12. a. This problem is difficult if you make it diffi- cult, but it’s easy if you make it easy. The eas- iest way to do this problem is to calculate the mean, median, and mode for the data set. Remember: ■ The mean is the same as the average. ■ The median is the middle number of data. First, you must order the num- bers from least to greatest. ■ The mode is the most frequently occurring number. So, the mean equals: ᎏ 5+7+6 5 +5+7 ᎏ = 6 The median, if found by rearranging the numbers in the data set as shown, is {5, 5, 6, 7, 7}. Therefore, the median is 6. The mode is the most frequently occur- ring number. In this data set, there are two numbers that appear most frequently: {5, 7}. Now, inspect the answers. You will quickly see that choice a is cor- rect: The mean = median, because 6 = 6. 13. c. You have to figure out if XY ៮ and YZ ៮ are per- pendicular. The key thing to remember here is that perpendicular lines intersect to form right angles. If you can find a right angle at the point that XY ៮ and YZ ៮ intersect, then you know that the two segments are perpendicular. In Figure 1, if XY ៮ and YZ ៮ are perpendi- cular, then ΔXYZ is a right triangle because it contains a right angle at Y. In ΔXYZ, you are given three sides. If ΔXYZ is a right triangle, then the Pythago- rean theorem should hold true for these three sides. (Leg 1) 2 + (Leg 2) 2 = (Hypotenuse) 2 (6) 2 + (8) 2 = (10) 2 Note: 10 is the hypotenuse because it is across from the largest angle of the triangle. 36 + 64 = 100 100 = 100 ΔXYZ is a right triangle and, likewise, X ෆ Y ෆ is perpendicular to YZ ៮ because the Pythagorean theorem is true for Figure 1. In Figure 2, you are given the two angles of ΔXYZ. If a third angle measures 90°, then ∠Y is a right triangle. Thus, X ෆ Y ෆ is perpendicular to Y ෆ Z ෆ . m∠X + m∠Y + m∠Z = 180°, since the sum of the angles of a triangle = 180. 25° + x + 65° = 180° 90° + x = 180° Therefore, x = 90°. ∠Y is a right angle and X ෆ Y ෆ is perpendicular to Y ෆ Z ෆ . Thus, X ෆ Y ෆ is perpendicular to Y ෆ Z ෆ in both figures. The answer is choice c. 14. d. The first thing that you should realize is that x and y are both greater than 0, but less than 1. So, x and y are going to be between 0 and 1 on the number line. Next, you see the formula for d; d = x – y. To solve for d, you must substitute a value in for x and y. However, you do not have a value. You should recognize however that x is less than y. Thus, whatever value you choose for x, the answer for d is going to be negative. Therefore, the answer is choice d. 15. a. This question involves calculating distance. The pieces of information that you are given or have to calculate are rate of speed and time. The formula for distance (with these specific given pieces of information) is: Distance = Rate × Time The first step is to calculate the rate traveled at by the car. Solving for rate, you have Rate = ᎏ D T is i t m an e ce ᎏ = ᎏ 1 2 10 ho m u i r le s s ᎏ = 55 mph Now, all you have to do is substitute into the formula above using the rate you just solved for. –THE SAT MATH SECTION– 162 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 162 Distance = Rate × Time = ( ᎏ 55 h m ou il r es ᎏ ) × (h hours) = 55h The answer is choice a. If you solve the formula above incor- rectly, the other choices might seem to be correct. Therefore, double-check that you are using the correct formula and you are solv- ing exactly what the question is asking for. 16. b. Inequalities can be solved just like equations. The difference between equations and inequalities is that equations have an equal sign and inequalities have a greater than (>) or less than (<) sign where the equal sign would be in an equation. The first part of this problem, then, is to figure out what type of number the answer is going to be. The prob- lem states that the solution set is the set of positive integers. Therefore, the answers will come from the set of numbers that include {1,2,3,4,5,6, }.Now,look at the answer choices. You notice that 0 is included in choices e and c. Zero is not a positive integer; therefore, you can eliminate those choices. The answer must be either choice b or d. Now, you must solve the equation in order to figure out the answer. 2x – 3 < 5 + 3 + 3 ᎏ 2 2 x ᎏ < ᎏ 8 2 ᎏ Thus, x < 4. The only positive integers that satisfy this statement are {1, 2, 3}. 4 is not less than 4. The answer is choice b. 17. c. You should remember the form of a rational number. A rational number is any number that can be expressed as ᎏ p q ᎏ where p and q are integers and q ≠ 0. You should recognize that π is an irrational number. It is a nonterminating, nonrepeating decimal. Choices b and e are incorrect. Choice a is ͙8 ෆ . This can be simplified to 2͙2 ෆ .The ͙2 ෆ is an irrational number. It is nonterminating and nonrepeating. There- fore, ͙8 ෆ is not rational. The same reasoning informs you that choice d cannot be rational; 6͙2 ෆ contains the number, ͙2 ෆ . It is irra- tional. Choice c is 5͙9 ෆ and ͙9 ෆ = 3. There- fore, 5͙9 ෆ = 5 × 3 = 15. 15 can be written as ᎏ 1 1 5 ᎏ . Thus, it is a rational number. The answer is choice c. 18. a. You should remember that the word quotient means the answer to a division problem. In this problem, you are dividing a polynomial by a monomial. Once you have realized all of this, you can divide each individual term in the numerator by the monomial in the denominator. Remember, you can only do this when there is a monomial, or single term, in the denominator. First, separate the fraction above into three separate fractions. ᎏ 6 3 x x 2 ᎏ + ᎏ 9 3 x x 2 ᎏ + ᎏ 3 3 x x ᎏ Next, you have to divide each mono- mial. This is accomplished by dividing the coefficients first. It is important to remember that sometimes you may have to simply reduce the fraction instead of dividing. After dividing the coefficients, you must divide the base part of the monomials. Remember: The way to divide terms with similar bases is to simply subtract the exponents. However, be careful! If you do not see an exponent, remember, that the exponent is implied to be 1. You should see the rules that were explained above in the following example. –THE SAT MATH SECTION– 163 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 163 Example: ᎏ 6 3 x x 3 1 ᎏ = 2x 2 (6 ÷ 3 = 2 AND x 3 – 1 = x 2 ) You should see that by following these rules, the answer to ᎏ 9 3 x x 2 ᎏ = 3x. Now, what is ᎏ 3 3 x x ᎏ ? Easy, what is anything divided by itself? The answer is 1. However, if you aren’t careful, you may simply cancel out these last terms. You cannot do this because you are dividing. The answer is choice a. If you haven’t realized this yet, choice b looks like the answer if you made a mental error and crossed out the 3x term. Don’t make this mistake! 19. c. First, you have to remember the formula for the volume of a rectangular prism. The formula is: Volume = length × width × height V = l × w × h Now, you have to interpret and write, in algebraic terms, what is happening to the dimensions of the prism. This is best achieved by using a key or legend. KEY: Let 2 × l = the length is doubled. Let 3 × w = the width is tripled. Let h = the height remains the same. Next, interpret the new volume based on the new dimensions. New Volume = (2l) × (3w) × (h) = 6(l × w × h) You should see that the original volume was equal to l × w × h. The new volume, 6(l × w × h) is six times the original volume. Therefore, the answer is choice c. Trying the formula for volume with sim- ple numbers inserted into it, like 1, then recalcu- lating the new volume using the changes mentioned in the problem may be an easy alternative. For example, V = l × w× h . Setting all three quantities equal to one yields a volume of 1 (one times one times one is one). Now if you double the length, the length is now two and tripling the width makes it three. Using the equation again with these new quantities gives: V = 2 × 3 × 1 = 6, the answer to the question. 20. a. This problem requires you to read carefully and determine what is actually given and what you are really trying to solve. You are told that the association is charged the following: Given: $20 charge for rental of the dining room. $2.50 charge for each dinner plate. Also, the association invited four non- paying guests and they must have enough money to pay the entire bill to the hotel. Four nonpaying guests cost the associa- tion $10 because 4 × $2.50 = $10. The association incurs the following costs: $30 + $2.50 × (# of paying people attending). The $30 comes from the $20 charge for the dining room and $10 fee for inviting the four nonpaying guests. The association charges: $3.00 × (# of paying people attending). So, if the association must have enough money to pay the hotel, what the association charges must be equal to what the hotel charges the association. Amount the association is charged = Amount the association charges guests $30 + $2.50 (# of paying people) = $3.00 (# of paying people) Let x = # of paying people. Thus: $30 + $2.50 x = $3.00x $30 = $.50x 60 = x Therefore, 60 guests must attend. The answer is choice a. –THE SAT MATH SECTION– 164 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 164 21. c. If you want to find the roots of an equation algebraically, you have to factor the equation and solve for the variable term. So, looking at the trinomial 2x 2 – x –15, you should notice the following: (1) There are no common factors between the three terms. (2) There are three terms. This elimi- nates the technique of factoring by difference of two perfect squares. (3) You can always use the quadratic formula to find the roots. This is sometimes difficult, especially if you do not remember the formula. Let’s try factoring into two binomials. After trial and error, you will see that the expression can be factored into (2x + 5)(x – 3) = 0. Now, you are multiplying two binomi- als together and the product is equal to zero. Thus, one of the binomial terms, if not both, equals zero. So, let’s set each term equal to zero and solve for x. 2x + 5 = 0 x – 3 = 0 – 5 – 5 + 3 + 3 ᎏ 2 2 x ᎏ = – ᎏ 5 2 ᎏ x = 3 x = – ᎏ 5 2 ᎏ and x = 3 The answer is choice c.However,watch out for the other choices because they are there to trick you; x = – ᎏ 5 2 ᎏ is an answer, how- ever, it is not listed. Only ᎏ 5 2 ᎏ is listed and that is not the same answer. 22. d. This question requires a different type of problem-solving technique. The most effec- tive way to solve this question is through trial and error. You start to eliminate wrong answers by testing their validity. Here is what that means. Choice a: 16 questions. The boy got 50% of the questions correct. An easy math calculation shows that he got 8 correct if there were only 16 questions on the test. However, you know that he had 10 out of the first 12 correct. This answer is not possible and cannot be true. You can rule out choice e, 18, using the same logic: Half of 18 is 9, and you know that the boy got at least 10 questions correct, so choice e is also incorrect. Choice b: 24 questions on the test. You have to set up a proportion in order to check this answer. The proportion is: = ᎏ 1 % 00 ᎏ The percentage that he got correct is 50%. Thus, the formula for choice b is: ᎏ x co 2 r 4 rect ᎏ = ᎏ 1 5 0 0 0 ᎏ If you solve for x by cross multiplying, the answer is x = 12. The boy got 10 out of the first 12 cor- rect. This means that he only had 2 out of the next 12 remaining questions correct; ᎏ 1 2 2 ᎏ is equal to .1666 and this is not equal to ᎏ 1 4 ᎏ ; ᎏ 1 4 ᎏ is the fraction of remaining questions cor- rect. Thus, choice b is incorrect. Choice c: 26 questions on the test. The proportion for choice c is: ᎏ x co 2 r 6 rect ᎏ = ᎏ 1 5 0 0 0 ᎏ After solving the proportion, you find that x = 13. Once again, the boy had 10 out of the first 12 correct. Therefore, he had only 3 questions correct out of the next 14 if there were 26 questions on the test; ᎏ 1 3 4 ᎏ = .214 . . . This answer is not equal to ᎏ 1 4 ᎏ or .25. There- fore, choice c is incorrect. # of questions correct ᎏᎏᎏ # of total questions –THE SAT MATH SECTION– 165 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 165 Choice d: 28 questions on the test. Hopefully, by process of elimination, this is the answer. You should still check it however. The proportion is: ᎏ x co 2 r 8 rect ᎏ = ᎏ 1 5 0 0 0 ᎏ You find that x = 14 after cross multi- plying. Therefore, the boy had 4 correct out of the 16 remaining questions. You know this because he had 10 out of the first 12 correct; ᎏ 1 4 6 ᎏ = ᎏ 1 4 ᎏ = .25. This is the answer. There were 28 questions on the test. The answer is choice d. 23. d. As a point of reference: A scalene triangle has three unequal sides. An acute triangle contains an angle less than 90 degrees. A right triangle contains an angle equal to 90 degrees. The first thing you should do when you encounter a word problem involving geome- try is to draw a diagram and create a legend. Legend: Let x = base angle. Let 3x = the vertex angle. Now that you have defined the angles, it is time to draw a diagram similar to the one below. You will see that in an isosceles triangle the base angles are equal. Next, in order to classify the triangle, you need to find out the exact angle measures. This is done by remembering the fact that the sum of the angles of a triangle is 180°. Step 1: x + x + 3x = 180 ᎏ 5 5 x ᎏ = ᎏ 18 5 0 ᎏ x = 36 Step 2: Since x = 36, the base angles are both 36° and the vertex angle is 3(36) = 108°. Step 3: This triangle is an obtuse triangle since there is one angle contained in the tri- angle that is obtuse. The obtuse angle is the vertex angle. The answer is choice d. 24. d. Real numbers have many properties. You need to remember a few of them. Let’s take a look at each one of the five choices in order to determine which one is the distributive property. Choice a: ᎏ 1 3 ᎏ + ᎏ 1 2 ᎏ = ᎏ 1 2 ᎏ + ᎏ 1 3 ᎏ Does this look familiar to you? It should. This is the commutative property. If the order of the terms is switched, but you still have the same answer when the opera- tion is performed, then the commutative property exists. Choice b: ͙3 + 0 ෆ = ͙3 ෆ This is known as the identity property for addition. Sometimes, it is called the zero property of addition. Either way, this is not the distributive property. Choice c: (1.3 × 0.07) × 0.63 = 1.3 × (0.07 × 0.63) This is the associative property. The parenthesis may be placed around different groups of numbers but the answer does not change. Multiplication is associative. Choice d: –3(5 + 7) = (–3)(5) + (–3)(7) This is the distributive property. You can multiply the term outside the parenthe- ses by each term inside the parentheses. The left side of the equation is equal to the right side. This is the answer and it is an impor- tant property to remember. The answer is choice d. D 3x x° x° G O –THE SAT MATH SECTION– 166 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 166 25. c. You have to factor this expression accord- ingly. Notice that there are only two terms and there is a subtraction sign between them. Sometimes, that is a clue to try to factor using the difference of two perfect squares technique. However, in this case, 3x 2 and 27 are not perfect squares. Therefore, you have to try a different method. First, notice that there is a common factor of 3 in both terms. Factor this term out of both terms. Once you do, the expres- sion is 3(x 2 – 9). The job is not done. You have to factor COMPLETELY! Look at the expression (x 2 – 9). This is a binomial with two perfect squares separated by a subtraction sign. Thus, this binomial can be factored according the difference of two perfect squares. The expres- sion now becomes: 3(x + 3)(x – 3). The answer is choice c. If you are not careful, you may select one of the alternate choices. Remember, factor completely and do not stop factoring until each term is sim- plified to lowest terms. 26. d. This is a word problem involving geometry and figures. The best way of solving a prob- lem like this is to read it carefully and then try to draw a diagram that best illustrates what is being described. You should draw a diagram similar to the one below. You are trying to find out the height of the ladder as it rests against the house. This height is represented as x. The ground and the house meet at a right angle because you are told that it is level ground. This makes the diagram a right tri- angle. Thus, in order to solve for x,you have to use the Pythagorean theorem. Remember, the Pythagorean theorem is a 2 + b 2 = c 2 , where c is the hypotenuse, or longest side, of a right triangle. It can also be written as (leg 1) 2 + (leg 2) 2 = (hypotenuse) 2 . In this case, the ladder is 5 feet from the house. This distance is leg 1 or a. The ladder is across from the right angle. This makes it the hypotenuse. The hypotenuse, or c, is 13 feet. Thus, you have to solve for leg 2, or b, the following way. 5 2 + b 2 = 13 2 25 + b 2 = 169 b 2 = 144 b = 12 feet The answer is choice d. Note: You could easily solve this equa- tion if you recognize that this right triangle is a Pythagorean triplet. It is a 5-12-13 right tri- angle and 12 feet had to be the length of leg 2 once you saw that 5 feet was leg 1’s length and 13 feet was the length of the hypotenuse. 27. b. You can outline all the possibilities that can occur. First, you have either boys or girls at the party. You also know that they are either wearing a mask or not wearing a mask. Therefore, you can start outlining the possi- ble events. You are told that 20 students did not wear masks. In addition, you know that 9 boys did not wear masks. Therefore, calculations tell you that 11 girls did not wear masks. Now, if 11 girls did not wear masks and 7 girls did wear masks, then 18 girls attended the party. 5 feet x feet 13 feet –THE SAT MATH SECTION– 167 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 167 . ᎏ 1 2 10 ho m u i r le s s ᎏ = 55 mph Now, all you have to do is substitute into the formula above using the rate you just solved for. THE SAT MATH SECTION 162 56 58 SAT2 006[04](fin).qx 11/21/ 05 6:44 PM Page. people. Thus: $30 + $2 .50 x = $3.00x $30 = $ .50 x 60 = x Therefore, 60 guests must attend. The answer is choice a. THE SAT MATH SECTION 164 56 58 SAT2 006[04](fin).qx 11/21/ 05 6:44 PM Page 164 21 total questions THE SAT MATH SECTION 1 65 5 658 SAT2 006[04](fin).qx 11/21/ 05 6:44 PM Page 1 65 Choice d: 28 questions on the test. Hopefully, by process of elimination, this is the answer. You