Solve the equation: 0.66X = 4 – 0.34X  X = 4. X3 = 3 x 2 = 6. 8. If X + 2Y = 24 and Y – 3X = 10, what is the value of X? From the second equation, Y = 3X + 10, replace this with Y in the first equation: X + 2(3X + 10) = 24  X + 6X + 20 = 24  7X = 4  X=4/7. 9. If X + Y = 15 and X – Y = -5, what is the value of X/Y? Add the equations to get: 2X = 10  X = 5. Y = 15 – X = 10. X/Y = 5/10 = ½. 10. The triangle in the figure above is not drawn to scale. If the measurement of angle 4 is 115.5 o , what is the measurement of angles 1 and 2 (in degrees) ? Notice that angles 3 and 4 are vertical angles and thus equal. The sum of the angles in the triangle is 180 o and therefore we can write the following connection: angle 1 + angle 2 + 115.5 o = 180 o  the sum of angle 1 and 2 is equal to (180 – 115.5 = 64.5 o ). 11. ABC is an isosceles triangle, AB = BC. If the measurement of angle ABC is between 102 and 105, what is the value of the measurement of angle BCA minus CAB? Draw a sketch of the triangle. Since the triangle is an isosceles, angles BCA = CAB and therefore the answer to the question is always zero no matter what the third angle is. 12. If the sum of two numbers is 6 and their difference is 2, what is the square of their product? Let X and Y be the two numbers. X + Y = 6 and Y – X = 2 are the two equations.  Y = 4 and X = 2. Their product is 8 and the square of their product is 8 2 = 64. Chapter 2: 1. If the product of two numbers is 9 and their difference is 0, what is their sum? Let X and Y be the numbers. We can write the following equations: XY = 9 and Y – X = 0. Y = X  X 2 = 9  X = Y = 3. The sum of the numbers is 3 + 3 = 6. 2. If 64 5 = 2 2X , what is the value of X? Rewrite the expression: 64 5 = 2 2X  (2 6 ) 5 = 2 2X  2 30 = 2 2X Compare the powers: 2X = 30  X = 15. 3. If 125 3 = 5 Y , what is the value of 2 3Y ? Rewrite the expression: 125 3 = 5 Y  (5 3 ) 3 = 5 Y  5 9 = 5 Y Compare the powers: Y = 9. The value of 3Y/2 = 27/2 = 13.5. 4. If the perimeter of a rectangle is four times the length of the rectangle, then the width of the rectangle is what percent of the length? Let W be the width and L the length of the rectangle. The perimeter of the rectangle is 2W + 2L. We can write the following connection: 2W + 2L = 4L  W = L (square). Therefore the width is 100% of the length and so the answer is 100. 5. In a certain rectangle, the length is three times the width and the perimeter is equal to 64. What is the value of the length of the rectangle? We can write the following connections: L = 3W and 2L + 2W = 64. Replace L with 3W and write: 2(3W) + 2W = 64  8W = 64  W=8. L = 3 x 8 = 24. 6. There are 50 blue balls and 120 red balls in a jar containing 170 balls only. If only blue balls are to be added to the jar so that the probability of randomly picking a blue ball from the jar becomes 1/2, how many blue balls must be added to the jar? Let X be the number of blue balls that must be added. We want the portion of the blue balls to be half of the entire amount of balls in the jar and therefore 50 + X (the new number of blue balls) divided be 170 + X (the entire number of balls) should be ½. 701702100 2 1 170 50    XXX X X . And so if 70 balls are added there’ll be 120 blue balls and 240 balls in general. 7. A bag contains 15 red marbles, 12 red marbles and 18 blue ones. What is the probability of drawing two red balls one after the other? The probability of drawing a red marble is the number of red marbles divided by the entire number of marbles in the bag. The probability of drawing the first red marble is (15)/(45) = 1/3. The probability of drawing the second red marble is (14)/(44) = 7/22. The joint probability is the multiplication of the probabilities, and therefore the answer is 66 7 22 7 3 1  . 8. What is the probability of getting a number larger than 3 tossing a fair dice? While throwing a dice there are 6 results: 1, 2, 3, 4, 5 and 6. Only three results are over 3: 4, 5 and 6 and therefore the probability is 3 out of 6 or ½. 9. The average (arithmetic mean) of 6 positive integers is 110. The value of two of the integers is 24 and 28 and the other integers are greater than 30. If all the numbers are different from one another, what is the greatest possible value for any of the 6 integers? We know the value of 2 integers. If we want one of the integers to be as large as possible, take all the others as small as possible. In other words, take the two integers that are given (24 and 28), take three more integers greater than 30: 31, 32 and 33 and the fourth one would be the greatest. Write the average formula: 110 6 3332312824       X  24+28+31+32+33+X = 660  X = 512, which is the largest possible value since we took the rest as small as possible. 10. If the sum of 4 consecutive numbers is 220, what is the average (arithmetic mean) of the first and the last among those numbers? Let x, x+1, x+2 and x+3 be the four numbers. We can write the equation: x + x + 1 + x + 2 + x + 3 = 220.  4x + 6 = 220  x = 52. The average arithmetic mean of the first and the last numbers is (52 + 55)/2 = 53.5. 11. What is the time elapsed from 12:12 to 23:43, in minutes? Start from 12:12, add 11 hours to reach 23:12. Add 31 more minutes to reach 23:43. Altogether, its 11 hours and 31 minutes. In minutes its: 11 x 60 + 31 = 691 minutes. 12. What is the angle between the large and the small hand of the clock at 12:30, in degrees? At 12:30, the angle is not 180 o since the hour hand (the small hand) rotated a bit clockwise. Every hour the small hand of the clock moves 30 o and so in 30 minutes, it moved 15 o . The angle between the hands of the clock is (180 – 15 = 165) degrees. It might go easier if you draw a sketch of a clock. Chapter 3: 1. If X > 6 and X 3 X 2.5 X Y = X 8 , what is the value of X? These questions are only solved by comparing the powers of both sides, in our case of X. X 3 X 2.5 X Y = X 5.5 + Y = X 8  5.5 + Y = 8  Y = 2.5. 2. If 2 X+2 = 4 X-1 , what is the value of X? These questions are only solved by comparing the powers of both sides. 2 X+2 = 4 X-1  2 X+2 = 2 2(X-1)  X+2=2(X-1)  X+2=2X – 2  X = 4. 3. If X = (0.5) 2 and Y = X 2 , what is the value of (X + 4Y)? X = (0.5) 2 = 0.25. Y = X 2 = (0.25) 2 = 0.0625. X + 4Y = 0.25 + 4(0.0625) = 0.25 + 0.25 = 0.5. 4. If A=1/X and B=X/Y and if X=1/4 and Y=1/5, what is the value of (A+B)? A = 1/X = 1/(1/4) = 4. B = X/Y = (1/4) / (1/5) = 5/4. A + B = 4 + 5/4 = 21/4. 5. Nikki and Mike bought a new house for $120,000. Their families paid 42% of the price and the rest was divided equally and annually across six years. How many thousand of dollars did Nikki and Mike pay each year? Their families paid 42% of $120,000 and so all they had to pay themselves is (100% - 42% = 58%) of $120,000. 0.58 x 120,000 = $69,600. Each year they would pay a sixth of that amount, thus (69,600/6 = 11,600) and so the answer is 11.6 thousands of dollars. 6. A new computer costs a thousand dollars including tax. If Travis paid for three quarters of his new computer every month for a year, how much did he spent each month assuming that the payments were equal? Travis paid for only 75% of his computer, thus $750. He paid that price in 12 equal payments, each ($750 / 12 = $62.5) and so the final answer is 62.5. 7. The following figure is of a parallelogram. What is the value of X + Y + Z ? Look at the upper left triangle, the sum of the angles there should equal 180 o and so X + Y + 115 = 180  X + Y = 65 o . Since the shape is a parallelogram, Z = 115 o and so X + Y + Z = 180. 8. An isosceles triangle was attached to a rectangle. If X = 3.5 and Y = 1.5, what is the perimeter of the figure above? The perimeter of the figure above is made from two sides of the triangle and three more sides of the rectangle, thus X + X + X + Y + Y = 3X + 2Y, which is also equal to 3(3.5) + 2(1.5) = 13.5. 9. If 100   YX and 752442 Y X Z Y X Z      , what is one possible value for Y 2 ? Simplify the expression: 752442 Y X Z Y X Z       31 1 Y X    1 = (XY)Y 2  Y 2 = 1/(XY) = 1/100  Y 2 = 0.01. 10. If AA   2 55125 , what is one possible value for A? Compare the powers of 5 in each side. AA   4 55125  5 3 5 A < 5 4-A  5 3+A < 5 4-A  3+A < 4-A  2A < 1  A < 0.5 and so one possible value would be anything smaller than 0.5, for example 0.25 or 1/6. 11. The volume of a cylinder is 3 cubic feet. The radius was increased by three times and the height was increased by 2 times, what is the new volume of the cylinder in cubic feet? o 115 o X o Y o Z X Y When the radius is increased by 3, the area of the base of the cylinder increases by 9 times. The new volume of the cylinder is 9 x 2 = 18 times greater than before and so the new volume is 3 x 18 = 54 cubic feet. 12. What is the volume of water needed in order to fill a third of the box in the figure above? Calculate the volume of the box: V = 12 x 4 x 7 = 336. A third of that volume is equal to 336/2 = 112. 12 4 7 Chapter 4: 1. A, B and C are digits between 0 to 9. CA and AB are double digit numbers and ABA is a three digit number. What is the value of ABA? In the tens digit, we can see that A + B = A and thus B=0. The sum of two double digit numbers is a three digit number and so its hundreds digit must be one, thus A=1. The number is therefore 101. 2. A and B are digits between 0 to 9. A 2 = 4B (4B is a double digit number) ?   BA The only number squared with a tens digit of 4 is 7 (7 2 = 49). And so A=7 and B=9. A x B = 7 x 9 = 63. 3. X and Y are two digits between 0 and 9. When 36 is multiplied by another double digit number, the result is 3XY. What is one possible value for Y? Start with an easy number, 36 times 10 = 360 add 36 to get 396. And therefore one answer is 0 and the other can be 6. 4. If A and B are positive integers, A < 34 – B and A > 17, what is the greatest possible value of (2A – B)? Since A + B < 34, take A as 32 and B as 1 and this way A will be the largest and B the smallest. (2A + B) would be equal to (2 x 32 - 1) = 63. 5. If Y 3065  , what is the value of Y? Compare the powers from both sides of the equation. Y 3065   Y 3030   Y 3030 5.0   Y = 0.5. 6. CA AB ABA   A B  In the figure above, B is the center of the circle and A is the center of the square. If the radius of the circle is 2 , what is the area of the square? The radius of the circle is actually half the diagonal of the square and so the diagonal is equal to 22 . The ratio between the sides of a square to its diagonal is 1: 2 and so each of the sides of the square are equal to 2. The area of the square is simply 2 x 2 = 4. 7. A chocolate box contains only white, sweet and bitter chocolate in the following ratio: 2:3:4 respectively. The sweet chocolate is either with or with out walnuts, and 4 times as many sweet chocolate are with walnuts than with out. If a chocolate is chosen at random, what is the probability that it would be a sweet chocolate with walnuts? In this question it is smart to plug in numbers. Say that there are 90 chocolates in the box. According to the ratio, there are (3/9) x (90) = 30 sweet chocolates. Since there are four times as many chocolates with walnuts as there are with out there are (4/5 x 30 = 24) walnut chocolate and (1/4 x 30 = 6) with out. The probability of pulling a sweet chocolate with walnuts is 24/90 = 4/15 or 0.266 or 0.267. 8. The area of a certain rectangle (which is not a square) is 25 inches squared, what is one possible length of its smaller side? If the rectangle was a square each side would be 5 and so 5 x 5 = 25. Since the rectangle is not a square, the larger side is bigger than 5 and the smaller side is smaller than 5. The acceptable answers are 4, 3, 2 and 1. 9. What is the area of a square if the sum of the diagonals is 24 [...]... 2  12  144 each side is (12/ 2 ) and the area is     72 2  2 10 The ratio of 8. 25 to 66 is the same as the ratio of X to Y What is the value of (X/Y)2 ? Since we want to know what is the value of X/Y replace X with 8. 25 and Y with 66  X/Y = 8. 25 /66 = 1 /8 (X/Y)2 = 1 /64 or 0.0 16 11 If 2 < X < 7 and -2 < Y < 8, what is the greatest possible value of (Y – X)? We want the greatest possible value... revolutions? (A) 260 (B) 540 (C) 780 (D) 1040 (E) 2520 The best answer is C The circumference of the wheel is 2R  2 130  260  After three whole revolutions, the wheel will cross a distance of 260 x 3 = 780 and so C is the right answer 9 For which of the following values of X will 5X + 8 be smaller than 14? (A) 4 (B) 3 (C) 2 (D) 1.5 (E) 1 The best answer is E We want 5X + 8 < 14  5X < 6  X < 6/ 5 or X... or 3.5 12 62 .5, 50, 40, … In the sequence above, each term after the 1’st term is 20% less of the term preceding it What is the value of the 5’Th term of this sequence? Each term in this sequence is 80 % of the previous term 40 is 80 % of 50 and so the fourth term will be 0 .8 x 40 = 32 and the fifth term will therefore be 0 .8 x 32 = 25 .6 and this is the right answer SAT Quantitative test 6 chapters... rectangle is (11 – 5 = 6) and the height is (12 – 5 = 7) The area of the rectangle is 6 x 7 = 42 10 100 square feet of a basketball floor parquet costs 5 dollars and 25 cents How much money will it cost to cover a court with the following dimensions: 60 feet on 100 feet? The area of 60 feet on 100 feet is 6, 000 square feet If 100 square feet cost 5.25 dollars, 6, 000 will cost (6, 000 / 100 = 60 ) x 5.25 = $315... 2(5.5 + 11) = 2 x 16. 5 = 33 8 There are 52 questions in a certain exam If the ratio between the easy questions to the hard questions is 6: 7, how many hard questions are there? If the number of easy questions is 6Q and the number of hard questions is 7Q, we can write: 6Q + 7Q = 52  13Q=52  Q=4 and so 4 x 7 = 28 is the number of hard questions in the exam 9 Every month Paul works 60 hours for 30 pounds... original number of boys in the class is therefore 9+3 = 12 8 If the sum of the first 4 out of 8 consecutive numbers is 40, what is the sum of the rest? Let the first 4 numbers be: X, X+1,X+2 and X+3 Their sum is 40  4X + 6 = 40  4X = 34 The rest of the numbers are X+4,X+5,X +6 and X+7 The sum of those numbers is 4X + 4 + 5 + 6 + 7 = 4X + 22 = 56 and this is the right answer 9 If the points A(5, 5), B(11,... every hour If during a whole day both girls sneezed 360 times, how many times did Reese sneeze each hour? Reese sneezed X times per hour and so Ana sneezed X+5 sneezes In one hour, they sneezed X + X + 5= 2X + 5 In a whole day (24 hours), they sneezed 24(2X + 5) = 48X + 120 48X + 120 = 360  48X = 240  X=5 and this is the answer 12 When Tim was 8 he was two times older than Rick during that time If... 17XY = 85 X  Divide by 17X  Y = 5 and therefore 3.5Y = 3.5 x 5 = 17.5 4 97 o 17o A Note: Figure not drawn to scale In the figure above, what is the value of A in degrees? A, 17o and 97o are all vertical angles to the inner angles of the triangle and therefore equal The sum of the angles in the triangle is 180 o and so we can write the following equation: 180 = 97 + 17 + A  A = 180 – 97 – 17 = 66 degrees... on Monday than on Sunday? (A) 3.5 5 7 2 (C) 6 3 (B) 5 (D) 7.5 (E) 9 The best answer is B Calculate the amount that each organization received each day: On Sunday: 180 /9 = $20 5 7 On Monday: 180 /7 = $ 25 , which is larger by 5 is the right answer 5 from Sunday and so B 7 6 The income of a doctor is X times larger than that of a teacher If we’ll cut 20% of the doctor’s income and we’ll increase the teacher’s... – 17 = 66 degrees 5 The bank gave Elaine a loan with an interest of 8% on the original amount per month If Elaine loaned $1,525, how much interest will she pay over a period of three months (in dollars)? Every month, there is an 8% interest on 1,525, which is 0. 08 x 1525 = $122 per month Elaine will pay 3 x 122 = $ 366 to the bank 6 If in a certain jar there are X+1 black marbles and 2X+2 white marbles, . (100% - 42% = 58% ) of $120,000. 0. 58 x 120,000 = $69 ,60 0. Each year they would pay a sixth of that amount, thus (69 ,60 0 /6 = 11 ,60 0) and so the answer is 11 .6 thousands of dollars. 6. A new computer. of 8. 25 to 66 is the same as the ratio of X to Y. What is the value of (X/Y) 2 ? Since we want to know what is the value of X/Y replace X with 8. 25 and Y with 66  X/Y = 8. 25 /66 = 1 /8. . equal to 64 . What is the value of the length of the rectangle? We can write the following connections: L = 3W and 2L + 2W = 64 . Replace L with 3W and write: 2(3W) + 2W = 64  8W = 64  W =8. L