thus, the number of arrangements for the first frame of choice (A) is: 5 × 4 × 3 × 2 = 120 By a similar logic, the number of arrangements in the second frame is 9 × 8 × 7 × 6 = 3,024. For each of the 120 possible arrangements in the first frame, there are 3,024 arrangements for the second. Because "for each" is a multiplication keyword, multiply these two results to get the total number of possibilities: 120 × 3,024 = 362,880. This is less than 500,000, so eliminate choice (A). Now apply the same logic to the other choices: Choice (B), First Frame: 6 × 5 × 4 × 3 = 360 Choice (B), Second Frame: 8 × 7 × 6 × 5 = 1,680 Choice (B), Total: 360 × 1,680 = 604,800 Choice (C), First Frame: 7 × 6 × 5 × 4 = 840 Choice (C), Second Frame: 7 × 6 × 5 × 4 = 840 Choice (C), Total: 840 × 840 = 705,600 Choice (D), First Frame: 10 × 9 × 8 × 7 = 5,040 Choice (D), Second Frame: 4 × 3 × 2 × 1 = 24 Choice (D), Total: 5,040 × 24 = 120,960 Only (B) and (C) have more than 500,000 possibilities, so the correct answers are choices (B) and (C).

Which of the following situations have a greater than 12 percent chance of occurring? Indicate all possible correct answers. A. Tossing a fair, six-sided die numbered 1 to 6 two times and getting a number greater than 4 and then a number less than 3 B. Being the second person chosen from a group of 8 people C. Opening randomly to a page from 43 to 82 in a 150-page book D.Winning a raffle that has 49 other entrants, where each entrant is allowed one entry and there is exactly one winner. E. Picking a tile with a number from 3 to 9 out of a bag with 52 tiles consisting of four groups of tiles of different colors consecutively numbered from 1 to 13.

b,c,e probablity = desired outcomes / total outcomes

In a certain college, the ratio of freshmen to sophomores is 1:3 and the ratio of sophomores to juniors is 3:5. Quantity A The ratio of freshmen to juniors Quantity B 0.3

quant b is greater

The radius of a circle was increased by 40 percent. Quantity A The percent increase in the area of the circle Quantity B 90%

quantity A is greater.

Lines u and v form 4 angles that each measure 90 degrees. The points (−4, 7) and (8, 23) are on line u. The points (−68, −16) and (−40, t) are on line v. What is the value of t ?

t = −37

Quantity A The circumference of a circle of radius r Quantity B One-half the circumference of a circle of radius (r + 1)

unable to determine Make both quantities look alike by stating them in terms of r. Since the circumference of a circle is 2πr, that is the value of Quantity A. Quantity B is (1/2)2π(r+1)= π(r+1). Divide both quantities by π (since π is a positive number, dividing by it does not change the relationship of the quantities). Now the comparison is between 2r and r + 1. Pick numbers to see whether different relationships can be found. Use positive numbers since a circle's radius is always positive. If r = 1, the quantities are equal, but if r = 3, Quantity A is greater. Therefore, (D) is correct. TAKEAWAY: When both quantities can be stated in terms of the same variable, do so to facilitate the comparison.