Hard Math #2

rt 48+ GMAT MATH QUIZ #2 Ever wondered what it feels like to take the GMAT and score in the top 10% on the quantitative section? Below is a 37question math workout to give you the experience of the kinds of questions you are likely to see if you ace the math portion of the test. If you are aiming for a perfect score, this is a great opportunity to practice your pacing and accuracy, as well as to work out on the kind of questions ETS is throwing at its top scorers these days. If you haven’t reached the top level yet, this is a great way to stretch yourself and to practice teasing your brain on the very hardest questions out there.

4. What does (1 + m)(1 – m + m2 – m3 ) = ?

2 blueberries and the rest 3 raspberries. Chen loves raspberries, so she added 12 quarts of raspberries to the salad. If the 5 mixture is now raspberries, how many quarts 7 of fruit salad were there to begin with? (A) 9 (B) 12 (C) 15 (D) 21 (E) 25 1. A fruit salad contains

2. What is the value of (1) x = 3 y + 8

(1) m6 = 64 (2) m -3 = −

1 8

5. If P = (n)(n – 1)(n – 2) . . . (1) and n > 2, what is the largest value of integer n where P has zero as its last 6 digits and a non-zero digit for its millions place? (A) 29 (B) 30 (C) 34 (D) 35 (E) 39

2x ? 8y

(2) y = 4

6. John is choosing a number n randomly from all integers from 56 to 150 inclusive. What is the probability that the number he chooses will be one where n(n + 1) is divisible by 5? 1 (A) 5 19 (B) 95 2 (C) 5 19 (D) 94 3 (E) 5

3. Susan is looking at her cell phone bill from the last 6 months. April’s bill was four times that of all the other bills. If the median amount due for a month was $32, what was the average amount due in a month during those six months? (A) $32 (B) $48 (C) $53 (D) $80 (E) $128

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If k = 20 x + 50 y and x + y = 1 , is k < 35 ? 1 (1) y > 2

7. Is x 3 > y ? (1)

3

12.

x>y

(2) x 2 > y

(2) y > x

8. What is the remainder when 332 is divided by 4? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

13.

If m < 0 , is n < 0 ?

(1) m = n − 3 (2) m n = k , where k > 0

14. There are 7 types of pizza toppings that Al can order put on his pizza: anchovies, broccoli, extra cheese, pepperoni, eggplant, peppers, and pineapple. Al hates the combination of anchovies and pineapple, but loves any other combination of toppings. How many different combinations of 3 toppings could Al order that he likes (assuming that he orders any topping no more than once in any given combination)? (A) 70 (B) 60 (C) 50 (D) 35 (E) 30

9. If m = 2 x 5 y 7 z and both 350 and 280 are factors of m , what is the minimum value of xyz ? (A) 700 (B) 70 (C) 24 (D) 6 (E) 5

10. Let M be the maximum value and N be the x2 + y minimum value of the expression . If y a ≤ x ≤ b and c ≤ y ≤ d , then what is the value of M – N?

15. Which of the following fractions has a decimal equivalent that terminates? 43 I. 256 35 II. 150 20 III. 99 (A) I only (B) II only (C) I and II (D) II and III (E) I, II, and III

(1) c = 3 (2) a = -2, b = 5 and d = 7 11. Olivier is an abstract painter who is working on a series of paintings. If each of these paintings has three identical blue vertical stripes, two identical red vertical stripes and two identical black vertical stripes spaced evenly across a square canvass, how many distinct paintings could Olivier’s series include? (A) 5040 (B) 720 (C) 210 (D) 96 (E) 6 2

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16. If p is the sum of the integers from 75 to 150 inclusive and q is the sum of integers from 75 to 151 inclusive, which of the following 1 1 expresses + in terms of q ? p q 2q − 151 (A) 2 q − 151q 2q + 151 (B) 2 q + 151q 1 (C) 2q + 151 1 (D) 2q − 151 q − 151 (E) 2 q − 151q

20. John, Aditi, and Karim are all stamp collectors. What is the ratio of the number of stamps Aditi has to the number of stamps John has. (1) Aditi has half as many stamps as John has, and Karim has three times as many stamps as John has. 1 the size collection of what 6 Karim has.

(2) Aditi has

21. Chef Gundy is creating a new dessert that will be made from 3 ingredients. If he has 8 cookies and one flavor of sorbet to choose from, what fraction of the possible arrangements will contain the sorbet? 1 (A) 84 1 (B) 56 1 (C) 9 1 (D) 3 2 (E) 3

17. If a = 2048 and b = 432 then what is the value 1

of (ab) 3 ? (A) 48 (B) 96 (C) 216 (D) 256 (E) 432 x an integer? yz (1) y is a factor of x more than once.

18. If x , y , and z are integers, is

22. If the grocery store is 10 miles west of the mall and 8 miles south of the office park, what is the approximate distance between the mall and the office park. (A) 6 (B) 7 (C) 9 (D) 11 (E) 13

(2) All of the prime factors of z are also factors of y .

19. Bob, Hans, and Ming are running a race what is the probability that Bob will win the race? (1) The chance of Hans winning is .3. (2) Seven tenths of the time either Bob or Ming will win.

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23. Is the distance between exits 4 and 5 longer than 12 miles?

26. What is the value of x ? (1) 13x > 12 x (2) x 2 = 9

(1) One car drives the distance between the two exits in 12 minutes at a rate of 55 miles per hour.

1 , which of the following 2 expresses the third smallest value? (A) x (B) x 2 (C) x −1 3 (D) x 4 1 (E) x − 2 27. If 0 < x <

(2) Another car takes less than 13 minutes to trave l the distance going 60 miles per hour. 24. In a particular college dormitory with 130 people, 57 students watch E. R. and 48 watch Law & Order. If there are twice as many students who watch neither show as who watch both, how many students watch both? (A) 25 (B) 35 (C) 45 (D) 50 (E) 70

28.

25.

Which of the following equations best expresses the line (drawn to scale) above? (A) x = 3 + 3 y 1 (B) x = 1 − y 3 (C) x = 3 y − 3 1 (D) x = y + 1 3 (E) x = 3 − 3 y

The circles above share the same center. If the radius of the smaller circle is r and the radius of 4 the larger circle is that of the smaller one, what 3 is the area of the shaded region, in terms of r ? 16 2 (A) πr 3 (B) πr 2 7 (C) πr 2 9 1 2 (D) πr 9 1 (E) πr 2 3

29. If rectangle A has width w and length l , and w > l , what is the value of w ? (1) The area of rectangle A = 24 (2) The perimeter of rectangle A = 20 4

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30. Set B has three positive integers with a median of 9. If the largest possible range of the three numbers is 19, given a certain mean, what is that mean? (A) 22 (B) 10 (C) 9.6 (D) 9 (E) It cannot be determined from the information given

35. How many even, three digit integers greater than 700 with distinct, non- zero digits are there? (A) 729 (B) 243 (C) 108 (D) 88 (E) 77

36. What is the value of x 2 ? (1) y (2 x + y ) = 32

31. What is the product of positive integers P and Q?

(2) ( x + y ) 2 = 36

(1) 18P + Q = 367 (2) Q < 18

37. The probability that a certain coin will fall heads up is 50%. What is the probability that it falls heads up on three of six tosses? 41 (A) 6 2 35 (B) 6 2 1 (C) 6 2 21 (D) 5 2 5 (E) 4 2

32. If xy ≠ 0 , is x < y ? (1) x 4 < y 4 (2) x −3 < y −3

33. In sequence P, P4 and P5 are 11 and 9 respectively. Each term after the first two terms in sequence P is either the sum of the previous two terms if that sum is odd, or half the sum of the previous two terms if the sum is even. What is the largest possible product of P1 and P2 ? (A) 40 (B) 14 (C) 12 (D) 10 (E) 7 34. Is x > y ? (1) x 2 > y 2 (2) xy < 0

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Answer key: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

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32 + 32 + 32 + 32 + 32 + 4(32) 9(32 ) = = 48 6 6 . The correct answer is (B).

1.

This question is a good candidate for a ratio box (convert the fractions talked about in the question to ratios). There will actually be two boxes--the first for the initial amounts of berries and the second for the final amounts. The boxes look as follows: Initial Ratio Multiplier Actual Blueberries 2 Raspberries 1 Total 3 Final Blueberries Raspberries Total

Ratio Multiplier Actual 2 5 7

Then to fill in the actual columns, plug in your answer choices. If you are paying attention, there is an easy way to ballpark the answer choice—the answers are the initial total amount, so the final total is the answer choice + 12. That amount must be divisible by 7. The only answer choice for which that is true is (A), the correct one. You can test that by plugging it into the boxes. 2.

3.

Before you look at the statements, get 2x 2x your base numbers to match: y = 3 y . 8 2 To evaluate statement 1, plug in the right half of the equation from the statement into the expression in the question: 2x 2 3 y +8 = 3 y = 2 3 y +8 − 3 y = 2 8 . Therefore, 3y 2 2 statement 1 is sufficient to solve. Statement 2 gives you no information about x. The median is the middle number, and since all but one bill are the same amount, the median is the same as the bill amount of the majority bills. So there are 5 months of bills at $32 and one month at 4 x $32. The average can be found this way:

4.

The issue with statement 1 is that you don't know whether m = 2 or -2. But don't stop here. If you plug both values into the expression in the question, you will see that both result in the same value. Statement 2 tells you that m = -2 and is therefore sufficient. The correct answer is (D).

5.

This one requires some canny thinking. P is the same as the factorial function. Factorials add a factor of ten, in other words, a zero, every fifth integer. You get 6 zeroes beginning with 30! and a seventh zero with 35!, so the largest integer that has only 6 zeros is 34!, so the correct answer is (C).

6.

When n equals a multiple of 5 or one less than a multiple of 5, n (n + 1) is divisible by 5. Two- fifths of all numbers fit that description. Since our range is a round set of 95 consecutive numbers, the probability that a number chosen at random from that set will fit the criteria required by the question is 2/5.

7.

Try plugging in numbers to this yes/no data sufficiency. For statement 1, if x = 8 and y = 1, you will get a "yes". On the other hand, if you plug in x = 1/8 and y = 1/3, you will get a "no", so statement 1 is not sufficient. For statement 2, the same first pair of numbers will work, giving a "yes", and x = -8 and y = 1 give you a "no". Therefore, statement 2 is insufficient. Even together the statements are not sufficient, although the range of numbers that will give you a "no" have been significantly decreased. But if you plug in x = 1/2 and y = 1/6, for example, you do get a "no". Therefore, the correct answer is (E). In general, for yes/no questions that have inequalities with

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exponents such as this one, exceptions can almost always be found between 0 and 1. 8.

Don't panic when you see a question like this that looks like it involves ridiculous amounts of calculating! There is always a pattern--all you have to do is find it. First of all, you can cross off answers (A), (C), and (E). For (A) or (E) to be correct, 332 would have to be divisible by 4, but it can't be (do you see why?). And for (C) to be correct, 332 would have to be even, but it won't be (again, can you explain why?). Now cycle through the first few exponents of 3 to find the pattern: 30 has a remainder of 1, 31 has a remainder of 3, 32 has a remainder of 1, and 33 has a remainder of 3. So the pattern that emerges is that all even exponents of 3 have a remainder of 1 when divided by 4, and all odd exponents have a remainder of 3. Since 32 is even, the correct answer is (B).

9.

One way of interpreting the question is to realize that they are asking you to find the least common multiple of 350 and 280. To do that, first find the prime factorization of both numbers: 350 = 2 × 5 2 × 7 and 280 = 2 3 × 5 × 7 . To then find the least common multiple, just take the largest exponent of each of the factors of the two numbers and multiply the result: 2 3 × 5 2 × 7 = 1400 . So the correct answer is (B).

10.

In order to find both the maximum values of the expression, we need to know the maximum of both x and y, found in statement 2. In order to find the minimum, if x could equal 0 (implied by its range covering both negative and positive number), then we know the minimum of the expression too: no matter what y equals, the result will be 1. We know all of this from statement 2, so the correct answer is (B).

11.

Here we have a permutations question with some interchangeable elements. You should start out by finding the number of ways to arrange 7 items: 7!. Then to account for the interchangeable items, you divide by the number of interchangeable items factorial, so the result looks as 7 × 6 × 5 × 4 × 3 × 2 ×1 follows: . After you 3 × 2 × 1× 2 × 1× 2 × 1 reduce the fraction, you should get 7 × 6 × 5 = 210 or (C).

12.

As usual with yes/no questions, you want to plug in to see what is going on. Here, however, the easiest way to see what is going on in the question is to start with statement 1, and rather than plug in a legal value for y, find out what happens at the boundary of the inequality. In other words, plug in y = 1/2, even though that is not allowed by the statement. If you do, you will find that x also equals 1/2 and 1 that k = 35. Therefore, if y > , k > 35 . 2 Even if you don't see that k has to be greater than 35, you know that is either going to be greater or less than 35--either way, statement 1 is sufficient. Statement 2 follows a similar logic: pretend for a moment that x = y . If the two variables are equal, then they both equal 1/2. This brings us back to the findings of statement 1, therefore statement 2 must also be correct. This means that the correct answer is (D).

13.

This is a straightforward yes/no data sufficiency question. Pick numbers for your variables. In statement 1, if m = -1 and therefore n = 2, yo u get a "no". If m = -5 and n = -2, you get a "yes". Therefore statement 1 is not sufficient. In statement 2, to get a negative base number to result in a positive number, the exponent must be even, but could be either positive or negative (try m = -1, k = 1, and n = either 2 or -2). Therefore the statement is insufficient. The problems

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remain when the two statements are combined, so the correct answer is (E). 14.

15.

16.

149 as your target answer. Only answer 150 choice (A) gives you that result.

First find the total number of three topping combos that are possible: 7 ×6×5 = 35 . Now find out how many 3 × 2 ×1 of those combine anchovies and pineapple: A, P, and 5 choices for the third topping makes a total of 5 illegal combinations. So how many legal combinations are there? 35 - 5 = 30 or (E). The key to this question is to factor the denominators. (Don't do the long division!) Statement I's denominator is just 28 . Any exponent of 2, no matter how large, can never create a non-terminating decimal. The other two statements have 3 as one of their factors. Since that 3 does not cancel into the numerator, the fractions will never terminate. Therefore the correct answer is (A). First of all, you need to recognize that 1 1 p = q − 151 . Replace p : + . q − 151 q From here, the best way to proceed is to realize that it no longer matters what q actually equals. Instead, you can plug in whatever number you please. If you glance at your answer choices, it should become immediately apparent that there will be a lot of calculation involved if you do not choose your number well, so this is an occasion where you could break one of the rules for choosing your number and set q = 1 --it is okay to do that once you have hit an advanced level, see that using 1 will make your life significantly easier, and check all five answer choices. You will have to make sure that you don't get more than one answer choice working out, but the ease of calculation makes the risk worth taking. If you do plug in 1, you get

17.

Clearly, you don’t want to try to actually do the math here instead rewrite the equation as 3 (2048)(432) . Once you see that you are actually dealing with a cubed root you should try to factor out perfect cubes, this yields 3 (8 • 256)(8 • 54) = 3 (8 • 8 • 32)(8 • 54) . Keep factoring and you will get 3 (8 • 8 • 8 • 4 )(8 • 2 • 27) then combine those numbers that are not perfect cubes. The answer is (B).

18.

For

19.

Here both statements tell you essentially the same thing. Neither one can tell you the probability that Bob will win--only the probability that Hans will win. Therefore the correct answer is (E).

20.

Only statement 1 relates Aditi to John, so the correct answer is (A).

21.

To answer this question you need to find out how many combinations of ingredients contain the sorbet and then

x to be an integer, both y and z yz must be proven to be factors of x . Statement 1 gives you part of that, but is not on its own sufficient. Statement 2 tells you that z is a factor of y , but not necessarily of x . Together, however, the statements are sufficient. To see this, statement one can be summarized as x follows: 2 + = an integer. Statement 2 y can be rewritten as y = zn where n is an integer greater than or equal to 1. The term zn can substitute for one of the y's in statement 1, and suddenly you have what the question is asking for. Therefore the correct answer is (C).

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how many combinations of ingredients there are total. If the dessert contains the sorbet, then there are 8 types of cookies to choose from for the last 2 ingredients, so 8×7 you get or 28 combinations. To 2 ×1 find all possible combinations, you have 9 9×8×7 items to choose for three slots: 3 × 2 ×1 or 84 combinations. The fraction that will 28 1 contain sorbet then is = or (D). 84 3 22.

23.

r in the answer choices, only (C) gives you your target answer.

Don't get fooled by the 8 and 10--this is not a 6:8:10 triangle! The side length 10 is not the hypotenuse. Instead, plug the two legs of the triangle into the Pythagorean formula: 64 + 100 = 164. The square root of 164 is approximately equal to 13 (132 = 169). The correct answer is (E). Statement 1 you do not need to calculate. Since they give you the time and the rate, we can find the distance. Either that distance will be longer or shorter than 12 miles, but either way, we will have an answer to our question. Statement 2, on inspection, tells us that the distance is less than 13 miles, but that does not tell us whether or not the distance is less than 12 miles. Therefore the correct answer to the question is (A).

24.

This question deals with the groups formula: Total = Group 1 + Group 2 Both + Neither. Once you fill in the parts of the formula you are given, plug in your answers. Answer choice (A), the correct answer, gives you 130 = 57 + 48 - 25 + 50, which works out correctly.

25.

Plug in. For example, if the radius of the small circle is 3, the radius of the larger one would be 4. The area of the shaded region in that scenario would be 16π − 9π = 7π . When you plug 3 in for

26.

Statement 1 tells you that x is positive, but not what it equals. Statement 2 tells you that x = ±3 . Together you have enough information to solve. Therefore, the correct answer is (C).

27.

Plug in. If you chose x =

28.

You could figure out what equation expresses and then translate all of the answer choices into the slope-intercept formula, but that would take a long time. The fastest way to do this question is to do POE by plugging in the points you do know. When y = 0, x must be positive, so that eliminates (C). When x = 0, y must be positive, so that eliminates (A) and (D). Now, the y-intercept is less than the xintercept. In (B), the y- intercept = 3, the x- intercept = 1, so that can't be the right answer either. Therefore (E) is the correct answer.

29.

Statement 1 can be translated into the equation wl = 24 . Because there are two variables and only one equation, it is not sufficient. For the same reason, statement 2 alone is not sufficient: 2 w + 2l = 20 . Together, however, it is possible to solve. This is not as obvious as it would be for two linear equations, because when you combine the two equations you get a quadratic (one way of combine the two equations results in 0 = w 2 − 10 w + 24 ), so there are two possible values for w --6 and 4. But only one of these, 6, is allowed, because of the restriction w > l . Therefore, the correct answer is (C).

1 , then (A) is 4 1 1 3 , (B) is , (C) is 4, (D) is , and (E) 2 8 16 1 is − . Therefore the correct answer is 4 (D).

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30.

Plug in your answers here. Don't start with (C), because 9.6 is an annoying number to calculate with. Start with (B) instead. If the mean is 10 and the median is 9, what would the largest possible range of the three integers be? To find that, our three integers must fit into the equation a+b+c = 10 . The median, b , equals 9, 3 so a + c = 21 . The range is defined as c − a , to make c − a as large as possible, given that a + c = 21 , we can set a = 1 and c = 20 . That does give us a range of 19, so (B) is the correct answer.

31.

Statement 1 on its own clearly will not give us the value of PQ, nor will statement 2 on its own. Together, however, since we know that both are positive integers, the statements are sufficient. Since Q < 18, the first expression defines P as the number of times 18 goes into 367 and Q as the remainder. The correct answer is (C).

32.

33.

and the third term or by summing the two terms and cutting them in half. Since it is impossible to add a positive integer to 11 to get 9, we must be adding an integer to 11 to get 18. Therefore the third term must equal 7. Your scratch paper should now look like this: _ _ 7 11 9. Now, the second term + 7 must equal either 22 or 11. If it equaled 22, the second term would have to be 15, but then there would be no legal possible value for the first term, so the two terms must sum to 11. Therefore, the second term is 4. Your scratch paper now looks like _ 4 7 11 9. Then by the same logic used to this point, the first term equals either 10 (to sum to 14) or 3 (to sum to 7). The largest possible product of the first two terms is therefore 40 or answer choice (A).

When you are dealing with inequalities and exponents to two major areas you need to test for exceptions are negative numbers and numbers between 0 and 1. Statement 1 is insufficient, because negative values for the variables will give you the opposite result as positive numbers. In Statement 2, negative numbers retain their sign, but we need to look closely at how fractions react. If you plug in x = 2 and y = 1 , that gives you a 1 1 "no". If you plug in x = , y = , you 2 4 also get a "no". Therefore statement 2 is sufficient and the correct answer is (B). This question looks really intimidating, but if you stick with it, it isn't all that hard. On your scratch paper to begin with, you should draw the set-up to look something like this: _ _ _ 11 9. You are told that, by the rules of the sequence, the fifth term, 9, is found either by summing 11

34.

Statement 1 does not tell us which variables are positive and which are negative and is therefore not sufficient. Statement 2 tells us one of the terms is positive and the other nega tive, but not which one is which. Together, the statements do not resolve the issue, so the correct answer is (E).

35.

This question is a royal pain, no question. The best way to approach it is to split up the problem, and treat the 700's, 800's and 900's separately. To find the number of numbers beginning with a 7 that fit our criteria, we have 7 choices for our tens digit (excluding 7, because no repeats are allowed, 0, and whichever even number is used as the units digit) and 4 choices (all non-zero, even digits) for our ones digit, giving a total of 28 choices. The 900's work exactly the same way, because its an odd hundreds digit as well. For the 800's, we have 7 choices for the tens and only 3 for the ones (8 is no longer a choice for the ones column). That leaves us with at total of 21 legal numbers beginning with an 8. Add 28 + 28 + 21 = 77, so the correct answer is (E).

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36.

Neither statements 1 nor 2 are sufficient on its own because we have no idea how large y is, so we can eliminate answers (A), (D) and (B). If you multiply statement 1 out, you get 2 xy + y 2 = 32 , which, if you note, is most of the square of a sum (i.e. statement 2): ( x + y ) 2 = x 2 + 2 xy + y 2 = 36 . If you subtract statement 1 from statement 2, you get x 2 = 4 , which is what you are looking for. Therefore, the correct answer is (C).

37.

First find the probability that it will rain on one arrangement of two rainy days and three sunny days (i.e. it will rain the first two days and then not rain for the last three days). This works out as follows: 4 4 6 6 6 2 2 33 × × × × = 5 . Now find 10 10 10 10 10 5 how many ways there are to arrange 2 rainy days and 3 sunny days: 5 × 4 × 3 × 2 ×1 = 10. Now multiply the 2 × 1× 3 × 2 × 1 probability of any one arrangement happening times the number of arrangements and you get 2 23 3 23 3 3 × 10 = or answer choice (E). 55 54

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