Hard Math #2 Explanations

EXPLANATIONS FOR GMAT Q UIZ II 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 amo unts. The boxes look as follows: Initial

Ratio

Blueberries

2

Raspberrie s

1

Total

3

Final

Ratio

Blueberries

2

Raspberries

5

Total

7

Multiplier

Multiplier

Actual

Actual

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. 2x 2x = . To 8 y 23 y evaluate statement 1, plug in the right half of the equation from the statement into the 2x 2 3 y +8 expression in the question: 3 y = 3 y = 2 3 y +8 − 3 y = 2 8 . Therefore, statement 1 is 2 2 sufficient to solve. Statement 2 gives you no information about x.

2. Before you look at the statements, get your base numbers to match:

3. 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: 32 + 32 + 32 + 32 + 32 + 4(32) 9(32 ) = = 48 . The correct answer is (B). 6 6 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 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. As always when you have variables in your answer choices, you want to plug in. A good number to choose is r = 2. If you plug it into the question correctly, your target answer should be 1/2. Only answer choice (A) gives you 1/2 when you plug in 2 for r.

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 that k = 1 35. Therefore, if y > , k > 35 . Even if you don't see that k has to be greater than 2 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, you 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 remain when the two statements are combined, so the correct answer is (E). 14. 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). 15. In order to answer this question, we need to know what proportion of the staff belong to each division in order to know how to weight the average. Statement 2 gives you that proportion, so the correct answer is (B). 1 1 + . From q − 151 q 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 149 calculation makes the risk worth taking. If you do plug in 1, you get as your 150 target answer. Only answer choice (A) gives you that result.

16. First of all, you need to recognize that p = q − 151 . Replace p :

17. To answer this question, first find the difference between the points we know: on scale R the difference is 6 and on scale S the difference is 24. The differences will remain proportional, so since the difference between 48 and 60 is 12, half of 24, the difference between the corresponding point on scale R and point 18 is half of 6--3. Add 3 to 18, and you get a corresponding reading of 21 or answer choice (A). x to be an integer, both y and z must be proven to be factors of x . Statement yz 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 x sufficient. To see this, statement one can be summarized as follows: 2 + = an y integer. Statement 2 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).

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. This is really a question of impossible triangles. Only statement III, 7, satisfies the condition that the sum of the distance of any two sides of a triangle must be larger than the distance of the third side. Therefore the correct answer is (B). 22. 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). 23. 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 r in the answer choices, only (C) gives you your target answer.

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 = −

1 1 1 3 , then (A) is , (B) is , (C) is 4, (D) is , and (E) is 4 2 8 16

1 . Therefore the correct answer is (D). 4

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 x- intercept. 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). 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 a+b+c integers must fit into the equation = 10 . The median, b , equals 9, so 3 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. 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

1 1 , y = , you also get a "no". Therefore statement 2 2 4 is sufficient and the correct answer is (B). you a "no". If you plug in x =

33. 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 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). 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 negative, but not which one is which. Together, the statements do not resolve the issue, so the correct answer is (E). 35. This permutations 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 8 choices for our tens digit (excluding 7, because no repeats are allowed, and 0, because the digits are non-zero) and 4 choices (all non-zero, even digits) for our ones digit, giving a total of 32 choices. This, however, includes the numbers 722, 744, 766, and 788 which are illegal, so our total for the 700's is actually 28. The 900's work exactly the same way. For the 800's, 8 is no longer a choice for the ones column, so we have 8 choices for the tens and only three for the ones with three illegal numbers included in that count. That leaves us with at total of 21 legal numbers beginning with an 8. Add 32 + 21 + 32 = 77, so the correct answer is (E). 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 stateme nt 1 from statement 2, you get x 2 = 4 , which is what you are looking for. Therefore, the correct answer is (C).

37. Here, as always with questions that have variables in the answer choices, the best technique is to plug in. Unlike most plug- in question, however, it is a good idea to plug in y = 1. This keeps the calculating down to a more manageable level. Be sure you test all of the answer choices however, because you may get more than one that

works. If you do plug in y = 1, x will equal 3. Only (A) returns 3, so it is the correct answer.