The sat math section 4

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5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 157 – THE SAT MATH SECTION – 39. Marguerite is remodeling her bathroom ﬂoor. 40. If Deirdre walks from Point A to Point B to Point 2 4 Each imported tile measures 1 7 inch by 1 5 inch. C at a constant rate of 2 mph without stopping, What is the area of each tile? what is the total time she takes? 8 a. 1 35 square inches x miles y miles 11 b. 1 35 square inches A B C 11 a. (x + y) × 2 c. 2 35 square inches b. 2x + 2y 3 d. 3 35 square inches c. xy 2 1 e. 4 32 square inches d. (x + y) 2 e. xy2 157
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 158 – THE SAT MATH SECTION – F ive-Choice Answers Second, the operation between the two terms is subtraction. Remember, the word difference means subtraction. 1. c. After reading the problem, you realize that Now, we have to factor 4x2 – 9. the amount of robins at the feeder is related 4x2 – 9 = (2x + 3) (2x – 3) to the number of cardinals. When this is the This is now factored and we can see case, you have to set up a legend or a key that one of the factors is choice b, (2x + 3). deﬁning the number of cardinals. Then, The answer is choice b. relate it to the number of robins. 3. a. The ﬁrst thing you should remember is that Step 1: Let x = number of cardinals the sum of the angles of a triangle equals 180°. Let 3x = number of robins m∠A + m∠B + m∠C = 180° Step 2: Now, you have to come up with a for- Now, you can substitute the values of mula to solve for x. each angle into the formula and solve for y. (number of cardinals) + (number of robins) = total number of birds Step 1: 90 + y + 40 + 3y – 10 = 180 x + 3x = 20 Step 2: 4y + 120 = 180 4x = 20 – 120 – 120 x=5 4y 60 Step 3: 4 = 4 Step 3: Take your answer for x and substitute Step 4: y = 15 it back into the legend. x = number of cardinals Next, you have to substitute the y value 3x = number of robins back into your angle measure in order to ﬁnd 5 = number of cardinals out the degree measure of each angle. 3(5) = number of robins m∠A = 90 15 = number of robins m∠B = y + 40 The answer is choice c. = (15) + 40 2. b. You should realize that since the question = 55 asks for a factor of 4x2 – 9, you have to begin m∠C = 3y – 10 factoring this expression. = 3(15) – 10 The question you should ask yourself = 35 is: Which method of factoring should I use? Looking at the expression 4x2 – 9, you The three angle measures are 90, 55, should notice that there are no common fac- and 35, respectively, and their sum is 180. tors between the two terms. Therefore, you Finally, you have to look at your answer cannot factor out a common factor. choices and determine what type of right tri- Also, this is not a trinomial. Thus, you angle this is. cannot factor this expression like a trinomial. Choice a is scalene. You remember that The method that you have to use in fac- a scalene right triangle is a right triangle that toring is the difference of two perfect squares. has three sides of different length and three There are a couple of hints in the prob- angles of different measure. The triangle in lem that clue you in on this. your problem ﬁts this deﬁnition, but check First, there are only two terms. the other three choices before you settle on choice a. 158
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 159 – THE SAT MATH SECTION – Choice b is isosceles. An isosceles right listed in a multiple-choice problem. Trust triangle has two base angles that are equal. your work if you have done it correctly. You This is not the correct answer. may have to manipulate your answer or sim- Choice c is equilateral. This is not an plify it so that it matches one of the answers. equilateral triangle since the three angles are The answer is choice d. 5. c. This involves multiple probabilities. The ﬁrst not equal. Choice d is obtuse. An obtuse right tri- thing you should try to ﬁgure out is what angle cannot exist since the angles of a trian- you are trying to ﬁnd. gle add up to 180°. An obtuse angle is The spinning of the spinner is an inde- between 90° and 180°, and if you add this to pendent event each time. This means that a right angle, you have a sum over 180°. the outcome on the ﬁrst trial does not inﬂu- Therefore, choice d is incorrect. ence the outcome of the second trial. The answer is choice a. Now, you must ﬁgure out the probabili- 4. d. Remember, when multiplying radical expres- ties of each event. When trying to ﬁgure out sions, multiply terms outside the radical the probabilities of multiple events, a tree together and multiply terms inside the radical diagram is most helpful. This is shown below. together. The formula is demonstrated below. B a b × c d = (a × c) b × d Therefore, ( x)( 2x) = x × 2x G ~B = 2x2 Of course, you remember that x × 2x = B ~G 2 because when multiplying terms with 2x like bases, you add the exponents. ~B This is your answer, 2x2; however, it The ﬁrst two branches represent the is not one of the choices. Therefore, you ﬁrst event, G or not G. The second set of must reduce this expression to simplest form branches represents what can happen on the in order to match your answer to one of the second spin of the spinner, B or not B. choices. You are interested in the events of the Remember, you can break a radical top branch, G then B. down into separate terms: These probabilities are P(G then B) = 1 1 1 2x2 = 2 × x2 4 × 2 = 8. 2 cannot be simpliﬁed, but x2 can You can calculate the P(G) by noticing 1 be simpliﬁed to x. that G is 4 of the circle. Likewise, you can 1 2 × x2 = 2 × x ﬁnd the P(B) by noticing B is 2 of the circle. = 2x Since you want the event (G then B), you 1 This answer is not listed as one of the multiply the probabilities to get 8 . four choices. Yet, x 2 is choice d. Multipli- You can also ﬁnd the probabilities of G cation is commutative, so 2(x) = x 2 . and B, respectively, by subdividing area B into Important: Many times, the answer that two sections as shown on the following page. you calculate will not be one of the answers 159
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 160 – THE SAT MATH SECTION – Choice c: 2 – 3 × 3 – 2 Therefore, choice c is false. Now, you know that choice d must be the answer. You should check it to be sure. Choice d: 2 + 3 = 3 + 2 1+2=2+1 5=5 3=3 1 Therefore, the answer is choice d. The P(G) = 4 . 2 7. a. The ﬁrst thing that you have to realize is that However, P(B) = 4 . the question requires you to simplify the If you do the problem this way, you will ﬁnd expression. that: If you are simplifying rational expres- P(G then B) = P(G) × P(B). 1 2 2 1 sions (rational expressions look like frac- 4 × 4 = 16 or 8 . tions), you have to factor the numerator and The answer is choice c. denominator if possible. 6. d. Method 1: There are two different ways you can approach this problem. The ﬁrst way is Step 1: Factor the numerator, x2 + 2x. to identify the four choices and see which There is a common factor of x between one looks like a mathematical property that the two terms. you remember. If you can do this, you will Then, factor out an x and place it out- clearly see that choice d is the commutative side a pair of parentheses. property and this is always true. If you can’t Then, divide x2 and 2x, respectively, by remember or don’t see that choice d is the x in order to ﬁnd out what terms are on the commutative property, don’t worry. You can inside of the parentheses. always solve the problem another way. So, x2 + 2x becomes x(x + 2). Method 2: You can solve this problem by sub- Step 2: The denominator cannot be fac- stituting values in for a and b. Then, see which tored. Therefore, you can now cancel out answer choice is true. Let’s see how this works: the like terms between the numerator and Let a and b equal 2 and 3, respectively. denominator. 2 3 Choice a: 3 ≠ 2 x(x + 2) = (x + 2) x The answer is choice a. Choice b: 2 + 2(3) = 3 + 2(2) 8. a. You should recognize this ﬁrst statement. 2+5=3+4 Think to yourself: Is there a triangle 7=7 that has two sides congruent and, thus, two This looks like this answer choice might angles opposite the sides congruent? be true. You should try two different values The answer is yes. It is an isosceles just to be sure. triangle. Let a and b equal 1 and 2, respectively. Now, look at the converse statement. 1 + 2(2) = 2 + 2(1) If two angles are congruent in a trian- 1+4≠2+2 gle, are the sides opposite these angles also 5≠4 congruent? So, by double-checking your choice, The answer is yes. you can see that choice b is not true. 160
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 161 – THE SAT MATH SECTION – An isosceles triangle has two sides con- y = x2 + 2 Coordinate 5: (4,18) gruent and two angles opposite these sides 18 = (4)2 + 2 congruent. 18 = 16 + 2 So, both statements are true. The 18 = 18 (True) answer is choice a. Therefore, all the coordinates make 9. b. One way that you could ﬁnd your answer is equation b true. The answer is choice b. to substitute the values of x and y into each 10. d. You have to solve for the variable, x, so you equation. The equation that is true in each need to get x by itself in the problem. There- ﬁve cases is the answer. The method of doing fore, eliminate the other terms on the same this is shown below. side of the equation as x by doing the inverse operation on both sides of the equal sign. Choice a: y=x+2 This is demonstrated below—ﬁrst add 2 to Coordinate 1: (0,2) 2=0+2 both sides: 2 = 2 (True) bx – 2 = K Coordinate 2: (1,3) 3=1+2 +2 +2 3 = 3 (True) bx = K + 2 You may think, at this point, choice a is Next, you have to divide both sides by b. the answer, but you should try the third bx K+2 = coordinate. b b K+2 Coordinate 3: (2,6) 6=2+2 x= b 6 = 4 (False) The answer is choice d. Therefore, by trying all the points, you can 11. b. You should remember that the formula for see that choice a is not the answer. the slope of a line is equal to: the change in the y-coordinate m= y = x2 + 2 Choice b: the change in the x-coordinate y = x2 + 2 Coordinate 1: (0,2) You should also remember the formula 2 = 02 + 2 (y2 – y1) m= (x2 – x1) 2 = 2 (True) If you look at the graph, you will see y = x2 + 2 Coordinate 2: (1,3) that the line crosses through exactly two 3 = (1)2 + 2 points. They are: 3=1+2 Point 1 (0,2) 3 = 3 (True) Point 2 (3,0) y = x2 + 2 Coordinate 3: (2,6) Now, all you have to do is substitute the 6 = (2)2 + 2 values of point 1 and point 2 into the for- 6=4+2 mula for slope. 6 = 6 (True) x1 = 0 y1 = 2 y = x2 + 2 Coordinate 4: (3,11) x2 = 3 y2 = 0 11 = (3)2 + 2 2 m = (0 – 2) or – 3 11 = 9 + 2 (3 – 0) The answer is choice b. 11 = 11 (True) At this point, you may believe that choice b is the answer. You should check in order to be sure. 161
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