The sat math section 5

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5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 162 – THE SAT MATH SECTION – ΔXYZ is a right triangle and, likewise, 12. a. This problem is difﬁcult if you make it difﬁ- XY is perpendicular to YZ because the cult, but it’s easy if you make it easy. The eas- Pythagorean theorem is true for Figure 1. iest way to do this problem is to calculate the In Figure 2, you are given the two mean, median, and mode for the data set. angles of ΔXYZ. If a third angle measures Remember: 90°, then ∠Y is a right triangle. Thus, XY is ■ The mean is the same as the average. perpendicular to YZ. ■ The median is the middle number of m∠X + m∠Y + m∠Z = 180°, since data. First, you must order the num- the sum of the angles of a triangle = 180. bers from least to greatest. 25° + x + 65° = 180° ■ The mode is the most frequently 90° + x = 180° occurring number. Therefore, x = 90°. ∠Y is a right angle and XY is perpendicular to YZ. 5+7+6+5+7 So, the mean equals: =6 5 Thus, XY is perpendicular to YZ in The median, if found by rearranging both ﬁgures. The answer is choice c. the numbers in the data set as shown, is {5, 5, 14. d. The ﬁrst thing that you should realize is that 6, 7, 7}. Therefore, the median is 6. x and y are both greater than 0, but less than The mode is the most frequently occur- 1. So, x and y are going to be between 0 and 1 ring number. In this data set, there are two on the number line. numbers that appear most frequently: {5, 7}. Next, you see the formula for d; d = x – y. Now, inspect the answers. To solve for d, you must substitute a You will quickly see that choice a is cor- value in for x and y. However, you do not have rect: The mean = median, because 6 = 6. a value. You should recognize however that x 13. c. You have to ﬁgure out if XY and YZ are per- is less than y. Thus, whatever value you choose pendicular. The key thing to remember here is for x, the answer for d is going to be negative. that perpendicular lines intersect to form right Therefore, the answer is choice d. angles. If you can ﬁnd a right angle at the 15. a. This question involves calculating distance. point that XY and YZ intersect, then you know The pieces of information that you are given that the two segments are perpendicular. or have to calculate are rate of speed and time. In Figure 1, if XY and YZ are perpendi- The formula for distance (with these cular, then ΔXYZ is a right triangle because it speciﬁc given pieces of information) is: contains a right angle at Y. Distance = Rate × Time In ΔXYZ, you are given three sides. If ΔXYZ is a right triangle, then the Pythago- The ﬁrst step is to calculate the rate rean theorem should hold true for these traveled at by the car. three sides. Solving for rate, you have (Leg 1)2 + (Leg 2)2 = (Hypotenuse)2 Distance 110 miles Rate = Time = 2 hours = 55 mph (6)2 + (8)2 = (10)2 Now, all you have to do is substitute Note: 10 is the hypotenuse because it is into the formula above using the rate you across from the largest angle of the triangle. just solved for. 36 + 64 = 100 100 = 100 162
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 163 – THE SAT MATH SECTION – 55 miles Distance = Rate × Time = ( hour ) × (h nonrepeating decimal. Choices b and e are hours) = 55h incorrect. The answer is choice a. Choice a is 8. This can be simpliﬁed If you solve the formula above incor- to 2 2. The 2 is an irrational number. It rectly, the other choices might seem to be is nonterminating and nonrepeating. There- correct. Therefore, double-check that you are fore, 8 is not rational. The same reasoning using the correct formula and you are solv- informs you that choice d cannot be rational; ing exactly what the question is asking for. 6 2 contains the number, 2. It is irra- 16. b. Inequalities can be solved just like equations. tional. Choice c is 5 9 and 9 = 3. There- fore, 5 9 = 5 × 3 = 15. The difference between equations and 15 inequalities is that equations have an equal 15 can be written as 1 . Thus, it is a sign and inequalities have a greater than (>) rational number. or less than (
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 164 – THE SAT MATH SECTION – alternative. For example, V = l × w × h. Setting all Example: three quantities equal to one yields a volume of 6x3 = 2x2 3x1 1 (one times one times one is one). Now if you (6 ÷ 3 = 2 AND x3 – 1 = x2) double the length, the length is now two and tripling the width makes it three. Using the You should see that by following these 9x2 equation again with these new quantities gives: rules, the answer to = 3x. 3x V = 2 × 3 × 1 = 6, the answer to the question. 3x Now, what is 3x ? 20. a. This problem requires you to read carefully Easy, what is anything divided by itself? and determine what is actually given and what The answer is 1. However, if you aren’t you are really trying to solve. You are told that careful, you may simply cancel out these last the association is charged the following: terms. You cannot do this because you are Given: dividing. $20 charge for rental of the dining room. The answer is choice a. If you haven’t $2.50 charge for each dinner plate. realized this yet, choice b looks like the answer Also, the association invited four non- if you made a mental error and crossed out paying guests and they must have enough the 3x term. Don’t make this mistake! money to pay the entire bill to the hotel. 19. c. First, you have to remember the formula for Four nonpaying guests cost the associa- the volume of a rectangular prism. tion $10 because 4 × $2.50 = $10. The formula is: The association incurs the following Volume = length × width × height costs: $30 + $2.50 × (# of paying people V=l×w×h attending). The $30 comes from the $20 charge for Now, you have to interpret and write, in the dining room and $10 fee for inviting the algebraic terms, what is happening to the four nonpaying guests. dimensions of the prism. This is best The association charges: $3.00 × (# of achieved by using a key or legend. paying people attending). KEY: So, if the association must have enough Let 2 × l = the length is doubled. money to pay the hotel, what the association Let 3 × w = the width is tripled. charges must be equal to what the hotel Let h = the height remains the same. charges the association. Next, interpret the new volume based Amount the association is charged = on the new dimensions. Amount the association charges guests New Volume = (2l) × (3w) × (h) $30 + $2.50 (# of paying people) = $3.00 = 6(l × w × h) (# of paying people) You should see that the original volume Let x = # of paying people. was equal to l × w × h. The new volume, 6(l × w × h) is six times the original volume. Thus: $30 + $2.50 x = $3.00x Therefore, the answer is choice c. $30 = $.50x Trying the formula for volume with sim- 60 = x ple numbers inserted into it, like 1, then recalcu- Therefore, 60 guests must attend. The lating the new volume using the changes answer is choice a. mentioned in the problem may be an easy 164
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 165 – THE SAT MATH SECTION – 21. c. If you want to ﬁnd the roots of an equation Choice a: 16 questions. The boy got 50% of the questions correct. An easy math algebraically, you have to factor the equation calculation shows that he got 8 correct if and solve for the variable term. there were only 16 questions on the test. So, looking at the trinomial 2x2– x –15, you However, you know that he had 10 out of the should notice the following: ﬁrst 12 correct. This answer is not possible (1) There are no common factors and cannot be true. between the three terms. You can rule out choice e, 18, using the (2) There are three terms. This elimi- same logic: Half of 18 is 9, and you know that nates the technique of factoring by the boy got at least 10 questions correct, so difference of two perfect squares. choice e is also incorrect. (3) You can always use the quadratic formula to ﬁnd the roots. This is Choice b: 24 questions on the test. You sometimes difﬁcult, especially if have to set up a proportion in order to check you do not remember the formula. this answer. The proportion is: Let’s try factoring into two binomials. # of questions correct % = After trial and error, you will see that 100 # of total questions the expression can be factored into The percentage that he got correct is (2x + 5)(x – 3) = 0. 50%. Thus, the formula for choice b is: Now, you are multiplying two binomi- x correct 50 = als together and the product is equal to zero. 24 100 Thus, one of the binomial terms, if not both, If you solve for x by cross multiplying, equals zero. the answer is x = 12. So, let’s set each term equal to zero and The boy got 10 out of the ﬁrst 12 cor- solve for x. rect. This means that he only had 2 out of the 2x + 5 = 0 x – 3 = 0 2 next 12 remaining questions correct; 12 is –5 –5 +3 +3 1 equal to .1666. . . . and this is not equal to 4 ; 2x 5 1 = –2 x=3 4 is the fraction of remaining questions cor- 2 rect. Thus, choice b is incorrect. 5 x = – 2 and x = 3 The answer is choice c. However, watch Choice c: 26 questions on the test. The out for the other choices because they are proportion for choice c is: 5 there to trick you; x = – 2 is an answer, how- 5 x correct 50 ever, it is not listed. Only 2 is listed and that = 26 100 is not the same answer. After solving the proportion, you ﬁnd 22. d. This question requires a different type of that x = 13. Once again, the boy had 10 out problem-solving technique. The most effec- of the ﬁrst 12 correct. Therefore, he had only tive way to solve this question is through trial 3 questions correct out of the next 14 if there and error. You start to eliminate wrong 3 were 26 questions on the test; 14 = .214 . . . answers by testing their validity. Here is what 1 This answer is not equal to 4 or .25. There- that means. fore, choice c is incorrect. 165
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 166 – THE SAT MATH SECTION – Choice d: 28 questions on the test. x = 36 Hopefully, by process of elimination, this is Step 2: Since x = 36, the base angles are both the answer. You should still check it however. 36° and the vertex angle is 3(36) = 108°. The proportion is: Step 3: This triangle is an obtuse triangle x correct 50 = 28 100 since there is one angle contained in the tri- You ﬁnd that x = 14 after cross multi- angle that is obtuse. The obtuse angle is the plying. Therefore, the boy had 4 correct out vertex angle. of the 16 remaining questions. You know this The answer is choice d. 24. d. Real numbers have many properties. You because he had 10 out of the ﬁrst 12 correct; 4 1 16 = 4 = .25. This is the answer. need to remember a few of them. Let’s take a There were 28 questions on the test. look at each one of the ﬁve choices in order The answer is choice d. to determine which one is the distributive 23. d. As a point of reference: property. A scalene triangle has three unequal sides. 1 1 1 1 Choice a: 3 + 2 = 2 + 3 An acute triangle contains an angle less Does this look familiar to you? It than 90 degrees. should. This is the commutative property. If A right triangle contains an angle equal to the order of the terms is switched, but you 90 degrees. still have the same answer when the opera- The ﬁrst thing you should do when you tion is performed, then the commutative encounter a word problem involving geome- property exists. try is to draw a diagram and create a legend. Choice b: 3 + 0 = 3 Legend: This is known as the identity property Let x = base angle. for addition. Sometimes, it is called the zero Let 3x = the vertex angle. property of addition. Either way, this is not Now that you have deﬁned the angles, it is the distributive property. time to draw a diagram similar to the one Choice c: (1.3 × 0.07) × 0.63 = 1.3 × (0.07 × below. 0.63) D This is the associative property. The 3x parenthesis may be placed around different groups of numbers but the answer does not x° x° O G change. Multiplication is associative. You will see that in an isosceles triangle the base angles are equal. Next, in order to Choice d: –3(5 + 7) = (–3)(5) + (–3)(7) classify the triangle, you need to ﬁnd out the This is the distributive property. You exact angle measures. can multiply the term outside the parenthe- This is done by remembering the fact ses by each term inside the parentheses. The that the sum of the angles of a triangle is 180°. left side of the equation is equal to the right side. This is the answer and it is an impor- Step 1: x + x + 3x = 180 tant property to remember. 5x 180 = The answer is choice d. 5 5 166
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 167 – THE SAT MATH SECTION – 25. c. You have to factor this expression accord- The ground and the house meet at a right angle because you are told that it is level ingly. Notice that there are only two terms ground. This makes the diagram a right tri- and there is a subtraction sign between them. angle. Thus, in order to solve for x, you have Sometimes, that is a clue to try to factor to use the Pythagorean theorem. Remember, using the difference of two perfect squares the Pythagorean theorem is a2 + b2 = c2, technique. However, in this case, 3x2 and 27 where c is the hypotenuse, or longest side, of are not perfect squares. Therefore, you have a right triangle. It can also be written as to try a different method. (leg 1)2 + (leg 2)2 = (hypotenuse)2. First, notice that there is a common In this case, the ladder is 5 feet from the factor of 3 in both terms. Factor this term house. This distance is leg 1 or a. out of both terms. Once you do, the expres- sion is 3(x2 – 9). The ladder is across from the right angle. This makes it the hypotenuse. The job is not done. You have to factor The hypotenuse, or c, is 13 feet. COMPLETELY! Look at the expression (x2 – 9). This is a binomial with two perfect Thus, you have to solve for leg 2, or b, the following way. squares separated by a subtraction sign. Thus, 52 + b2 = 132 this binomial can be factored according the 25 + b2 = 169 difference of two perfect squares. The expres- b2 = 144 sion now becomes: 3(x + 3)(x – 3). b = 12 feet The answer is choice c. If you are not The answer is choice d. careful, you may select one of the alternate Note: You could easily solve this equa- choices. Remember, factor completely and tion if you recognize that this right triangle is do not stop factoring until each term is sim- a Pythagorean triplet. It is a 5-12-13 right tri- pliﬁed to lowest terms. 26. d. This is a word problem involving geometry angle and 12 feet had to be the length of leg 2 once you saw that 5 feet was leg 1’s length and and ﬁgures. The best way of solving a prob- 13 feet was the length of the hypotenuse. lem like this is to read it carefully and then 27. b. You can outline all the possibilities that can try to draw a diagram that best illustrates occur. First, you have either boys or girls at what is being described. the party. You also know that they are either You should draw a diagram similar to wearing a mask or not wearing a mask. the one below. Therefore, you can start outlining the possi- ble events. You are told that 20 students did not 13 feet wear masks. In addition, you know that 9 boys x feet did not wear masks. Therefore, calculations tell you that 11 girls did not wear masks. 5 feet Now, if 11 girls did not wear masks and 7 girls did wear masks, then 18 girls attended You are trying to ﬁnd out the height of the party. the ladder as it rests against the house. This height is represented as x. 167