The sat math section 6

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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
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 168 – THE SAT MATH SECTION – If 18 girls attended the party and 15 Thus, the product of bc is even. Since a is odd, a × (even #) = even number. Therefore, boys were at the party, then 33 students attended the school costume party overall. this expression is even and not the answer The answer is choice b. you are searching for. 28. c. You have to be able to read and interpret the Another way you could try this prob- lem (if you do not remember the even × odd wording in this problem in order to develop an equation to solve. = even number rule) is to substitute num- Let x = the number. bers for a, b, c. Let’s say a = 5; b = 6; c = 9. 1 Then a(bc) = 5(6 × 9) = 5(54) = 270. Now, “One-half of a number” is 2 x. The word “is” means equals. So, you have This is an even number and not the answer 1 written 2 x = . that you are looking for. The last phrase is “8 less than two-thirds of Choice b: acb0 the number.” The phrase “less than” means This expression requires that you eval- uate b0 ﬁrst. This is an important rule to to subtract and switch the order of the num- remember. Any term raised to the zero bers. The reason for reversing the order of power is 1. Well, a × c is an odd number 2 the terms is that 8 is deducted from 3 of the times an odd number. The product of any 2 number. Thus, the last part is 3 x – 8. two odd numbers is an odd number. Thus, 1 2 The equation to solve is 2 x = ( 3 )x – 8. an odd number times 1 is an odd number. Choice b is an odd number. There is no need Finally, you have to solve the equation. to try the other expressions. 1 2 2 x = ( 3 )x – 8 30. a. This question fortunately, or unfortunately, 1 1 – 2x – 2x requires simple memorization. You must Find a common → denominator in order to 0 = 64x – 8 remember the properties of a parallelogram in order to get this question correct. There are subtract the like terms. 3 – 6x six basic properties of every parallelogram. 1 0 = 6x – 8 They are: +8 +8 1. The opposite sides of a parallelogram 6 → Multiply both sides by 1 8 = 6x 1 are congruent. in order to solve for x. 2. The opposite sides of a parallelogram 48 = x are parallel. The answer is choice c. 3. The opposite angles of a parallelogram 29. b. This problem can be difﬁcult if you simply are congruent. look at it and try to guess. It becomes easier 4. The consecutive angles of a parallelo- if you try each answer by substituting into gram are supplementary. the expression. Here is a way of doing it. 5. The diagonals of a parallelogram bisect Choice a: a(bc) each other. You are told that a and c are odd and b 6. The diagonal of a parallelogram is even. Following order of operations, you divides the parallelogram into two multiply bc ﬁrst. Remember, that an even × congruent triangles. odd = even number. This is always true. 168
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 169 – THE SAT MATH SECTION – Every parallelogram has these six proper- 36. c. The total amount of proﬁt according to the ties. However, speciﬁc types of parallelograms, graph is 9% of the year’s income. Therefore, 225,198 × .09 = 20,267.2. such as rectangles, rhombus, and squares, have additional properties. One of the properties 37. b. First, solve for x: shared by both rectangles and squares happens x2 – 1 = 36 Add 1 to both sides. to be that the diagonals are congruent. So, the +1 +1 answer is choice a. Not every parallelogram x2 = 37 has this property, only speciﬁc parallelograms x2 = 37 such as rectangles or squares. Take the square root of both 31. d. 52% is the same as .52 (drop the % sign and sides. x2 = move the decimal point two places to the 37 13 26 52 left); 25 = 50 = 100 ; 52 × 100 = .52; And 52 × x= 37 10–2 = 52 × .01 = .52. Obviously, .052 does not equal .52, so your answer is d. 37 is an irrational number. Irrational 32. c. The mean is the average. First, you add 80 + numbers cannot be expressed as a ratio of 85 + 90 + 90 + 95 + 95 + 95 + 100 + 100 = two integers. (Simply put, irrational num- 830. Divide by the number of tests: 830 ÷ 9 = bers have decimal extensions that never ter- 92.22, which shows that statement I is false. minate or extensions that never repeat.) The median is the middle number, which is A prime number has only two positive 95. And the mode is the number that appears factors, itself and 1. Rational numbers can be most frequently, which is also 95; therefore, expressed as a ratio of two integers. The set statement II is correct. of integers is: { . . . –3, –2, –1, 0, 1, 2, 3, . . . }. 33. d. An obtuse angle measures greater than 90°. A 37 is not prime, rational, or an integer. square has four angles that are 90° each, as You can use your calculator to see that it is 6 does a rectangle and cube. The angles inside a with a decimal extension that neither termi- triangle add up to 180°, and one angle in a nates nor repeats. right triangle is 90°, so the other two add up to 38. c. An effective way ﬁgure out this question is to 90°, so there cannot be one angle that alone plug in some low, easy numbers to see what has more than 90 degrees. Therefore, the will happen. Below we picked (1,7) as our answer is d. point A and (5,15) as our point B. (Note that 5 20 34. c. Set up a proportion: 100 = x . Cross multi- the x-coordinate of our point B is 4 greater ply: 5x = 2,000. Then divide both sides by 5 than the x-coordinate of our point A.) to get x = 400. This is only the ﬁrst part of x y the problem. If you chose answer d, you for- 0 5 got to do the next step, which is to ﬁnd what 1 7 pick as A 50 x number is 50% of 400; 100 = 400 , or reduce 2 9 1 x to 2 = 400 . Then again, cross multiply: 400 = 3 11 2x. Divide both sides by 2 to get x = 200. 4 13 35. d. If you look at the pattern, you will see it is 5 15 pick as B 3x – 1. Plug in some numbers, like 3(1) – 1 = 6 17 2, 3(2) – 1 = 5, 3(3) – 1 = 8, etc. You can see that since every other number is even, of the As you can see, the y-coordinate of B is ﬁrst 100 terms, half will be even. 8 greater than the y-coordinate of A. 169
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 170 – THE SAT MATH SECTION – 39. c. Converting mixed numbers into improper denominator (35) into the numerator (81). Any remainder becomes part of the mixed fractions is a two-step process. First, multiply number (35 goes into 81 twice with a the whole number by the denominator (bot- 11 remainder of 11, hence 2 35 ). tom number) of the fraction. Then add that 40. d. We use D = RT, and rearrange for T. Divid- number to the numerator of the fraction. So 2 9 4 9 ing both sides by R, we get T = D ÷ R. The 1 7 becomes 7 and 1 5 becomes 5 . Since Area total distance, D = (x + y), and R = 2 mph. 9 9 81 11 = length × width, 7 × 5 = = 2 35 . Remem- 35 Thus, T = D ÷ R becomes T = (x + y) ÷ 2. 81 ber, to convert the improper fraction ( 35 ) back into a mixed number, you divide the 170
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 171 – THE SAT MATH SECTION – P art 2: Grid-in Questions provided that the answer ﬁts. If you are entering a decimal, do not begin with a 0. For example, sim- ply enter .5 if you get 0.5 for an answer. “Grid-in”questions are also called student-response ques- Enter mixed numbers as improper fractions or ■ tions because no answer choices are given; you, the stu- decimals. This is important for you to know when dent, generate the response. Otherwise, grid-in questions working on the grid-in section. As a math stu- are just like ﬁve-choice questions. In responding to the dent, you are used to always simplifying answers grid-in questions on the SAT, there are several things you to their lowest terms and often converting will need to know about the special four-column grid. improper fractions to mixed numbers. On this Become familiar with the answer grid below. section of the test, however, just leave improper fractions as they are. For example, it is impossible 1 3 to grid 1 2 in the answer grid, so simply grid in 2 / / instead. You could also grid in its decimal form of 1.5. Either answer is correct. • • • • If the answer ﬁts the grid, do not change its form. ■ 0 0 0 If you get a fraction that ﬁts into the grid, do not 1 1 1 1 waste time changing it to a decimal. Changing the 2 2 2 2 form of an answer can result in a miscalculation 3 3 3 3 4 4 4 4 and is completely unnecessary. 5 5 5 5 Enter the decimal point ﬁrst, followed by the ﬁrst ■ 6 6 6 6 three digits of a long or repeating decimal. Do not 7 7 7 7 round the answer. It won’t be marked as wrong if 8 8 8 8 you do, but it is not necessary. 9 9 9 9 If the answer is a fraction that requires more than ■ 17 four digits, like 25 , write the answer as a decimal The above answer grid can express whole numbers 17 instead. The fraction 25 does not ﬁt into the grid from 0 to 9999, as well as some fractions and decimals. and it cannot be reduced; therefore, you must To grid an answer, write it in the top row of the column. turn it into a decimal by dividing the numerator If you need to write a decimal point or a fraction bar, by the denominator. In this case, the decimal skip a column and ﬁll in the necessary oval below it. would be .68. If a grid-in answer has more than one possibility, ■ Very important: No grid-in questions will have a ■ enter any of the possible answers. This can occur negative answer. If you get a negative number, when the answer is an inequality or the solution you have done something wrong. to a quadratic equation. For example, if the Write the answer in the column above the oval. ■ answer is x < 5, enter a 4. If the answer is x = ± 3, The answer you write will be completely disre- enter positive 3, since negative numbers cannot garded because the scoring machine will only be entered into the grid. read the ovals. It is still important to write this If you are asked for a percentage, only grid the ■ answer, however, because it will help you check numerical value without the percentage sign. your work at the end of the test and ensure that There is no way to grid the symbol, so it is simply you marked the appropriate ovals. not needed. For example, 54% should be gridded Answers that need fewer than four columns, except ■ as .54. Don’t forget the decimal point! 0, may be started in any of the four columns, 171
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 172 – THE SAT MATH SECTION – 3 4 / 7 3 4 . 7 . 3 4 7 / / / / / • • • • • • • • • • 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 Remember these important tips: Be especially careful that a fraction bar or deci- ■ If you write in the correct answer but do not mal point is not marked in the same column as a ﬁll in the oval(s), you will get the question digit. marked wrong. If you know the correct answer but ﬁll in the Now it is time to do some grid-in practice prob- wrong oval(s), you will get the question lems. Be sure to review the strategies listed above to marked wrong. ensure that you fully understand the grid system. If you do not fully erase an answer, it may be Remember: You will never be penalized for an incorrect marked wrong. answer on the grid-in questions—so go ahead and guess. Check your answer grid to be sure you didn’t Good luck! mark more than one oval per column. 172