Hard Math #1

RT 48+ GMAT MATH QUIZ #1 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. −1

1. Stock Y starts at $5 per share and then increases to $6 per share. What is the percent increase of the price per share of stock Y? (A) 15% (B) 16.6% (C) 18.3% (D) 20% (E) 83.3%

5.

(A) (B)

2. If 5 25 × 413 = 2 ×10 k , what is the value of k? (A) 13 (B) 24 (C) 25 (D) 26 (E) 38 3. What is the value of (1) m2 – n2 = 40

    1   =  1 + 1   −1  6 7 −1 

(C) (D) (E)

m−n = m 2 − n2

1 42 1 13 42 13 13 42

6. If there are 400 women at College X and 200 students live off- campus, how many students does College X have? (1) 120 women live off-campus.

(2) m + n = 10

(2) There are half as many men at College X as there are students who live on-campus.

4. If the two sets have an equal number of numbers, is the mean of set Q lower than the mean of set P? (1) Set Q consists of consecutive even integers and set P of consecutive odd integers. (2) The median of Q is higher than the mean of P.

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7.

m ? n (1) mn is a positive integer.

10. Is mn < C B

(2) n is a negative number.

A

11.

If r ≠ 0 and r * is defined as

is the value of − (r *) + (r − 1) * ? (A)

If the figure above is a cube, and A, B and C are midpoints on the edges of the cube, what is the measure of ∠ ABC? (A) 30 (B) 45 (C) 60 (D) 90 (E) 120

(B) (C) (D) (E)

8. Goldenrod and No Hope are in a horse race with 6 contestants. How many different arrangements of finishes are there if No Hope always finishes before Goldenrod and if all of the horses finish the race? (A) 720 (B) 360 (C) 120 (D) 24 (E) 21

9. (A) (B) (C) (D) (E)

4 80 +

1 − 1 , what r

1 r −r r2 + r −1 r2 − r 2r 2 + 2r + 1 r − r2 2 − 3r r 2r − 1 r2 − r 2

12.

Which of the following is a possible equation for the above graph? A) x 3 B) x 3 -1 C) 3x3 + 2x D) 3x 3 – 2x E) x 3 + 3x2 – x + 2

4 = ? 9+4 5

4 5 9 2 6 4 5 +2 3+ 2 5 36

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13. If 1 < a < b < c < d, what is the value of d?

17. y°

(1) a + b + c + d = 170 (2) a, b, c, and d are all positive integers in the set of numbers that can be written in the form of 2n , where n is also a positive integer.

x°

z° w°

What is the value of y? 14. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together? (A) 3510 (B) 2620 (C) 1404 (D) 700 (E) 635

(1) w – x = 65 (2) x + z = 115

18. Given that x& = x 2 + 3, what is the value of integer k if there is a remainder of 3 when k& is divided by 4? (1) 16 < k& < 37 (2) k is a factor of 24.

 a  b  c   b  c  15. Is     >    ?  d  e  f   d  e 

19. If xy ≠ 0 , which quadrant is point (x, y) in?

(1) bc > de (2) a > f

(1)

1− x >0 1− y

(2) x + y = 20 16. Six people are on an elevator that stops at exactly 6 floors. What is the probability that exactly one person will push the button for each floor? 6! (A) 6 6 66 (B) 6! 6 (C) 6! 6 (D) 6 6 1 (E) 6 6

20. The value of â is the product of all positive, odd integers less then a but greater than 0. If e = 16 and o = 14 what is the largest prime factor of ê + 2ô? (A) 23 (B) 19 (C) 17 (D) 13 (E) 11

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21. Amy is planting rose bushes in her garden. Her pink, red and white bushes are not planted in a straight line. If the pink bush is 5 yards from the red, and the red bush is 11 yards from the white. Which of the following is a possible distance of the white bush from the pink? I. 5 II. 6 III. 7

24. The probability that it will rain in NYC on any given day in July is 40%. What is the probability that it will rain in NYC on exactly 2 days out of 5 in July? 2 2 33 (A) 55 2 3 33 (B) 55 2 2 33 (C) 54 23 32 (D) 54 2 3 33 (E) 54

(A) None of the above (B) III only (C) II and III (D) I and II (E) I, II, and III

(

)

25. If ab = – 1 and a = 4 − 17 , what is the value of b? 1 (A) 4 − 17 4 − 17 (B) 4 (C) 4 + 17 (D) 1 + 17 (E) − 4 − 17

22. In quadrilateral WXYZ, is XY perpendicular to YZ? (1) WXY is a right angle (2) WZY is a right angle 23. A tourist has travelers’ checks in $20 and $100 denominations. How many $20 checks are there? (1) If half of the $20 checks is spent, the remaining amount is $520.

26. A circle is inscribed in equilateral triangle ABC so that point D lies on the circle and on line segment AC and point E lies on the circle and on line segment AB. If line segment AB = 6, what is the area of the figure created by line segments AD, AE and minor arc DE? 9 (A) 3 3 − π 4 (B) 3 3 − π (C) 6 3 − 3π (D) 9 3 − 3π (E) It cannot be determined from the information given.

(2) The total value of the checks is $740.

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27. There are 16 stones in a bowl: 4 are painted orange, 3 green and the rest purple. What is the probability of getting first an orange and then a purple stone if you choose two stones at random? 9 (A) 80 3 (B) 64 9 (C) 64 1 (D) 20 3 (E) 20

31. In corporation X, the finance division has an average annual salary of $102,000. The consulting division has an average annual salary of $73,000. What is the average annual salary of the two divisions combined? (1) The two divisions have a total of 42 workers. (2) There are twice as many employees in the consulting division as there are in the finance division. 32. If in sequence R, rn +1 − rn =

(−1) n

, then what n is the order, from smallest to largest of r5 , r6 and r7 ? (A) r5 < r6 < r7 (B) r6 < r5 < r7 (C) r5 < r7 < r6 (D) r6 < r7 < r5 (E) r7 < r5 < r6

28. If f (a + b) = f (a) + f (b), then which of the following could be f (x) for all distinct values of a and b? A) x 2 + 1 B) 2 x C) x − 4 5 D) x E) − 7 x

33. There are 3 black balls and 7 white balls in a box. If two balls are chosen randomly, what is the probability that at least one will be black? 1 (A) 15 7 (B) 30 7 (C) 15 8 (D) 15 23 (E) 30

29. If mn ≠ 0 , what is the ratio of m to n2 ? 7 (1) The ratio of m2 to 1 is 5 7 (2) The ratio of m2 to n is 5 30. Jon’s cat consumes an average of 2.5 lbs of food per day. Yesterday, the cat ate only .5 lbs of food. Approximately what percent of the average weekly food did the cat eat yesterday? (A) 28% (B) 2.8% (C) 2% (D) .28% (E) .028%

34. If x = 274 – 270 , what is the largest prime factor of x? (A) 2 (B) 3 (C) 5 (D) 7 (E) 11

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35. A distinct three-digit pin number is distributed to each member of a video club. All digits 0-9 are allowed, and digits can be repeated to create distinct pin numbers. If 327 pin numbers are not used, how many members belong to the video club? (A) 1000 (B) 940 (C) 673 (D) 613 (E) 402

36. (A) (B) (C) (D) (E)

0.08 −5 =? 0.04 − 4 5 64 5 32 25 32 32 25 50

37. If x = 6 xy − 9 y 2 , what is the value of x in terms of y? (A) 3 y 1 (B) − y 3 y2 (C) 3 + 6y (D) − 3 y (E) 6 y − 9 y 2

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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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1.

2.

5.

(D). This is a basic percent increase question. The formula you should use is difference × 100 = %change . That works original 6 −5 1 out to be × 100 = × 100 = 20% . 5 5

(B). This one looks a lot scarier than it actually is. If you know your exponent rules well, it should be no problem:     1    1 + 1   −1  6 7 −1 

(C). When you have questions that deal with variables in the exponent, your goal should always be to convert all your base numbers to the same number, if possible. Dealing with the left side of the equation first, it should look as follows:

6.

−1

 1  =  6+7

From Statement 1:

13

Offcampus Oncampus Total

5 25 × 226 =

(5 × 2)25 × 2 = 2 × 10 25 therefore k = 25 .

4.

−1

1 =   = 13 .  13 

(C). This is a question that should be set up with a groups grid:

5 25 × 413 = 5 25 × (2 2 ) =

3.

−1

Women 120

Men

400

Total 200

Total

from statement 2:

(B). Remember to rewrite the expression m 2 − n 2 as (m + n )(m − n ) , whenever you see it. If you do that, then it is easy to see that the piece of the puzzle you are looking for is the value of (m + n ) . Statement 2 gives it to you, but statement 1 does not.

Women Offcampus Oncampus Total

Men

Total 200 (2) x

(2) Total ½ x Therefore you can set up 2 equations that allow you to solve for x.

(C). For statement 1, it is easy to come up with examples that will give you a "yes" and those that will give you a "no": if you plug in Set P {1, 3, 5} and Set Q {10, 12, 14} you get a "no", but if you plug in Set P {11, 13, 15} and Set Q {2, 4, 6}, you will get a "yes". For statement 2, it gets a little bit tricky, but it is still possible to get both answers: if you plug in Set P {1, 3, 5} and Set Q {10, 12, 14} you still get a "no", but if you keep Set P the same and change Set Q to {-100, 12, 14} the median of Q is higher than the mean of P, but Q's mean is less than P's, so you get a "yes". When you combine the information, however, with consecutive sets the mean is equal to the median, so you can only come up with "no". 8

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400

7.

(C). The key to this question, as with many geometry questions, is to draw an additional line on the diagram. If you draw a line between points A and C, you get an equilateral triangle.

8.

(B). Start out by finding the total number of possible runner finish arrangements: 6 × 5 × 4 × 3 × 2 × 1 = 720 . Of those arrangements, half of them have No Hope beating Goldenrod and the other half have Goldenrod winning. Since Goldenrod never comes in before No Hope, the 1 correct answer is  720 = 360 . 2

9.

scenarios will give us a "no". Statement 1 rules out one but not the other variable being negative, but it does not address the issue of n being a fraction less than one. Statement 2 does even less, since we do not know if m < 0 . Together, the statements still do not tell us if 0 < n < 1 .

(E). This one is a beast if you do the simplification. On the other hand, it can be solved quite easily if you ballpark. If you round 80 up to 81, and 5 down to 4, 4 you get: 4( 9) + which reduces 9 + 4( 2) to something very close to 6, the correct answer. To prove it mathematically, the steps you take look like this: 4 80 +

4

11.

(A). 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.

(D). Use the points you can identify to eliminate answer choices. Since the curve passes through the origin, when x = 0, the result should equal 0 also. By that logic, you can cross off (B) and (E). Then, in order to have the curve dip below the xaxis to the right of the y-axis as it does, some positive values of x must result in a negative value for the expression. The only remaining answer that could ever do that is the correct answer, (D).

13.

(C). Statement 1 is certainly not enough information on its own. Nor is statement 2--it simply tells you that all of the variables are exponents of 2 (i.e. 2, 4, 8, 16, etc.). Together, however, you need to slow down and watch out. Is there only one possible value for d? Yes. First list all exponents of 2 less than 170: 2, 4, 8, 16, 32, 64, 128. Is there any way to combine four of the numbers to add up to 170 without using 128? If you add up 8 + 16 + 32 + 64, you will see that even that is too small. Therefore, d must equal 128.

14.

(E). This question has two wrinkles: we don't know exactly how many men and women are serving on the committee and we have some illegal combinations. Simplest thing to do would be to ballpark. There are two ways to distribute

=

9+4 5

  1 4  80 +  =  9 + 4 5   1 44 5 +  =  9+4 5

(

)

 4 5 9 + 4 5 +1   = 4   9+4 5    36 5 + 80 + 1  4  =  9+4 5   4 5 +9 4 × 9  =  9+4 5  36

10.

(E). As with all yes/no data sufficiency, it is a good idea to get some sense of what kind of numbers will give you a "yes" and which will give you a "no" before you look at the statements. In this question, m mn < when one or the other variable is n negative or when 0 < n < 1 . All other 9

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the committee slots between men and women: 3 of each or 2 me n and 4 women. We must treat the two scenarios separately. If there are 3 of each sex on the committee, then the total number of possible combinations given this scenario is found as follows: 8 • 7 •6 5• 4 • 3 • = 560 . . If there are 2 3 • 2 • 1 3 • 2 •1 men and 4 wo men, then the total number of possible combinations: 8 • 7 5 • 4 • 3• 2 = 140. Together you • 2 •1 4 • 3 • 2 • 1 have a total of 700, since your answer excludes some of these options the correct answer is smaller than 700. To actually calculate how the number of people lets look at the 3 men 3 women situation. First you would have to deal with the two separate possible scenarios; the scenario where neither of the refusers are on the committee and the scenario where exactly one is on the committee. Where neither is on the committee the number of 6 •5• 4 5 • 4 • 3 • combinations is . 3• 2 •1 3 • 2 • 1 Where exactly one of the refusers is on 2 •6• 5 5 • 4 • 3 the committee is • . Then 2 •1 3•2 •1 calculate the combinations for the 2 men 3 women. 15.

a > 1 if the f right side of the inequality is positive, a < 1 if it is negative. To see how this f b c works, set    = −1 and then 1.  d  e  Statement 2 ends up telling us very little, a because we can choose values for that f result in numbers on either side of 1. a ( < 1 when one or both variables is f negative). Taking both statements together does little to resolve any of these issues. above inequa lity to hold true,

(E). Since this is a yes/no type data sufficiency question, take stock of the question before you dive into the statements. To understand what is really being asked of you, it is helpful to rearrange the terms in the left side of the inequality to match better those on the  a  b  c   b  c  right:     >    . When you  f  d  e   d  e  do that rearrangement, it becomes clear that the only difference between the two a sides is the term. At this point, it f becomes clear that statement 1 does nothing to answer the question. For the

16.

(A). For probability questions, the # favorable equation you should use is . # total Here you should use your knowledge of permutations to find the elements of the formula. There are 6! ways that one person is going to each floor, and each of 6 people have 6 choices as to which button to push, which means that there are 66 total ways people's destinations are distributed.

17.

(D). As with all questions dealing the angles of a triangle, here the key to the question is remembering that the interior angles of a triangle add up to 180. For statement 1, you also know that w + z = 180 or z = 180 - w. Plug that into the expression x + y + z = 180 and you get x + y + 180 - w = 180. Simplify and solve for y, and you get y = w - x. Therefore, statement 1 is sufficient. To evaluate statement 2, plug 115 in for x + z in the equation x + y + z = 180. When you do that, you get an equation with only one variable, which can always be solved.

18.

(A). In statement 1, there are two possible values for k&: 19 and 28. Only 19 has a

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remainder of 3 when divided by 4, however, so statement 1 is sufficient. All even values of k will have a remainder of 3 when divided by 4 (can you explain why this is so?), so statement 2 does not give you enough information. 19.

(C). This question is really asking us if we know for sure whether x and y are positive or negative. Statement 1 is not sufficient because (x, y) could equal points in all four quadrants such as (2, 2), (-1, -1), (-1, 0.5), (0.5, -1). Statement 2 is not sufficient because again, one can find points in all quadrants but III that would work: (10, 10), (-1, 21), (21, -1). Together, however, it is sufficient, because the only quadrant that now works is I (points such as (10, 10) or (1, 19)).

20.

(C). This function should remind you of factorials, and indeed, the expression ê + 2ô can be rewritten as (15 •13 •11• 9...) + 2(13 •11• 9...) . (Note that the function is defined as the product of all even numbers less than a, not less than or equal to a.) The common factors of the two terms can be factored out, so the expression is rewritten as (13 • 11• 9...)(15 + 2) . The largest prime factor is 17.

21.

(B). 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.

22.

statements don’t resolve it because it could be a square or it could be a shape with angles 90, 90, 110 and 80 (try to draw this you will see its possible).

(E). If you choose to draw the quadrilateral, you must realize that you don’t know whether it’s a regular shape such as a parallelogram or whether it’s a stranger shape, also when you label points do so clockwise. Statement 1 only tells about one angle so its insufficient since the others could be anything. Statement 2 has the exact same problem. Together the

23.

(C). Statement 1 can be expressed as 100H + 0.5 (20T) = 520 (where H = number of $100 checks and T = number of $20 checks). There are 2 variables and only one equation, so on its own, statement 1 is not sufficient. Statement 2 can be expressed as 100H + 20T = 740. This also is not sufficient on its own. Together, however, the statements are sufficient, since you have two equations and two unknowns.

24.

(E). 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 = . 55 54

25.

(C). If you plug in the choices, starting with the simplest of the choices and math that seems ugly will become very easy. Answer (C), which is one of the easier choices, when plugged in should remind you of the quadratics, thinking of the GMATs 3 favorite quadratic equations. Here the one we need is x 2 – y2 .

26.

(B). In general terms the area of the portion you are looking for can be triangle − circle described as . The area 3

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of the triangle is fairly straight forward: the base equals 6 and the height equals 3 3 , so the area equals 9 3 . In order to find the area of the circle, we need to find the radius, and this is a bit tricky, but if you draw the right lines on your diagram, it is doable. First draw lines bisecting each of the three angles in the triangle. This forms six smaller 30:60:90 triangles. The shortest side of each of those triangles is equal to the radius of the circle. This means that the radius on the circle is 3 , and the area of the circle is 3π . Plug all of this informa tion into our initial equation and we get the area of the portion described by the minor arc DE and the line segments AD and AE is equal to 9 3 − 3π = 3 3 −π. 3 27.

28.

(E). The probability of getting an orange and then a purple stone is  4  9  3 .    =  16  15  20

29.

(C). Statement 1 provides no info about n so its insufficient. Statement two almost gives you a ratio of m to n (when m is 7 ≈ 2.64 , n is 5, and all multiples of this), however because you are told about m2 its insufficient because m could be a negative or positive number. Together however we know m is positive 2.64, when n is 5.

30.

(B). If you sent up percents as simple fractions they are easy to reduce and .5 5 1 1 = = = solve. , from 2.5 • 7 25 • 7 5 • 7 35 here you can either ball park or divide.

31.

(B). 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 its sufficient.

32.

(D). Here, plugging in for n is key. If n = 5, then you have the following expression: (− 1)5 = − 1 . This means that r r6 − r5 = 5 5 5 is larger than r6 by 1/5. Eliminate answer choices (A), (C), and (E). Now plug in n = 6 to see how r7 relates to everything. When you do, you get (− 1)6 = 1 . This means that r is r7 − r6 = 7 6 6 larger than r6 but less than r5 .

33.

(D). When you see the words "at least" in a probability question, chances are it is easier to find the probability of the opposite happening and then subtract from 1. The opposite of at least one ball being black is both balls being white. The probability of drawing two white balls is 7 6 7 × = . Don't forget to subtract that 10 9 15 from 1!

(E). This one is hard to sort out at first, but if you plug in for a and b as specific values of x, the question begins to make more sense. For example, if a = 2, b = 3, and a + b = 5, then the answer choices respond as follows:

(

) (

)

(A) 5 2 + 1 ≠ 2 2 + 1 + 3 2 + 1

(

) ( )

(B) 2 5 ≠ 2 2 + 2 3 Etc.

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34.

(D). This looks scary, but if you factor out your common factors from each term, it gets easier: 2 74 − 2 70 = 2 70 2 4 − 1 = 2 70 (15 ) , so the largest prime factor is 5.

(

)

35.

(B). The total number of possible pin numbers are 10 x 10 x 10 = 1000. If all but 327 of them are in use, that means that there are 673 club members.

36.

(C). This one is all about exponents: −2 0.08 −5 0.04 4 (4 × 10 ) = = = 0.04 −4 0.085 (8 × 10 −2 )5 4

(2 ) × 10 (2 ) × 10 2 4

−8

3 5

− 10

= 2 8−15 × 10 −8− (− 10) =

102 25 = 27 32 37.

(A). If you square both sides in order to rid yourself of the radical, you get x 2 = 6xy – 9y2 , you should notice this is a quadratic equation. Reshuffle the equation to standard form and factor and you get (x – 3y)(x – 3y) = 0 and therefore (x – 3y) = 0.

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