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23 Cards in this Set
- Front
- Back
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A data set with a mean of 60 has a standard deviation of 3.5. Which of the following is the smallest number that falls within one standard deviation of the mean?
53 56 59 63.5 65 |
One standard deviation is both subtracted from and added to the mean in order to get the range of one standard deviation.
In this problem, the mean is 60 and the standard deviation is 3.5. The range of numbers within one standard deviation, therefore, is 56.5 (60 – 3.5) to 63.5 (60 + 3.5). The smallest answer choice given which falls within this range is 59. The correct answer is C. |
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What is the largest angle in triangle ABC?
(1) The sum of angles A and B is 70 degrees. (2) Angle C is 110 degrees. |
Before we analyze each statement, let's recall that the sum of all angles in any triangle is 180 degrees.
(1) SUFFICIENT: If angles A and B add up to 70 degrees, the remaining angle, angle C, must be 110 degrees (i.e., 180 – 70 = 110). Note that we do not need to know the individual values of angles A and B to answer the question. Since the sum of angles A and B is 70 degrees, none of them can be greater than 110 degrees, implying that angle C is the largest angle. (2) SUFFICIENT: If angle C is 110 degrees, then the sum of the remaining two angles, A and B, must be 70 degrees (i.e., 180 – 110 = 70), implying that none of them can be greater than angle C. Therefore, angle C is the largest angle. The correct answer is D. |
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If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?
8 9 16 23 24 |
The definition given tells us that when x is divided by y a remainder of (x # y) results. Consequently, when 16 is divided by y a remainder of (16 # y) results. Since (16 # y) = 1, we can conclude that when 16 is divided by y a remainder of 1 results.
Therefore, in determining the possible values of y, we must find all the integers that will divide into 16 and leave a remainder of 1. These integers are 3 , 5, and 15. The sum of these integers is 23. The correct answer is D. |
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Store X sold 1/6 more television sets in March than in February, and in February, it sold 3/8 fewer television sets than in January. If the store sold 480 television sets in January, how many sets were sold in March?
210 300 350 550 770 |
In January, store X sold 480 television sets.
In February, store X sold 3/8 fewer television sets than in January: 480 – 3/8 × 480 = 480 – 180 = 300 In March, store X sold 1/6 more television sets than in February: 300 + 1/6 × 300 = 350 The correct answer is C. |
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#8
If , √xy = xy what is the value of x + y? (1) x = -1/2 (2) y is not equal to zero |
Let’s start by rephrasing the question. If we square both sides of the equation we get:
√(XY)^2 = xy^2 xy = (xy)^2 Now subtract xy from both sides and factor: (xy)2 – xy = 0 xy(xy – 1) = 0 So xy = 0 or 1 To find the value of x + y here, we need to solve for both x and y. If xy = 0, either x or y (or both) must be zero. If xy = 1, x and y are reciprocals of one another. While we can’t come up with a precise rephrasing here, the algebra we have done will help us see the usefulness of the statements. (1) INSUFFICIENT: Knowing that x = -1/2 does not tell us if y is 0 (i.e. xy = 0) or if y is -2 (i.e. xy = 1) (2) INSUFFICIENT: Knowing that y is not equal to zero does not tell us anything about the value of x; x could be zero (to make xy = 0) or any other value (to make xy = 1). (1) AND (2) SUFFICIENT: If we know that y is not zero and we have a nonzero value for x, neither x nor y is zero; xy therefore must equal 1. If xy = 1, since x = -1/2, y must equal -2. We can use this inf |
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#9
If 274x + 2 × 162-2x × 36x × 96 – 2x = 1, then what is the value of x? -9 -6 3 6 9 |
An effective strategy for problems involving exponents is to break the bases of all the exponents into prime factors. This technique will allow us to combine like terms:
274x + 2 × 162-2x × 36x × 96 – 2x = 1 (33)4x + 2 × (2 × 34)-2x × (22 × 32)x × (32)6 – 2x = 1 312x + 6 × 2-2x × 3-8x × 22x × 32x × 312 – 4x = 1 2-2x + 2x × 312x + 6 – 8x + 2x + 12 – 4x = 1 20 × 3 2x + 18 = 1 3 2x + 18 = 1 3 2x + 18 = 30 2x + 18 = 0 2x = -18 x = -9 The correct answer is A. |
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7 teams compete in a track competition. If there are 20 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?
3 4 5 6 7 |
There are 7 teams and 20 events. We want to make each team win the most number of events. Since the max is 3, 6 teams have to win 3 events 6*3=18, thus leaving two events for the 7th team. That means that all the 7 teams won at least one event.
In order to get the smallest possible number of teams as winners (of at least one event), we want to have as many teams as possible not win any events. How can we accomplish this? Since there are 20 events, there are going to be 20 events won. We want as few teams as possible to be the winners of those 20 events. To accomplish this, we will make each "winning" team win as many events as possible. We are told that no team wins more than 3 events. Thus, the maximum number of events that a team wins is 3. Team A B C D E F G # of wins 3 3 3 3 3 3 2 The chart shows that even when we award teams 3 wins each, the final team (G) still wins 2 events. No team ends up without a win. The correct answer is E. |
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If |a| = |b|, which of the following must be true?
I. a = b II. |a| = -b III. -a = -b I only II only III only I and III only None |
Because we know that |a| = |b|, we know that a and b are equidistant from zero on the number line. But we do not know anything about the signs of a and b (that is, whether they are positive or negative). Because the question asks us which statement(s) MUST be true, we can eliminate any statement that is not always true. To prove that a statement is not always true, we need to find values for a and b for which the statement is false.
I. NOT ALWAYS TRUE: a does not necessarily have to equal b. For example, if a = -3 and b = 3, then |-3| = |3| but -3 ≠ 3. II. NOT ALWAYS TRUE: |a| does not necessarily have to equal -b. For example, if a = 3 and b = 3, then |3| = |3| but |3| ≠ -3. III. NOT ALWAYS TRUE: -a does not necessarily have to equal -b. For example, if a = -3 and b = 3, then |-3| = |3| but -(-3) ≠ -3. The correct answer is E. |
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What is the area of a circle with circumference 14?
7 49 7 14 49 |
The question mentions two measurements for which we know formulas: area and circumference. The area of a circle = r2 and the circumference = 2r, where r is the radius.
We're told the circumference is 14. We can use this to find the radius. 14 = 2r, therefore r = 7. Thus, area = (7)2 = 49. The correct answer is E. |
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Bob bikes to school every day at a steady rate of x miles per hour. On a particular day, Bob had a flat tire exactly halfway to school. He immediately started walking to school at a steady pace of y miles per hour. He arrived at school exactly t hours after leaving his home. How many miles is it from the school to Bob's home?
(x + y) / t 2(x + t) / xy 2xyt / (x + y) 2(x + y + t) / xy x(y + t) + y(x + t) |
Let b be the number of hours Bob spends biking. Then (t – b) is the number of hours he spends walking. Let d be the distance in miles from his home to school. Since he had the flat tire halfway to school, he biked d/2 miles and he walked d/2 miles. Now we can set up the equations using the formula rate x time = distance. Remember that we want to solve for d, the total distance from Bob's home to school.
1) xb = d/2 2) y(t – b) = d/2 Solving equation 1) for b gives us: 3) b = d/2x Substituting this value of b into equation 2 gives: 4) y(t – d/2x) = d/2 Multiply both sides by 2x: 5) 2xy(t – d/2x) = dx Distribute the 2xy 6) 2xyt – dy = dx Add dy to both sides to collect the d's on one side. 7) 2xyt = dx + dy Factor out the d 8) 2xyt = d(x + y) Divide both sides by (x + y) to solve for d 9) 2xyt / (x + y) = d The correct answer is C. |
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If the positive integer x is rounded to the nearest ten, will the result be greater than x ?
(1) If x is divided by 10, the remainder is even. (2) If x is divided by 5, the remainder is odd. |
(1): INSUFFICIENT. If a positive integer is divided by 10, the remainder is the same as the integer’s units digit. This statement thus implies that the units digit of x is 0, 2, 4, 6, or 8.
If the units digit of x is 6 or 8, then rounding to the nearest ten will result in an increase; otherwise, it will not. The statement is insufficient. (2): INSUFFICIENT. If a positive integer is divided by 5, the remainder is: 0, if the integer’s units digit is 0 or 5 1, if the integer’s units digit is 1 or 6 2, if the integer’s units digit is 2 or 7 3, if the integer’s units digit is 3 or 8 4, if the integer’s units digit is 4 or 9 According to this statement, then, the integer’s units digit must be 1, 6, 3, or 8. If the units digit of x is 6 or 8, then rounding to the nearest ten will result in an increase; otherwise, it will not. The statement is insufficient. (1) AND (2): SUFFICIENT. If both statements are true, the units digit of the integer must be 6 or 8 (the only values common |
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Is x > 0?
(1) x2 > 0 (2) x + 2 > 0 |
The question asks whether x is positive (i.e. greater than zero). We cannot do much to simplify the question, so we must turn to the statements.
(1) INSUFFICIENT: This tells us that x2 is positive. If you square a positive number, you get a positive result. If you square a negative number, you also get a positive result. Therefore, we cannot tell from this information whether x is positive or negative. (2) INSUFFICIENT: This tells us that x + 2 > 0. If we subtract 2 from both sides, we get the following inequality: x > -2. According to this inequality, x could be positive (in which case it would definitely be greater than -2) or negative (x could be -1, for example). (1) AND (2) INSUFFICIENT: Both statements tell us that x could be either positive or negative. Therefore, we cannot answer the question. The correct answer is E. |
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If x represents the sum of the interior angles of a regular hexagon and y represents the sum of the interior angles of a regular pentagon, then the difference between x and y is equal to the sum of the interior angles of what geometric shape?
Triangle Square Rhombus Trapezoid Pentagon |
Recall that the sum of the interior angles of a polygon is computed according to the following formula: 180(n – 2), where n represents the number of sides in the polygon. Let's use this formula to find the values of x and y:
x = the sum of interior angles of a regular hexagon = 180(6 – 2) = 720 y = the sum of interior angles of a regular pentagon = 180(5 – 2) = 540 x – y = 720 – 540 = 180 Thus, the value of (x – y), i.e.180, is equal to the sum of interior angles of a triangle. The correct answer is A. |
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In an increasing sequence of 8 consecutive even integers, the sum of the first four integers is 172. What is the sum of the last four integers?
180 188 192 200 204 |
Let the eight consecutive even integers be represented by x, x + 2, x + 4, x + 6, x + 8, x + 10, x + 12, and x + 14. Thus, the first four integers are x, x + 2, x + 4, and x + 6. Since the sum of these four integers is 172, it follows that
4x + 12 = 172, so 4x = 160, and x = 40. The first integer in the sequence is 40 and the last integer in the sequence is x + 14, or 54. The sum of the last four integers is 48 + 50 + 52 + 54 = 204. The correct answer is E. |
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#26
Look at Cax exam page |
#26
Look at cat exam |
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A clock store sold a certain clock to a collector for 20 percent more than the store had originally paid for the clock. When the collector tried to resell the clock to the store, the store bought it back at 50 percent of what the collector had paid. The shop then sold the clock again at a profit of 80 percent on its buy-back price. If the difference between the clock's original cost to the shop and the clock's buy-back price was $100, for how much did the shop sell the clock the second time?
$270 $250 $240 $220 $200 |
If p is the price that the shop originally paid for the clock, then the price that the collector paid was 1.2p (to yield a profit of 20%). When the shop bought back the clock, it paid 50% of the sale price, or (.5)(1.2)p = .6p. When the shop sold the clock again, it made a profit of 80% on .6p or (1.8)(.6)p = 1.08p.
The difference between the original cost to the shop (p) and the buy-back price (.6p) is $100. Therefore, p – .6p = $100. So, .4p = $100 and p = $250. If the second sale price is 1.08p, then 1.08($250) = $270. (Note: at this point, if you recognize that 1.08p is greater than $250 and only one answer choice is greater than $250, you may choose not to complete the final calculation if you are pressed for time.) The correct answer is A. |
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look at #30
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look at #30
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look at 31
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look at 31
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If a – b > a + b, where a and b are integers, which of the following must be true?
I. a < 0 II. b < 0 III. ab < 0 I only II only I and II only I and III only II and III only |
We are given the inequality a – b > a + b. If we subtract a from both sides, we are left with the inequality -b > b. If we add b to both sides, we get 0 > 2b. If we divide both sides by 2, we can rephrase the given information as 0 > b, or b is negative.
I. FALSE: All we know from the given inequality is that 0 > b. The value of a could be either positive or negative. II. TRUE: We know from the given inequality that 0 > b. Therefore, b must be negative. III. FALSE: We know from the given inequality that 0 > b. Therefore, b must be negative. However, the value of a could be either positive or negative. Therefore, ab could be positive or negative. The correct answer is B. |
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A convenience store currently stocks 48 bottles of mineral water. The bottles have two sizes of either 20 or 40 ounces each. The average volume per bottle the store currently has in stock is 35 ounces. How many 40 ounce bottles are in stock?
20 24 28 32 36 |
Let x = the number of 20 oz. bottles
48 – x = the number of 40 oz. bottles The average volume of the 48 bottles in stock can be calculated as a weighted average: {[ x(20)+(48-x)(40) ] / 48} = 35 20x + (40)(48) – 40x = (35)(48) 20x = (40)(48) – (35)(48) 20x = (48)(40 – 35) 20x = (48)5 20x = 240 x = 12 Therefore there are 12 twenty oz. bottles and 48 – 12 = 36 forty oz. bottles in stock. The correct answer is E. |
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look at 34
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look at 34
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A certain galaxy is known to comprise approximately 4 x 1011 stars. Of every 50 million of these stars, one is larger in mass than our sun. Approximately how many stars in this galaxy are larger than the sun?
800 1,250 8,000 12,000 80,000 |
50 million can be represented in scientific notation as 5 x 107. Restating this figure in scientific notation will enable us to simplify the division required to solve the problem. If one out of every 5 x 107 stars is larger than the sun, we must divide the total number of stars by this figure to find the solution:
(4 x 10^11) / (5 x 10^7) = 4/5 x 10^(11-7) = 0.8 x 10^4 The final step is to move the decimal point of 0.8 four places to the right, with a result of 8,000. The correct answer is C. |
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A palindrome is a number that reads the same forward and backward, such as 121. How many odd, 4-digit numbers are palindromes?
40 45 50 90 2500 |
Use the “slot method” to count all the 4-digit, odd palindromes. __ __ __ __
Since the last digit must be odd our only choices are 1, 3, 5, 7, or 9 for the first/last digit. There are no restrictions on the inner digits, we have 10 choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. the outer two numbers must match and the inner two numbers must match, creating numbers such as 1221 or 5665. We have 5 choices for the outer two digits and 10 choices for the inner two digits. Our “slot method” diagram looks like this: 5 10 1 1. Once a digit is selected for the left outer digit, there is only one possible choice for the right outer digit, which must match it. Similarly for the two inner digits, the left choice determines the right. Using the counting principle, we have 5 × 10 × 1 × 1 = 50 choices for our 4-digit number we do not set the problem up as 5 10 10 5 and multiply, giving 2500. Only two choices to be made:number of possibilities for inner digits and number of possibilities for outer digits C |
