Base AB of the triangle is 4 units (because from A to the y-axis is 2 units and from the y-axis to B is another 2 units). Height B of the triangle is 6 units (3 units from B to the x-axis and another 3 units to C). Note that-/B is a right angle. So area of triangle ' x 4 x 6 2 x24 12 6. (D) You may wish to sketch a figure. 50 16 The volume of the swimming pool is equal to its length x width x depth. The volume of the swimming pool is also the volume of the water needed to completely fill the swimming pool. Then volume of water = 50 x 20 x 6 = 1,000 x 6 = 6,000 cubic feet Since each cubic foot of water costs 30 cents 0.30), total cost = .30 x 6,000 = $1,800 2 7. (E) In the formula for the area of a circle (area irr the radius is squared. So if the radius is multiplied by 3, the area 2 will be increased by 3 1 or 9 times. Another way to solve this problem is to plug in your own values. Assume that the initial radius is 2. Then area of circle = zr2 = 7r(2)2 = 47r Now, if the radius is multiplied by 3, the new radius is 3x 2 6. So area of new circle = nr2 = 7C(6)2 = 367r The second area (367r) is 9 times the first area (47r). 8. (D) Constructing a simple chart may be the fastest method to answer this question. The chart would look something like this. I year 6 years 5 years The answer can be derived from this chart; only I and 11 must be true. You could also work the problem mathematically, like this. Since Sue is 5 years older than Roberta, you can write S=R+5 And since Tim is 6 years older than John, T=J+6 You also know that Tim is I year older than Sue. That is, T=S+l From the first equation, you know that S = R + 5. You can su stitute this value of S in the third equation. Then T=S+I=R+5+1 T=S+I=R+6 You also know that T = J + 6. So you can write T=S+I=R+6=J+6 Now you can use an elimination strategy by looking at the roman numerals. Numeral I says that Tim is 6 years older Roberta, which is true because T = R + 6. So roman numeral must be true. At this point, knock out any answer choice that doesn't have roman numeral I in it-in this case, choices (B) and (C). You're left with choices (A), (D), and (E). Roman numeral 11 says that John and Roberta are the same The last part of the equation above is R + 6 = J + 6. If you tract 6 from both sides of this equation, you get R=J. So rom numeral 11 is also true. Again, knock out any answer choice doesn't have roman numeral 11 in it. You can now knock out choice (A), and you're left with only choices (D) and (E) as posse ilities. Roman numeral III says that Sue is 7 years older than John. From the equation above, you see that S+l=J+6 S=J+5 So Sue is 5 years older than John, not 7, and roman numeral III is false. Since you can knock out choice (E), the right answer is choice (D). 9. (C) To find the time when the toys will beep together again, YOU need to find the common multiple of 8, 10, and 12. To find the common multiple, take the greatest of the three numbers, 12, and find its multiple that is also a multiple of the other two. Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, and 120. Notice that 120 is the first multiple of 12 that is also a multiple of both 8 (because 8 x 15 = 120) and IO (IO x 12 = 120). So the toys will next beep together after 120 minutes, that is, after 2 hours. Two hours after 1:00 a.m. is 3:00 a.m. IO. (D) To find the perimeter of the triangle, you need the lengths of the three sides. You know that radius OB is 3 units long. Then OA and OC are each 3 units because they are also radii. Therefore, side AC of the triangle is 6 units. y A o 10 x B (3, 0) c In triangle A OB, you know that OA is 3 and OB is 3. From the Pythagorean theorem, a 2 + b2 = Cl (OA)2 + (OB)2 = (AB)2 3 2 + 32 = (AB)2 9+9=(AB)2 18 = (AB)2 @18 =AB 9@x 2 =AB 3,F2 = AB (If you spotted that triangle A OB is an isosceles right triangle with sides in the ratio I: 1 : -F2, you wouldn't have needed to us the Pythagorean theorem.) By symmetry, you know that AB = CB. So CB = 3,V2 and perimeter = CA + AB + CB = 6 + 3,F2 + 3,F2 6 + 6,F2 11. (D) Numbers that are divisible by both 4 and 6 are 12, 24, 36, and so forth. Don't forget that d can be a negative number as well. For exarnple,-12, -24, and-36 are alidivisiblebyboth4 and 6. To answer this question, you might want to use an eliminatio strategy. Because d can be either positive or neg you c rule out roman numeral 111, which says that d must be greater than 0. That is, roman numeral III cannot be a part of the fma answer. At this point, you can knock out choices (C) and (E) because each one contains roman numeral III. So you're left with choices (A), (B), and (D). Because d is divisible by 6, d must also be divisible by 3. So roman numeral I must be part of the answer. At this point, you can rule out choice (B) because it doesn't have roman numeral 1. Roman numeral 11 says that d must be a multiple of 12. If d is divisible by both 4 and 6, the smallest positive value for d must be 12 because no other smaller positive number is divisible by both 4 and 6. The next greater number is 24, which is another multiple of 12. (In problems like this, take the greater of the two numbers-in this case, 6-and think about its multiples, for example, 6, 12, 18, and 24. Of these numbers, you'll see that only 12 and 24 are divisible by both 4 and 6.) So roman numeral 11 must also be true, which means that both I and 11 must be part of the answer. So (D) is the right answer. 12. (B) In AABC, Z-A is 30'and IC is 90'. Then IB = 1800- (ZA + IC) = 180 - (30 + 90) = 180 -120 = 60' A In the smaller triangle, ABDC, you see that So ID+ZB=95+60 155 Then Z-x must be 180 - 155 = 25'. 13. (E) Plug in your own values for x and n. If x is a negative n her (say, -2) and n is a prime number (say, 3), then the value p is (-2)3 =-2x-2x-2=-8 So choice (A) is incorrect; p is not necessarily a positive number. And p is not greater than 1 if x is a fraction (say if x is 2 2 and n is 3, then p is (,)3 So choice (B) is also incorrect 2 is not necessarily greater than 1. If x is 2 and n is 3, thenp is 2 3 = 8, which is not prime. Choi (C) is also incorrect. If x is 3 and n is 3, thenp isp =2 3 = 8, which is not an even number. Choice (D) is also incorrect. Because all prime numbers are also whole numbers, notice any whole-number power of x will always be a multiple of x That is, if x is 2 and n is 3, then p = 23= 8, which is a multipi )3 2. This is true even if x is a fraction (say, Then p 2 which is also a multiple of '2' 14. (D) You know that there are 20 Japanese cars in the lot and that this number represents IO% of all cars in the lot. If 20 is 1 0% of the total, then the total (which is I 00%) must be 20 x 10 = 200 cars If the total number of cars in the lot is 200, then American cars + Japanese cars + European cars = 200 60 + 20 + E = 200 80 + E = 200 E=200-80 E 120 15. (A) You need to find the value of AB 1 which can be written as AB x - c c Then AB I ABx- c q 2 P x x- q r Since I r 2 S s r2 Then I P x q 2 2 AB x- - C q Canceling gives I p q pqr AB X-= x X-- C s s 16. (D) If you roll the sheet of paper as shown in the diagram, y will get a cylinder of height 8. 6 8 8 To find the volume of the cylinder, you need to first find the radius of the cylinder. Since the width of the sheet of paper is the circumference of the cylinder will also be 6. (If you had rolled the sheet of paper around its length, its circumference would have been 8.) If the circumference of the cylinder is 6, then you can find its radius using the formula C 2ar. That i C=2nr 6 = 2irr Dividing each side by ir, 3 The volume of the cylinder is, where h is the height, 8, and 3 r = -, gr2h =*X 3 x 3 x 8 1 $ ir I 72 17. (A) Let a, b, and c be the three numbers in the set, from least to greatest, respectively. Then c must be 5 (because you're told that the greatest number is 5). The average of the two smallest numbers is a+b 2 You're told that this average is 5 less than the average of all three numbers in the set. The average of all three numbers is a+b+5 3 Then you can form the equation a+b a+b+5 2 3 Multiplying both sides of the equation by 6 (a multiple of both 2 and 3, to eliminate the denominators), you get 3 a+b 2 a+b+5 6 = 3(a + b) = 2(a + b + 5) - 30 3a + 3b = 2a + 2b + 10 - 30 3a + 3b = 2a + 2b - 20 Subtract 2a + 2b from each side, which leaves a + b = -20 Since a + b is the sum of the two smallest numbers, their sum is -20. In this problem, you could also work from the answers. 18. (A) Because Ralph kept the car for 6 days, his daily rate was $30. Furthermore, he paid for only 5 days. So the total amount he paid was 5 x 30 = $150 Because he actually kept the car for 6 days, his daily rate was 150 -- 6 = $25 Bill had to pay $40 per day. So his average daily rate was $40. So the average daily rate paid by Ralph is 25 x 100 = 62.5% 40 of that paid by Bill. 19. (B) In AABC,,ZB is a right angle and AB and BC are equal. (Note that you are told that this figure is not drawn to scale, an consequently, you must ignore the fact that AB and BC or AE and EC may not appear to be equal.) Then using the Pythagorean theorem, 2 2 2 c =a +b C)2 )2 C)2 (A = (AB + (B 2)2 + = 2)2 2 + 2 (AC)2 4 AC=2 You're told that AC = AD, so AD = 2. A 2 300 @2 t] B 42- Right triangle AED has a 60' angle, which means that ZEDA must be 30'@ is, is a 300- 600- 900 @e. You know that a 300- 600- 900 triangle has sides in the ratio 1, NF3, and 2, respectively. You know that side AD, which is opposite the 900 angle, is 2. So side AE, which is opposite the 300 angle, must be 1, and side ED, which is opposite the 600 angle, must be -F3. Also, because AC = 2, and AE = 1, then EC must be 2 -I = 1. Now, to find the area of ADEC because you know the height, ED= -r3, and you know the base, EC = 1, then area of ADEC x base x height 2 xlx@3 @3 2 20. (B) First simplify the equation by multiplying (x + 4)(x + 1). (x + 4)(x + 1) = x 2+4x+lx+4 =x2 +5x+4 Next multiply (x - 5)(x - 2). (x - 5)(x - 2) = x 2-5x-2x+10 = X' - 7x + 10 Now take a negative of the second expression because in the original equation there is a negative sign in front of it. This operation leaves +7x-10 The original equation now looks like this. 2 2 x +5x+4 _ X +7x-10=0 Simplifying the left side by combining like terms leaves 12x - 6 = 0 Adding 6 to each side gives 12x = 6 6 X =12 x = 1 2 But the question asks for the value of x2x, which is (1)2 _ I 1 2 2 1. (E) You know that =5* = 5 + the lowest prime number > 5 = 5 + 7 (because 7 = the lowest prime number > 5) = 12 You now need to fmdj*. Then = 12 + the lowest prime number > 12 = 12 + 13 = 25 22. (B) If d - 2 is the length and d - 5 is the width, then the area of the photograph is length x width. area = (d - 2)(d - 5) = d 2 -7d+10 But you know that the area is 40. So 40 = d 2 -7d+10 O=d 2 -7d-30 Factoring gives 0 = (d - 10)(d + 3) Setting each one equal to 0, O=d-10 O=d+3 From this equation, d can be either -3 or IO. But since d can't be negative, d has to be I 0. If d is I 0, then the width is d-5=5 For this problem, working from the answers is also efficient. If you start from the middle answer (C), if the width (d - 5) is 8, then the length (d - 2) is I 1. Does 8 x I I = 40? No. (C) is incoffect; 8 x I I = 88, a value that is too high, so try (B). If the width (d - 5) is 5, then the length (d - 2) is 8. Does 5 x 8 = 40? Yes. So (B) is correct, and you need go no farther. is to plug in the answers. Ive this problem 23. (D) one way to so distance between the two m choice (C) 360. if he Start fro 'les, then, traveling at 40 miles per hour, Manuel towns is 360 ml and 366-00 = 6 hours, for a total Of uld have taken 3,6-00 = 9 hours wo - vou, re told that the entire trip low because hours. This is too can see, 360 is too low, you ok him 20 hours. Since, as VOu 0 low, so are to knock out choice (C). if choice (C) is to ither choice (D) can (B). So the answer has to be e choices (A) and or choice (E). (D) 480. if the distance is 480, then the trip to Now trY choice 480 = 12 hours, and the return trip e lasted @ Bayville would ha@ r a total of 20 hours - SO (D) would have lasted 6800 = 8 hours, fO is the right answer. roblem is to assume that it took Another waY Of solving this p Bayville. Then the distance manuel x hours to make the trip to e distance @ speed from Albertville to Bayville is 40(x) (becaus total time is 20 hours, then the return trip must x time). if the - x hours. So the distance from Bayville to have lasted 20 st be Albertville mu 1,200 - 60x 60(20 So ces are equal But the two distan 1,200 - 60x 40x 1,200 40x + 60x ioox 1,200 x = 12 if x 12, the distance from Albertville to Bayville is 40 x 12 == 480 miles 24. (B) If x represents the total value of the estate, then W'x represents the amount that was distributed among the charities. Because four charities received equal amounts, ainount received by each charity =3x@4 4 3x I =-x- 4 4 3x =- 16 You know that each charity spent 3 of the ainount it received. 4 That is, amount spent by each charity3x3x 4 16 9x 64 25. (B) To find the area of A OAB, you need its base and its height. You can take side OB as the base of the triangle. Because OP is 3 (you know this from the coordinates of P), and OP is the radius of the circle, OB is also 3 (because OB is another radius) So the base of the triangle is 3. You now need its height. 0) To find the height of the triangle, you can project its base OB t point Q. Then AQ is the height of the triangle. Note that because Z-AOP is 30', ZQOA must be 90'- 30'= 60' And because OA is also a radius, it is 3. So now you have A OQA 30'- 601- 90' right triangle in which ZAQO = 90' IQOA = 60' -IOAQ = 30 In a triangle, the three corresponding sides are in the ratio 1, -r3, and 2, respectively. In AOAQ, side OA, which is opposite 3 the 90' angle, is 3, which is i times 2. Then side AQ, which is opposite the 60'angle, should be 32 times -r3. That is, the height of AOAB is 3-r3. 2 2 x F3 Q A 0 Then area of AOAB = -2(base x height) = -1243 x 23@3) = 2 2 3) 9,F3 4