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T ᎏ r ᎏ y ᎏ ᎏ a ᎏ ᎏ s ᎏ p ᎏ e ᎏ c ᎏ i ᎏ f ᎏ i ᎏ c ᎏ ᎏ n ᎏ u ᎏ m ᎏ b ᎏ e ᎏ r ᎏ . ᎏ Let b ϭ 1. Then a ϭ 4b ϭ 4. So the average ϭ ᎏ l ϩ 2 4 ᎏ ϭ ᎏ 5 2 ᎏ . Look at choices where b ϭ 1. The only choice that gives ᎏ 5 2 ᎏ is Choice C. EXAMPLE 5 The sum of three consecutive even integers is P. Find the sum of the next three consecutive odd integers that follow the greatest of the three even integers. (A) P ϩ 9 (B) P ϩ 15 (C) P ϩ 12 (D) P ϩ 20 (E) None of these. Choice B is correct. T ᎏ r ᎏ y ᎏ ᎏ s ᎏ p ᎏ e ᎏ c ᎏ i ᎏ f ᎏ i ᎏ c ᎏ ᎏ n ᎏ u ᎏ m ᎏ b ᎏ e ᎏ r ᎏ s ᎏ . ᎏ Let the three consecutive even integers be 2, 4, 6. So, 2 ϩ 4 ϩ 6 ϭ P ϭ 12. The next three consecutive odd integers that follow 6 are: 7, 9, 11 So the sum of 7 ϩ 9 ϩ 11 ϭ 27. Now, where P ϭ 12, look for a choice that gives you 27: (A) P ϩ 9 ϭ 12 ϩ 9 ϭ 21—NO (B) P ϩ 15 ϭ 12 ϩ 15 ϭ 27—YES EXAMPLE 6 If 3 Ͼ a, which of the following is not true? (A) 3 Ϫ 3 Ͼ a Ϫ 3 (B) 3 ϩ 3 Ͼ a ϩ 3 (C) 3(3) Ͼ a(3) (D) 3 Ϫ 3 Ͼ 3 Ϫ a (E) ᎏ 3 3 ᎏ Ͼ ᎏ a 3 ᎏ Choice D is correct. T ᎏ r ᎏ y ᎏ ᎏ s ᎏ p ᎏ e ᎏ c ᎏ i ᎏ f ᎏ i ᎏ c ᎏ ᎏ n ᎏ u ᎏ m ᎏ b ᎏ e ᎏ r ᎏ s ᎏ . ᎏ Work backward from Choice E if you wish. Let a ϭ 1. Choice E: ᎏ 3 3 ᎏ Ͼ ᎏ a 3 ᎏ ϭ ᎏ 1 3 ᎏ TRUE STATEMENT Choice D: 3 Ϫ 3 Ͼ 3 Ϫ a ϭ 3 Ϫ l or 0 Ͼ 2 FALSE STATEMENT EXAMPLE 7 In the figure of intersecting lines above, which of the fol- lowing is equal to 180 Ϫ a? (A) a ϩ d (B) a ϩ 2d (C) c ϩ b (D) b ϩ 2a (E) c ϩ d Choice A is correct. T ᎏ r ᎏ y ᎏ ᎏ a ᎏ ᎏ s ᎏ p ᎏ e ᎏ c ᎏ i ᎏ f ᎏ i ᎏ c ᎏ ᎏ n ᎏ u ᎏ m ᎏ b ᎏ e ᎏ r ᎏ . ᎏ Let Then 2a ϭ 40° Be careful now—all of the other angles are now deter - mined, so don’t choose any more. Because vertical angles are equal, 2a ϭ b,so . Now c ϩ b ϭ 180°, so c ϩ 40 ϭ 180 and . Thus, (vertical angles are equal). Now look at the question: 180 Ϫ a ϭ 180 Ϫ 20 ϭ 160 Which is the correct choice? (A) a ϩ d ϭ 20 ϩ 140 ϭ 160—that’s the one! d ϭ 140° c ϭ 140° b ϭ 40° a ϭ 20° STRATEGY SECTION • 89 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 89 EXAMPLE 1 If p is a positive integer, which could be an odd integer? (A) 2p ϩ 2 (B) p 3 Ϫ p (C) p 2 ϩ p (D) p 2 Ϫ p (E) 7p Ϫ 3 Choice E is correct. Start with Choice E first since you have to test out the choices. Method 1: Try a number for p. Let p ϭ 1. Then (starting with choice E) 7p Ϫ 3 ϭ 7(1) Ϫ 3 ϭ 4. 4 is even, so try another number for p to see whether 7p Ϫ 3 is odd. Let p ϭ 2. 7p Ϫ 3 ϭ 7(2) Ϫ 3 ϭ 11. 11 is odd. Therefore, Choice E is correct. Method 2: Look at Choice E. 7p could be even or odd, depending on what p is. If p is even, 7p is even. If p is odd, 7p is odd. Accordingly, 7p Ϫ 3 is either even or odd. Thus, Choice E is correct. Note: By using either Method 1 or Method 2, it is not necessary to test the other choices. EXAMPLE 2 If y ϭ x 2 ϩ 3, then for which value of x is y divisible by 7? (A) 10 (B) 8 (C) 7 (D) 6 (E) 5 Choice E is correct. Since you must check all of the choices, start with Choice E: y ϭ 5 2 ϩ 3 ϭ 25 ϩ 3 ϭ 28 28 is divisible by 4 (Answer) If you had started with Choice A, you would have had to test four choices, instead of one choice before finding the correct answer. EXAMPLE 3 Which fraction is greater than ᎏ 1 2 ᎏ ? (A) ᎏ 4 9 ᎏ (B) ᎏ 1 3 7 5 ᎏ (C) ᎏ 1 6 3 ᎏ (D) ᎏ 1 2 2 5 ᎏ (E) ᎏ 1 8 5 ᎏ Choice E is correct. L ᎏ o ᎏ o ᎏ k ᎏ ᎏ a ᎏ t ᎏ ᎏ C ᎏ h ᎏ o ᎏ i ᎏ c ᎏ e ᎏ ᎏ E ᎏ ᎏ f ᎏ i ᎏ r ᎏ s ᎏ t ᎏ . ᎏ Is ᎏ 2 l ᎏ Ͼ ᎏ 1 8 5 ᎏ ? Use the cross-multiplication method. ᎏ 1 2 ᎏ ᎏ 1 8 5 ᎏ 15 16 15 Ͻ 16 So, ᎏ 1 2 ᎏ Ͻ ᎏ 1 8 5 ᎏ You also could have looked at Choice E and said ᎏ 1 8 6 ᎏ ϭ ᎏ 1 2 ᎏ and realized that ᎏ 1 8 5 ᎏ Ͼ ᎏ 1 2 ᎏ because ᎏ 1 8 5 ᎏ has a smaller denominator than ᎏ 1 8 6 ᎏ . 90 • STRATEGY SECTION When Each Choice Must Be Tested, Start with Choice E and Work Backward If you must check each choice for the correct answer, start with Choice E and work backward. The reason for this is that the test maker of a question in which each choice must be tested often puts the correct answer as Choice D or E. In this way, the careless student must check all or most of the choices before finding the correct one. So if you’re trying all the choices, start with the last choice, then the next to last choice, etc. MATH STRATEGY 8 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 90 EXAMPLE 4 If n is an even integer, which of the following is an odd integer? (A) n 2 Ϫ 2 (B) n Ϫ 4 (C) (n Ϫ 4) 2 (D) n 3 (E) n 2 Ϫ n Ϫ 1 Choice E is correct. L ᎏ o ᎏ o ᎏ k ᎏ ᎏ a ᎏ t ᎏ ᎏ C ᎏ h ᎏ o ᎏ i ᎏ c ᎏ e ᎏ ᎏ E ᎏ ᎏ f ᎏ i ᎏ r ᎏ s ᎏ t ᎏ . ᎏ n 2 Ϫ n Ϫ 1 If n is even n 2 is even n is even 1 is odd So, n 2 Ϫ n Ϫ 1 ϭ even Ϫ even Ϫ odd ϭ odd. EXAMPLE 5 Which of the following is an odd number? (A) 7 ϫ 22 (B) 59 Ϫ 15 (C) 55 ϩ 35 (D) 75Ϭ 15 (E) 4 7 Choice D is correct. L ᎏ o ᎏ o ᎏ k ᎏ ᎏ a ᎏ t ᎏ ᎏ C ᎏ h ᎏ o ᎏ i ᎏ c ᎏ e ᎏ ᎏ E ᎏ ᎏ f ᎏ i ᎏ r ᎏ s ᎏ t ᎏ . ᎏ 4 7 is even, since 4 ϫ 4 ϫ 4 is even So now look at Choice D: ᎏ 7 5 5 ᎏ ϭ 5, which is odd. EXAMPLE 6 3 Ե 2 ϫ 8 28 ଙ 6 If Ե and ଙ are different digits in the correctly calculated multiplication problem above, then Ե could be (A) 1 (B) 2 (C) 3 (D) 4 (E) 6 Choice E is correct. T ᎏ r ᎏ y ᎏ ᎏ C ᎏ h ᎏ o ᎏ i ᎏ c ᎏ e ᎏ ᎏ E ᎏ ᎏ f ᎏ i ᎏ r ᎏ s ᎏ t ᎏ . ᎏ 3 Ե 232 ϫ 8 ϫ 8 28 ଙ 9289 9 and 6 are different numbers, so Choice E is correct. EXAMPLE 7 Which choice describes a pair of numbers that are un equal? (A) ᎏ 1 6 ᎏ , ᎏ 1 6 1 6 ᎏ (B) 3.4, ᎏ 3 1 4 0 ᎏ (C) ᎏ 1 7 5 5 ᎏ , ᎏ 1 5 ᎏ (D) ᎏ 3 8 ᎏ , 0.375 (E) ᎏ 8 2 6 4 ᎏ , ᎏ 4 1 2 0 ᎏ Choice E is correct. L ᎏ o ᎏ o ᎏ k ᎏ ᎏ a ᎏ t ᎏ ᎏ C ᎏ h ᎏ o ᎏ i ᎏ c ᎏ e ᎏ ᎏ E ᎏ ᎏ f ᎏ i ᎏ r ᎏ s ᎏ t ᎏ . ᎏ ᎏ 8 2 6 4 ᎏ ? ᎏ 4 1 2 0 ᎏ Cross multiply: ᎏ 8 2 6 4 ᎏᎏ 4 1 2 0 ᎏ 860 ends in 0 24 ϫ 42 ends in 8 Thus, the numbers must be different and unequal. EXAMPLE 8 ଙ 3 4 ଙ ଙ 1 6 ଙ ଙ 3 2 ෆ 0 ෆ 3 ෆ In the above addition problem, the symbol ଙ describes a particular digit in each number. What must ଙ be in order to make the answer correct? (A) 7 (B) 6 (C) 5 (D) 4 (E) 3 Choice E is correct. Try substituting the number in Choice E first for the ଙ . ଙ 333 4 ଙ 43 ଙ 131 6 ଙ 63 ଙ 333 2 ෆ 0 ෆ 3 ෆ 2 ෆ 0 ෆ 3 ෆ Since you get 203 for the addition, Choice E is correct. 6 9 STRATEGY SECTION • 91 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 91 92 • STRATEGY SECTION EXAMPLE 1 The diagram below shows two paths: Path 1 is 10 miles long, and Path 2 is 12 miles long. If Person X runs along Path 1 at 5 miles per hour and Person Y runs along Path 2 at y miles per hour, and if it takes exactly the same amount of time for both runners to run their whole path, then what is the value of y? (A) 2 (B) 4 1 / 6 (C) 6 (D) 20 (E) 24 Choice C is correct. Let T ϭ Time (in hours) for either runner to run the whole path. Using R ϫ T ϭ D, for Person X, we have (5 mi/hr)(T hours) ϭ 10 miles or 5T ϭ 10 or T ϭ 2 For Person Y, we have ( y mi/hr)(T hours) ϭ 12 miles or yT ϭ 12 Using y(2) ϭ 12 or y ϭ 6 EXAMPLE 2 A car traveling at 50 miles per hour for two hours travels the same distance as a car traveling at 20 miles per hour for x hours. What is x? (A) ᎏ 4 5 ᎏ (B) ᎏ 5 4 ᎏ (C) 5 (D) 2 (E) ᎏ 1 2 ᎏ Choice C is correct. U ᎏ s ᎏ e ᎏ ᎏ R ᎏ ᎏ ϫ ᎏ ᎏ T ᎏ ᎏ ϭ ᎏ ᎏ D ᎏ . ᎏ Call distance both cars travel, D (since distance is same for both cars). So we get: 50 ϫ 2 ϭ D(ϭ100) 20 ϫ x ϭ D(ϭ100) Solving you can see that x ϭ 5. EXAMPLE 3 John walks at a rate of 4 miles per hour. Sally walks at a rate of 5 miles per hour. If both John and Sally both start at the same starting point, how many miles is one person from the other after T hours of walking? (Note: Both are walking on the same road in the same direction.) (A) ᎏ 2 t ᎏ (B) t (C) 2t (D) ᎏ 4 5 ᎏ t (E) ᎏ 5 4 ᎏ t Choice B is correct. 2 2 1 1 1 Know How to Solve Problems Using the Formula R ϫ T ϭ D Almost every problem involving motion can be solved using the formula R ϫ T ϭ D or rate ϫ elapsed time ϭ distance MATH STRATEGY 9 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 92 Draw a diagram: John (4 mph) Sally (5 mph) Let D J be distance that John walks in t hours. Let D S be distance that Sally walks in t hours. Then, using R ϫ T ϭ D, for John: 4 ϫ t ϭ D J for Sally: 5 ϫ t ϭ D S The distance between Sally and John after T hours of walking is: D S Ϫ D J ϭ 5t Ϫ 4t ϭ t EXAMPLE 4 A man rode a bicycle a straight distance at a speed of 10 miles per hour and came back the same distance at a speed of 20 miles per hour. What was the man’s total number of miles for the trip back and forth, if his total traveling time was 1 hour? (A) 15 (B) 7 1 / 2 (C) 6 1 / 3 (D) 6 2 / 3 (E) 13 1 / 3 Choice E is correct. Always use R ϫ T ϭ D (Rate ϫ Time ϭ Distance) in problems like this. Call the first distance D and the time for the first part, T 1 . Since he rode at 10 mph: 10 ϫ T 1 ϭ D Now for the trip back. He rode at 20 mph. Call the time it took to go back, T 2 . Since he came back the same dis- tance, we can call that distance D also. So for the trip back using R ϫ T ϭ D, we get: 20 ϫ T 2 ϭ D Since it was given that the total traveling time was 1 hour, the total traveling time is: T 1 ϩ T 2 ϭ 1 Now here’s the trick: Let’s make use of the fact that T 1 ϩ T 2 ϭ 1. Dividing Equation by 10 we get: T 1 ϭ ᎏ 1 D 0 ᎏ Dividing Equation by 20 we get: T 2 ϭ ᎏ 2 D 0 ᎏ Now add T 1 ϩ T 2 and we get: T 1 ϩ T 2 ϭ 1 ϭ ᎏ 1 D 0 ᎏ ϩ ᎏ 2 D 0 ᎏ Factor D: 1 ϭ D ᎏ 1 1 0 ᎏ ϩ ᎏ 2 1 0 ᎏ Add ᎏ 1 1 0 ᎏ ϩ ᎏ 2 1 0 ᎏ . Remember the fast way of adding fractions? ᎏ 1 1 0 ᎏ ϩ ᎏ 2 1 0 ᎏ ϭ ᎏ 2 2 0 0 ϩ ϫ 1 1 0 0 ᎏ ϭ ᎏ 2 3 0 0 0 ᎏ So: 1 ϭ (D) ᎏ 2 3 0 0 0 ᎏ Multiply by 200 and divide by 30 and we get: ᎏ 2 3 0 0 0 ᎏ ϭ D; D ϭ 6 ᎏ 2 3 ᎏ Don’t forget, we’re looking for 2D: 2D ϭ 13 ᎏ 1 3 ᎏ EXAMPLE 5 What is the average rate of a bicycle traveling at 10 mph a distance of 5 miles and at 20 mph the same distance? (A) 15 mph (B) 20 mph (C) 12 1 / 2 mph (D) 13 1 / 3 mph (E) 16 mph Choice D is correct. Ask yourself, what does average rate mean? It does not mean the average of the rates! If you thought it did, you would have selected Choice A as the answer (averaging 10 and 20 to get 15)—the “lure” choice. Average is a word that modifies the word rate in this case. So you must define the word rate first, before you do anything with averaging. Since Rate ϫ Time ϭ Distance, Rate ϭ ᎏ D T is i t m an e ce ᎏ 2 1 2 1 STRATEGY SECTION • 93 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 93 94 • STRATEGY SECTION Then average rate must be: Average rate ϭ ᎏ TO T T O A T L AL dis ti t m an e ce ᎏ The total distance is the distance covered on the whole trip, which is 5 ϩ 5 ϭ 10 miles. The total time is the time traveled the first 5 miles at 10 mph added to the time the bicycle traveled the next 5 miles at 20 mph. Let t 1 be the time the bicycle traveled first 5 miles. Let t 2 be the time the bicycle traveled next 5 miles. Then the total time ϭ t 1 ϩ t 2 . Since R ϫ T ϭ D, for the first 5 miles: 10 ϫ t 1 ϭ 5 for the next 5 miles: 20 ϫ t 2 ϭ 5 Finding t 1 : t 1 ϭ ᎏ 1 5 0 ᎏ Finding t 2 : t 2 ϭ ᎏ 2 5 0 ᎏ So, t 1 ϩ t 2 ϭ ᎏ 1 5 0 ᎏ ϩ ᎏ 2 5 0 ᎏ ϭ ᎏ 1 2 ᎏ ϩ ᎏ 1 4 ᎏ ϭ ᎏ 4 ϩ 8 2 ᎏ (remembering how to quickly add fractions) ϭ ᎏ 6 8 ᎏ ϭ ᎏ 3 4 ᎏ Average rate ϭ ϭ ϭ (5 ϩ 5) ϫ ᎏ 4 3 ᎏ ϭ 10 ϫ ᎏ 4 3 ᎏ ϭ ᎏ 4 3 0 ᎏ ϭ 13 ᎏ 1 3 ᎏ (Answer) Here’s a formula you can memorize: If a vehicle travels a certain distance at a mph and trav- els the same distance at b mph, the average rate is ᎏ a 2 ϩ ab b ᎏ . Try doing the problem using this formula: ᎏ a 2 ϩ ab b ᎏ ϭ ᎏ 2 ϫ 1 ( 0 10 ϩ ) ϫ 20 (20) ᎏ ϭ ᎏ 4 3 0 0 0 ᎏ ϭ 13 ᎏ 1 3 ᎏ Caution: Use this formula only when you are looking for average rate and when the distance is the same for both speeds. 5 ϩ 5 ᎏ ᎏ 3 4 ᎏ TOTAL DISTANCE ᎏᎏᎏ TOTAL TIME Know How to Use Units of Time, Distance, Area, or Volume to Find or Check Your Answer EXAMPLE 1 What is the distance in miles covered by a car that trav- eled at 50 miles per hour for 5 hours? (A) 10 (B) 45 (C) 55 (D) 200 (E) 250 Choice E is correct. Although this is an easy “R ϫ T ϭ D” problem, it illustrates this strategy very well. Recall that rate ϫ time ϭ distance (50 mi./hr.)(5 hours)ϭ distance Notice that when I substituted into R ϫ T ϭ D, I kept the units of rate and time (miles/hour and hours). Now I will treat these units as if they were ordinary variables. Thus, distance ϭ (50 mi./hr.)(5 hours) By knowing what the units in your answer must be, you will often have an easier time finding or checking your answer. A very helpful thing to do is to treat the units of time or space as variables (like “x” or “y”). Thus, you should substitute, multiply, or divide these units as if they were ordi- nary variables. The following examples illustrate this idea. MATH STRATEGY 10 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 94 I have canceled the variable “hour(s)” from the numerator and denominator of the right side of the equation. Hence, distance ϭ 250 miles The distance has units of “miles” as I would expect. In fact, if the units in my answer had been “miles/hour” or “hours,” then I would have been in error. Thus, the general procedure or problems using this strategy is: Step 1. K ᎏ e ᎏ e ᎏ p ᎏ ᎏ t ᎏ h ᎏ e ᎏ ᎏ u ᎏ n ᎏ i ᎏ t ᎏ s ᎏ ᎏ g ᎏ i ᎏ v ᎏ e ᎏ n ᎏ ᎏ i ᎏ n ᎏ ᎏ t ᎏ h ᎏ e ᎏ ᎏ q ᎏ u ᎏ e ᎏ s ᎏ t ᎏ i ᎏ o ᎏ n ᎏ . ᎏ Step 2. T ᎏ r ᎏ e ᎏ a ᎏ t ᎏ ᎏ t ᎏ h ᎏ e ᎏ ᎏ u ᎏ n ᎏ i ᎏ t ᎏ s ᎏ ᎏ a ᎏ s ᎏ ᎏ o ᎏ r ᎏ d ᎏ i ᎏ n ᎏ a ᎏ r ᎏ y ᎏ ᎏ v ᎏ a ᎏ r ᎏ i ᎏ a ᎏ b ᎏ l ᎏ e ᎏ s ᎏ . ᎏ Step 3. M ᎏ a ᎏ k ᎏ e ᎏ ᎏ s ᎏ u ᎏ r ᎏ e ᎏ ᎏ t ᎏ h ᎏ e ᎏ ᎏ a ᎏ n ᎏ s ᎏ w ᎏ e ᎏ r ᎏ ᎏ h ᎏ a ᎏ s ᎏ ᎏ u ᎏ n ᎏ i ᎏ t ᎏ s ᎏ ᎏ t ᎏ h ᎏ a ᎏ t ᎏ ᎏ y ᎏ o ᎏ u ᎏ ᎏ w ᎏ o ᎏ u ᎏ l ᎏ d ᎏ e ᎏ x ᎏ p ᎏ e ᎏ c ᎏ t ᎏ . ᎏ EXAMPLE 2 How many inches is equivalent to 2 yards, 2 feet, and 7 inches? (A) 11 (B) 37 (C) 55 (D) 81 (E) 103 Choice E is correct. Remember that 1 yard ϭ 3 feet 1 foot ϭ 12 inches Treat the units of length as variables! Divide by 1 yard, and by 1 foot, to get 1 ϭ ᎏ 1 3 y fe a e rd t ᎏ 1 ϭ ᎏ 12 1 in fo c o h t es ᎏ We can multiply any expression by 1 and get the same value. Thus, 2 yards ϩ 2 feet ϩ 7 inches ϭ (2 yards)(1)(1) ϩ (2 feet)(1) ϩ 7 inches Substituting and into , 2 yards ϩ 2 feet ϩ 7 inches ϭ 2 yards ᎏ 3 ya fe r e d t ᎏ ᎏ 12 f i o n o c t hes ᎏ ϩ2 feet ᎏ 12 f i o n o c t hes ᎏ ϩ7 inches ϭ 72 inches ϩ 24 inches ϩ 7 inches ϭ 103 inches Notice that the answer is in “inches” as I expected. If the answer had come out in “yards” or “feet,” then I would have been in error. EXAMPLE 3 A car wash cleans x cars per hour, for y hours at z dollars per car. How much money in cents did the car wash receive? (A) ᎏ 10 xy 0z ᎏ (B) ᎏ 1 x 0 yz 0 ᎏ (C) 100xyz (D) ᎏ 10 yz 0x ᎏ (E) ᎏ 10 yz 0x ᎏ Choice C is correct. Use units: ᎏ x h c o a u r r s ᎏ ( y hours ) ᎏ z d c o a ll r ars ᎏ ϭ xyz dollars Multiply by 100. We get 100xyz cents. EXAMPLE 4 There are 3 feet in a yard and 12 inches in a foot. How many yards are there altogether in 1 yard, 1 foot, and 1 inch? (A) 1 ᎏ 1 3 ᎏ (B) 1 ᎏ 1 3 3 6 ᎏ (C) 1 ᎏ 1 1 1 8 ᎏ (D) 2 ᎏ 1 5 2 ᎏ (E) 4 ᎏ 1 1 2 ᎏ Choice B is correct. Know how to work with units. Given: 3 feetϭ 1 yard 12 inches ϭ 1 foot Thus, 1 yard ϩ 1 foot ϩ 1 inch ϭ 1 yard ϩ 1 foot ᎏ 1 3 y fe a e rd t ᎏ ϩ 1 inch ᎏ 12 1 in fo c o h t es ᎏ ϭ ᎏ 1 3 y fe a e rd t ᎏ ϭ 1 ϩ ᎏ 1 3 ᎏ ϩ ᎏ 3 1 6 ᎏ yards ϭ 1 ϩ ᎏ 1 3 2 6 ᎏ ϩ ᎏ 3 1 6 ᎏ yards ϭ 1 ᎏ 1 3 3 6 ᎏ yards 5 1 1 5 43 4 3 2 1 2 1 STRATEGY SECTION • 95 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 95 96 • STRATEGY SECTION EXAMPLE 1 If the symbol is defined by the equation a b ϭ a Ϫ b Ϫ ab for all a and b, then Ϫ ᎏ 1 3 ᎏ (Ϫ3) ϭ (A) ᎏ 5 3 ᎏ (B) ᎏ 1 3 1 ᎏ (C) Ϫ ᎏ 1 3 3 ᎏ (D) Ϫ4 (E) Ϫ5 Choice A is correct. All that is required is substitution: a b ϭ a Ϫ b Ϫ ab Ϫ ᎏ 1 3 ᎏ (Ϫ3) Substitute Ϫ ᎏ 1 3 ᎏ for a and Ϫ3 for b in a Ϫ b Ϫ ab: Ϫ ᎏ 1 3 ᎏ (Ϫ3) ϭϪ ᎏ 1 3 ᎏ Ϫ (Ϫ3) Ϫ Ϫ ᎏ 1 3 ᎏ (Ϫ3) ϭϪ ᎏ 1 3 ᎏ ϩ 3 Ϫ 1 ϭ 2 Ϫ ᎏ 1 3 ᎏ ϭ ᎏ 5 3 ᎏ (Answer) EXAMPLE 2 Let ϭ Ά ᎏ 5 2 ᎏ (x ϩ 1) if x is an odd integer ᎏ 5 2 ᎏ x if x is an even integer Find , where y is an integer. (A) ᎏ 5 2 ᎏ y (B) 5y (C) ᎏ 5 2 ᎏ y ϩ 1 (D) 5y ϩ ᎏ 5 2 ᎏ (E) 5y ϩ 5 Choice B is correct. All we have to do is to substitute 2y into the definition of . In order to know which defini- tion of to use, we want to know if 2y is even. Since y is an integer, then 2y is an even integer. Thus, ϭ ᎏ 5 2 ᎏ (2y) or ϭ 5y (Answer) 2y 2y x x 2y x Use New Definitions and Functions Carefully Some SAT questions use new symbols, functions, or definitions that were created in the question. At first glance, these questions may seem difficult because you are not familiar with the new symbol, function, or definition. However, most of these questions can be solved through simple sub- stitution or application of a simple definition. MATH STRATEGY 11 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 96 EXAMPLE 3 As in the previous Example 1, ø is defined as a ø b ϭ a Ϫ b Ϫ ab. If, a ø 3ϭ 6, a ϭ (A) ᎏ 9 2 ᎏ (B) ᎏ 9 4 ᎏ (C) Ϫ ᎏ 9 4 ᎏ (D) Ϫ ᎏ 4 9 ᎏ (E) Ϫ ᎏ 9 2 ᎏ Choice E is correct. a ø b ϭ a Ϫ b Ϫ ab a ø3ϭ 6 Substitute a for a, 3 for b: a ø 3 ϭ a Ϫ 3 Ϫ a(3) ϭ 6 ϭ a Ϫ 3 Ϫ 3a ϭ 6 ϭϪ2a Ϫ 3 ϭ 6 2a ϭϪ9 a ϭϪ ᎏ 9 2 ᎏ EXAMPLE 4 The symbol is defined as the greatest integer less than or equal to x. (A) 16 (B) 16.6 (C) 17 (D) 17.6 (E) 18 Choice C is correct. is defined as the greatest integer less than or equal to Ϫ3.4. This is Ϫ4, since Ϫ4 ϽϪ3.4. is defined as the greatest integer less than or equal to 21. That is just 21, since 21 ϭ 21. Thus, Ϫ4 ϩ 21 ϭ 17 EXAMPLE 5 is defined as xz Ϫ yt ϭ Choice E is correct. ϭ xz Ϫ yt; ϭ ? Substituting 2 for x, 1 for z, 1 for y, and 1 for t, ϭ (2)(1) Ϫ (1)(1) ϭ 1 Now work from Choice E: (E) ϭ xz Ϫ yt ϭ (3)(1) Ϫ (1)(2) ϭ 3 Ϫ 2 ϭ 1 EXAMPLE 6 If for all numbers a, b, c the operation ᭹ is defined as a ᭹ b ϭ ab Ϫ a then a ᭹ (b ᭹ c) ϭ (A) a(bc Ϫ b Ϫ 1) (B) a(bc ϩ b ϩ 1) (C) a(bc Ϫ c Ϫ b Ϫ 1) (D) a(bc Ϫ b ϩ 1) (E) a(b Ϫ a ϩ c) Choice A is correct. STRATEGY SECTION • 97 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 97 98 • STRATEGY SECTION a ᭹ b ϭ ab Ϫ a a ᭹ (b ᭹ c) ϭ ? Find (b ᭹ c) first. U ᎏ s ᎏ e ᎏ ᎏ s ᎏ u ᎏ b ᎏ s ᎏ t ᎏ i ᎏ t ᎏ u ᎏ t ᎏ i ᎏ o ᎏ n ᎏ : ᎏ a ᭹ b ϭ ab Ϫ a ↑↑ b ᭹ c Substitute b for a and c for b: b ᭹ c ϭ b(c) Ϫ b Now, a ᭹ (b ᭹ c) ϭ a ᭹ (bc Ϫ b) Use definition a ᭹ b ϭ ab Ϫ a Substitute a for a and bc Ϫ b for b: a ᭹ b ϭ ab Ϫ a a ᭹ (bc Ϫ b) ϭ a (bc Ϫ b) Ϫ a ϭ abc Ϫ ab Ϫ a ϭ a(bc Ϫ b Ϫ 1) In many of the examples given in these strategies, it has been explicitly stated that one should not calculate complicated quantities. In some of the examples, we have demonstrated a fast and a slow way of solving the same problem. On the actual exam, if you find that your solution to a problem involves a tedious and complicated method, then you are probably doing the problem in a long, hard way.* Almost always there will be an easier way. Examples 3, 7, and 8 can also be solved with the aid of a calculator and some with the aid of a calculator allowing for exponential calculations. However, to illustrate the effectiveness of Math Strategy 12, we did not use the calculator method of solving these examples. Try Not to Make Tedious Calculations Since There Is Usually an Easier Way EXAMPLE 1 If y 8 ϭ 4 and y 7 ϭ ᎏ 3 x ᎏ , what is the value of y in terms of x? (A) ᎏ 4 3 x ᎏ (B) ᎏ 3 4 x ᎏ (C) ᎏ 4 x ᎏ (D) ᎏ 4 x ᎏ (E) ᎏ 1 x 2 ᎏ Choice A is correct. Don’t solve for the value of y first, by finding y ϭ 4 ᎏ 1 8 ᎏ *Many times, you can DIVIDE, MULTIPLY, ADD, SUBTRACT, or FACTOR to simplify. Just divide the two equations: (Step 1) y 8 ϭ 4 (Step 4) y ϭ 4 ϫ ᎏ 3 x ᎏ (Step 2) y 7 ϭ ᎏ 3 x ᎏ (Step 5) y ϭ ᎏ 4 3 x ᎏ (Answer) (Step 3) ᎏ y y 8 7 ᎏ ϭ EXAMPLE 2 If x ϭ 1 ϩ 2 ϩ 2 2 ϩ 2 3 ϩ 2 4 ϩ 2 5 ϩ 2 6 ϩ 2 7 ϩ 2 8 ϩ 2 9 and y ϭ 1 ϩ 2x, then y Ϫ x ϭ (A) 2 7 (B) 2 8 (C) 2 9 (D) 2 10 (E) 2 11 Choice D is correct. I hope you did not calculate 1 ϩ 2 ϩ. 2 9 . If you did, then you found that x ϭ 1,023 and y ϭ 2,047 and y Ϫ x ϭ 1,024. 4 ᎏ ᎏ 3 x ᎏ MATH STRATEGY 12 1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 98 [...]...1FM-pg116.qxd :22 678_0000-FM.qxd 5/1/08 3:53 PM Page 99 STRATEGY SECTION • 99 Here is the FAST method Instead of making these tedious calculations, observe that since x ϭ 1 ϩ 2 ϩ 22 ϩ 23 ϩ 24 ϩ 25 ϩ 26 ϩ 27 ϩ 28 ϩ 29 If (a2 ϩ a)3 ϭ (a ϩ 1)3x, where a ϩ 1 1 then 2x ϭ 2 ϩ 22 ϩ 23 ϩ 24 ϩ 25 ϩ 26 ϩ 27 ϩ 28 ϩ 29 ϩ 21 0 2 and y ϭ 1 ϩ 2x ϭ 1 ϩ 2 ϩ 2 ϩ 2 ϩ 2 ϩ 25 ϩ 26 ϩ 27 ϩ 28 ϩ 29 ϩ 21 0 3 2 3 EXAMPLE 5 4... Thus, calculating 3 Ϫ 1 , we get y Ϫ x ϭ 1 ϩ 2 ϩ 22 ϩ 23 ϩ 24 ϩ 25 ϩ 26 ϩ 27 ϩ 28 ϩ 29 ϩ 21 0 Ϫ(1 ϩ 2 ϩ 22 ϩ 23 ϩ 24 ϩ 25 ϩ 26 ϩ 27 ϩ 28 ϩ 29 ) ϭ 21 0 (Answer) EXAMPLE 3 (A) a (B) a2 (C) a3 aϩ1 (D) ᎏᎏ a a (E) ᎏᎏ aϩ1 Choice C is correct Isolate x first: (a2 ϩ a)3 x ϭ ᎏᎏ (a ϩ 1)3 x3 x 3 Now use the fact that ᎏ ᎏ ϭ ᎏᎏ : 3 y y (a ϩ a)3 a2 ϩ a ᎏᎏ ϭ ᎏᎏ 3 (a ϩ 1) aϩ1 2 Use factoring to make problems simpler... Pythagorean Theorem, a2 ϩ a2 ϭ (a ϩ b )2 where a ϩ b is hypotenuse of right triangle We get: 2a2 ϭ (a ϩ b )2 Divide by (a ϩ b )2: 2a 2 ᎏᎏ ϭ 1 (a ϩ b )2 1FM-pg116.qxd :22 678_0000-FM.qxd 5/1/08 3:53 PM Page 103 STRATEGY SECTION • 103 Divide by 2: EXAMPLE 3 2 a 1 ᎏᎏ ϭ ᎏᎏ (a ϩ b )2 2 Take square roots of both sides: a 1 ᎏᎏ ϭ ᎏᎏ ϭ (a ϩ b) 2 ෆ 2 1 ͙ෆ ϭ ᎏᎏ ᎏᎏ 2 ͙ෆ ෆ 2 ͙ෆ 2 ϭ ᎏᎏ 2 (Answer) EXAMPLE 2 In the figure... Choice B is correct Divide by 8: ᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏ 3 ϭ a3 ϭ ͙ෆ ϫ ͙4 8 82 ෆ ϭ 88 ϫ 2 ϭ 176 1 ᎏᎏ 8 1 ᎏᎏ 4 1 ᎏᎏ 2 2 4 a(a ϩ 1) 3 ᎏ ϭ ΄ᎏϩ 1 ΅ a (88 )2 ϩ (88 )2( 3) ϭ 8 82( 1 ϩ 3) ϭ 8 82( 4) If 16r Ϫ 24 q ϭ 2, then 2r Ϫ 3q ϭ 3 Now factor a2 ϩ a ϭ a(a ϩ 1) ᎏᎏᎏᎏᎏᎏ So: (A) 88 (B) 176 (C) 348 (D) 350 (E) 3 52 EXAMPLE 4 0, then x ϭ 16r Ϫ 24 q 2 ᎏᎏ ϭ ᎏᎏ 8 8 1 2r Ϫ 3q ϭ ᎏᎏ 4 Get rid of the fraction Multiply both sides of the... Choice D is correct 81 Given: ᎏϫ y ϭ 21 ᎏ 27 y2 Ϫ 7y ϩ 10 Given: ᎏᎏ y 2 (Answer) 1 Factor and reduce: ᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏ Factor the numerator of 1 We get ( y Ϫ 5) ( y Ϫ 2) ᎏᎏ ϭ y Ϫ 5 2 yϪ 2 ϫ 6 ϫ 2 ϫ ϫ 2 ϫ 9 7 12 ϫ 14 ϫ 18 ᎏᎏ ϭ ᎏᎏᎏ 7 6 ϫ ϫ 8 ϫ 9 6ϫ7ϫ8ϫ9 2 2 2 8 ϭ ᎏᎏ ϭ ᎏᎏ ϭ 1 8 8 2 3 5 6 16 Substitute 8.000001 in 2 We have 8.000001 Ϫ 5 ϭ 3.000001 Ϸ 3 2 117-174.qxd 5/1/08 11:55 AM Page... 3 x3 ϭ ᎏyᎏ ϭ 3 ϭ 27 x So ᎏᎏ y 3 3 EXAMPLE 7 10 12 Ϫ10 Ϫ 12 0 m 3 m 4 n If ᎏᎏ ϭ ᎏᎏ and ᎏᎏ ϭ ᎏᎏ, then ᎏᎏ ϭ n q q 7 8 Choice B is correct Multiply x ϩ 2y ϭ 4 by 5 to get: 5x ϩ 10y ϭ 20 Now subtract 8: 5x ϩ 10y Ϫ 8 ϭ 20 Ϫ 8 ϭ 12 EXAMPLE 5 1 If 6x ϭ y and x ϭ ᎏᎏ, then y ϭ y 5 1 (A) (B) (C) (D) (E) 2 (A) x6 x5 (B) ᎏᎏ 6 12 (A) ᎏᎏ 15 12 (B) ᎏᎏ 56 56 (C) ᎏᎏ 12 32 (D) ᎏᎏ 21 21 (E) ᎏᎏ 32 Choice D is correct... Multiply both sides by 27 to get 81 ϫ y ϭ 21 ϫ 27 Choice A is correct 4 ϩ4 ϩ4 ᎏᎏ ϭ 33 ϩ 33 ϩ 33 2 2 27 ϫ 21 y ϭ ᎏᎏ 81 2 3( 42) ᎏᎏ ϭ 3(33) 16 ᎏᎏ 27 Factor and reduce: ᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏ Factor and reduce: ᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏᎏ EXAMPLE 3 yϭ7 12 ϫ 14 ϫ 18 If 6 ϫ 7 ϫ 8 ϫ 9 ϭ ᎏᎏ, then x ϭ x (A) (B) (C) (D) (E) 3 и7 ϫ 3и9 y ϭ ᎏᎏ 9и 9 и 7 ϫ 3 3 ϭ ᎏᎏ 3 и 3 EXAMPLE 5 1 ᎏᎏ 2 1 4 8 12 y2 Ϫ 7y ϩ 10 Find the... ABC has BC ϭ 10, using 2 ͙ෆ ෆ 2 statement VIII (the right triangle ᎏᎏ, ᎏᎏ, 1), multiply 2 2 by 10 to get a right triangle: (Note: Figure is not drawn to scale.) In triangle ABC, if a Ͼ c, which of the following is true? (A) (B) (C) (D) (E) 10 2 10 2 ෆ ෆ ᎏᎏ, ᎏᎏ, 10 2 2 10 2 ෆ ෆ Thus side AB ϭ ᎏᎏ ϭ 5 2 2 BC ϭ AC AB Ͼ BC AC Ͼ AB BC Ͼ AB BC Ͼ AC 10 2 ෆ side AC ϭ ᎏᎏ ϭ 5 2 ෆ 2 Choice D is correct (Remember... (D) (E) 20 25 26 45 48 Choice C is correct Method 1: Use VII above Then, x2 ϭ 24 2 ϩ 1 02 ϭ 576 ϩ 100 ϭ 676 Thus, x 5 26 (Answer) 1FM-pg116.qxd :22 678_0000-FM.qxd 5/1/08 3:53 PM Page 114 114 • STRATEGY SECTION EXAMPLE 4 The given information translates into the diagram above Note Statement VIII on p 113 The triangle above is a multiple of the special 5, 12, 13 right triangle 50 ϭ 10(5) 120 ϭ 10( 12) Thus,... Pythagorean Theorem for triangle AED: h2 ϩ 32 ϭ 52 h2 ϭ 52 Ϫ 32 h2 ϭ 25 Ϫ 9 h2 ϭ 16 hϭ4 So the perimeter is 3 ϩ h ϩ 6 ϩ 5 ϭ 3 ϩ 4 ϩ 6 ϩ 5 ϭ 18 (Answer) 1 Lines ᐉ1 and 2 are parallel AB ϭ ᎏᎏAC 3 The area of triangle ABD ᎏᎏᎏ ϭ The area of triangle DBC 1FM-pg116.qxd :22 678_0000-FM.qxd 5/1/08 3:53 PM Page 104 104 • STRATEGY SECTION EXAMPLE 5 1 (A) ᎏᎏ 4 1 (B) ᎏᎏ 3 3 (C) ᎏᎏ 8 1 (D) ᎏᎏ 2 (E) Cannot be determined Choice . 2 4 ϩ 2 5 ϩ 2 6 ϩ 2 7 ϩ 2 8 ϩ 2 9 then 2x ϭ 2 ϩ 2 2 ϩ 2 3 ϩ 2 4 ϩ 2 5 ϩ 2 6 ϩ 2 7 ϩ 2 8 ϩ 2 9 ϩ 2 10 and y ϭ 1 ϩ 2x ϭ 1 ϩ 2 ϩ 2 2 ϩ 2 3 ϩ 2 4 ϩ 2 5 ϩ 2 6 ϩ 2 7 ϩ 2 8 ϩ 2 9 ϩ 2 10 Thus, calculating. 2 10 Thus, calculating Ϫ , we get y Ϫ x ϭ 1 ϩ 2 ϩ 2 2 ϩ 2 3 ϩ 2 4 ϩ 2 5 ϩ 2 6 ϩ 2 7 ϩ 2 8 ϩ 2 9 ϩ 2 10 Ϫ(1 ϩ 2 ϩ 2 2 ϩ 2 3 ϩ 2 4 ϩ 2 5 ϩ 2 6 ϩ 2 7 ϩ 2 8 ϩ 2 9 ) ϭ 2 10 (Answer) EXAMPLE 3 U ᎏ s ᎏ e ᎏ ᎏ f ᎏ a ᎏ c ᎏ t ᎏ o ᎏ r ᎏ i ᎏ n ᎏ g ᎏ ᎏ t ᎏ o ᎏ ᎏ m ᎏ a ᎏ k ᎏ e ᎏ ᎏ p ᎏ r ᎏ o ᎏ b ᎏ l ᎏ e ᎏ m ᎏ s ᎏ ᎏ s ᎏ i ᎏ m ᎏ p ᎏ l ᎏ e ᎏ r ᎏᎏ . ͙(88) 2 ϩ ෆ (88) 2 ෆ (3) ෆ ϭ (A). ϫ ᎏ 3 x ᎏ (Step 2) y 7 ϭ ᎏ 3 x ᎏ (Step 5) y ϭ ᎏ 4 3 x ᎏ (Answer) (Step 3) ᎏ y y 8 7 ᎏ ϭ EXAMPLE 2 If x ϭ 1 ϩ 2 ϩ 2 2 ϩ 2 3 ϩ 2 4 ϩ 2 5 ϩ 2 6 ϩ 2 7 ϩ 2 8 ϩ 2 9 and y ϭ 1 ϩ 2x, then y Ϫ x ϭ (A) 2 7 (B) 2 8 (C)
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