The sat math section 1

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5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 142 – THE SAT MATH SECTION – If given a percentage, write it in the numerator Finding what percentage one number is of ■ position of the number column. If you are not another: given a percentage, then the variable should be What percentage of 75 is 15? placed there. # % The denominator of the number column repre- 15 x ■ __ = ___ sents the number that is equal to the whole, or 75 100 100%. This number always follows the word “of ” Cross multiply: in a word problem. 15(100) = (75)(x) The numerator of the number column represents ■ 1,500 = 75x the number that is the percent. 1,500 75x = 75 In the formula, the equal sign can be inter- ■ 75 20 = x Therefore, 20% of 75 is 15. changed with the word “is.” Ratio and Variation Examples: A ratio is a comparison of two quantities measured in Finding a percentage of a given number: the same units. It is symbolized by the use of a colon—x:y. What number is equal to 40% of 50? Ratio problems are solved using the concept of # % multiples. x 40 __ = ___ 50 100 Example: A bag contains 60 red and green candies. The Solve by cross multiplying. ratio of the number of green to red candies is 7:8. 100(x) = (40)(50) How many of each color are there in the bag? 100x = 2,000 100x 2,000 = 100 From the problem, it is known that 7 and 8 100 x = 20 Therefore, 20 is 40% of 50. share a multiple and that the sum of their prod- uct is 60. Therefore, you can write and solve the Finding a number when a percentage is given: following equation: 40% of what number is 24? 7x + 8x = 60 # % 15x = 60 24 40 __ = ___ 15x 60 = 15 15 100 x x=4 Therefore, there are (7)(4) = 28 green candies Cross multiply: and (8)(4) = 32 red candies. (24)(100) = (40)(x) 2,400 = 40x Variation 2,400 40x = 40 40 Variation is a term referring to a constant ratio in the 60 = x Therefore, 40% of 60 is 24. change of a quantity. A quantity is said to vary directly with another if ■ they both change in an equal direction. In other words, two quantities vary directly if an increase 142
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 143 – THE SAT MATH SECTION – in one causes an increase in the other. This is also Rate Problems You will encounter three different types of rate prob- true if a decrease in one causes a decrease in the lems on the SAT: cost, movement, and work-output. other. The ratio, however, must be the same. Rate is deﬁned as a comparison of two quantities with Example: different unites of measure. Assuming each child eats the same amount, if x units 300 children eat a total of 58.5 pizzas, how Rate = y units many pizzas will it take to feed 800 children? Examples: Since each child eats the same amount of pizza, miles dollars cost hour , hour , pound you know that they vary directly. Therefore, you can set the problem up the following way: Cost Per Unit Some problems on the SAT will require you to calcu- 300 800 Pizza = 58.5 = x Children late the cost of a quantity of items. Cross multiply to solve: (800)(58.5) = 300x Example: 46,800 = 300x If 60 pens cost $117.00, what will the cost of 46,800 300x = four pens be? 300 300 156 = x 117 $1.95 total cost = = # of pens 60 pen Therefore, it would take 156 pizzas to feed 800 To ﬁnd the cost of 4 pens, simply multiply children. $1.95 × 4 = $7.80. If two quantities change in opposite directions, ■ they are said to vary inversely. This means that as Movement one quantity increases, the other decreases, or as When working with movement problems, it is impor- one decreases, the other increases. tant to use the following formula: Example: If two people plant a ﬁeld in six days, how may (Rate)(Time) = Distance days will it take six people to plant the same ﬁeld? (Assume each person is working at the same rate.) Example: A scooter traveling at 15 mph traveled the As the number of people planting increases, the 1 length of a road in 4 of an hour less than it took days needed to plant decreases. Therefore, the when the scooter traveled 12 mph. What was relationship between the number of people and the length of the road? days varies inversely. Because the ﬁeld remains constant, the two expressions can be set equal First, write what is known and unknown. to each other. Unknown = time for scooter traveling 2 people × 6 days = 6 people × x days 12 mph = x 2 × 6 = 6x Known = time for scooter traveling 15 mph = 12 6x 1 6=6 x– 4 2=x Then, use the formula, (Rate)(Time) = Thus, it would take six people two days to plant Distance to make an equation. The distance of the same ﬁeld. 143
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 144 – THE SAT MATH SECTION – the road does not change; therefore, you know Rate Time = Part of Job Completed 1 to make the two expressions equal to each other: Danette x = 1 car 3 1 12x = 15(x – 1 ) Judy x = 1 car 2 4 15 12x = 15x – 4 Since they are both working on only one car, –15x –15x you can set the equation equal to one: –15 –3x 4 = 1 1 –3 –3 3x + 2x = 1 5 1 x= 4 , or 1 4 hours Solve by using 6 as the LCD for 3 and 2: Be careful, 1 1 is not the distance; it is the time. 1 1 6( 3 x) + 6( 2 x) = 6(1) 4 Now you must plug the time into the formula: 2x + 3x = 6 (Rate)(Time) = Distance. Either rate can be used. 5x 6 = 5 5 1 x = 15 12x = distance 1 5 Thus, it will take Judy and Danette 1 5 hours to 12( 4 ) = distance wash and wax one car. 15 miles = distance Special Symbols Problems Work-Output Problems The SAT will sometimes invent a new arithmetic oper- Work-output problems are word problems that deal ation symbol. Don’t let this confuse you. These prob- with the rate of work. The following formula can be lems are generally very easy. Just pay attention to the used of these problems: placement of the variables and operations being performed. (rate of work)(time worked) = job or part of job completed Example: Given a ∇ b = (a × b + 3)2, ﬁnd the value of 1 ∇ 2. Example: Danette can wash and wax two cars in six Fill in the formula with 1 being equal to a and 2 hours, and Judy can wash and wax the same being equal to b. two cars in four hours. If Danette and Judy (1 × 2 + 3)2 = (2 + 3)2 = (5)2 = 25 work together, how long will it take to wash and So, 1 ∇ 2 = 25. wax one car? Example: Since Danette can wash and wax two cars in six 2 cars hours, her rate of work is , or one car b 6 hours a−b a−c b−c every three hours. Judy’s rate of work is there- = _____ + _____ + _____ If c b a 2 cars fore 4 hours , or one car every two hours. In this a c problem, making a chart will help: 2 Then what is the value of . . . 1 3 144
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 145 – THE SAT MATH SECTION – Fill in variables according to the placement of To solve, you will perform a special type of calcu- number in the triangular ﬁgure; a = 1, b = 2, lation known as a combination. The formula to use is: and c = 3. nPr nCr = 1 1 1–2 1–3 2–3 r! + + = –3 + –1 + –1 = –2 3 3 2 1 For example, if there are six students (A, B, C, D, Counting Principle E, and F), and three will be chosen to represent the Some word problems may describe a possibilities for school in a nationwide competition, we calculate the one thing and b possibilities for another. To quickly number of possible combinations with: solve, simply multiply a × b. For example, if a student has to choose one of 8 nPr nCr = r! different sports to join and one of ﬁve different com- munity service groups to join, we would ﬁnd the total Note that here order does NOT matter. number of possibilities by multiplying 8 × 5, which Here, n = 6 and r = 3. gives us the answer: 40 possibilities. 120 nPr 6P3 120 nCr = = 6C3 = = = = 20 3×2×1 6 r! 3! Permutations Probability Some word problems may describe n objects taken r at Probability is expressed as a fraction and measures the a time. In these questions, the order of the objects matters. likelihood that a speciﬁc event will occur. To ﬁnd the To solve, you will perform a special type of calcu- probability of a speciﬁc outcome, use this formula: lation known as a permutation. The formula to use is: Number of speciﬁc outcomes Probability of an event = n! Total number of possible outcomes nPr = (n – r)! Example: For example, if there are six students (A, B, C, D, E, If a bag contains 5 blue marbles, 3 red marbles, and F), and three will be receiving a ribbon (First and 6 green marbles, ﬁnd the probability of Place, Second Place, and Third Place), we can calcu- selecting a red marble. late the number of possible ribbon winners with: Probability of an event = n! nPr = Number of speciﬁc outcomes 3 (n – r)! = 5+3+6 Total number of possible outcomes Here, n = 6, and r = 3. Therefore, the probability of selecting a red 3 n! 6! 6! marble is 14 . nPr = = 6P3 = = = (n – r)! (6 – 3)! (3)! 6×5×4×3×2×1 = 6 × 5 × 4 = 120 Multiple Probabilities 3×2×1 To ﬁnd the probability that two or more events will Combinations occur, add the probabilities of each. For example, in the Some word problems may describe the selection of r problem above, if we wanted to ﬁnd the probability of objects from a group of n. In these questions, the order drawing either a red or blue marble, we would add the of the objects does NOT matter. probabilities together. 145
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 146 – THE SAT MATH SECTION – 3 The probability of drawing a red marble = 14 Helpful Hints about Probability 5 ■ If an event is certain to occur, the probability is 1. and the probability of drawing a blue marble = 14 . So, 3 ■ If an event is certain not to occur, the probability the probability for selecting either a blue or a red = 14 5 8 is 0. + 14 = 14 . ■ If you know the probability of all other events occurring, you can ﬁnd the probability of the remaining event by adding the known probabili- ties together and subtracting from 1. 146
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 147 – THE SAT MATH SECTION – math section on a past SAT. The distribution of P art 1: Five-Choice Questions questions on your test will vary. 1. 1 8. 2 15. 3 22. 3 The ﬁve-choice questions in the Math section of the 2. 1 9. 3 16. 5 23. 5 SAT will comprise about 80% of your total math score. 3. 1 10. 2 17. 4 24. 5 Five-choice questions test your mathematical reason- 4. 1 11. 3 18. 4 25. 5 ing skills. This means that you will be required to apply 5. 2 12. 3 19. 4 several basic math techniques for each problem. In the 6. 2 13. 3 20. 4 math sections, the problems will be easy at the begin- 7. 1 14. 3 21. 4 ning and will become increasingly difficult as you From this list, you can see how important it is progress. Here are some helpful strategies to help you to complete the ﬁrst ﬁfteen questions before get- improve your math score on the ﬁve-choice questions: ting bogged down in the complex problems that follow. After you are satisﬁed with the ﬁrst ﬁfteen Read the questions carefully and know the ■ questions, skip around the last ten, spending the answer being sought. In many problems, you will most time on the problems you ﬁnd to be easier. be asked to solve an equation and then perform Don’t be afraid to write in your test booklet. an operation with that variable to get an answer. ■ That is what it is for. Mark each question that In this situation, it is easy to solve the equation you don’t answer so that you can easily go back to and feel like you have the answer. Paying special it later. This is a simple strategy that can make a attention to what each question is asking, and lot of difference. It is also helpful to cross out the then double-checking that your solution answers answer choices that you have eliminated. the question, is an important technique for per- Sometimes, it may be best to substitute in an forming well on the SAT. ■ answer. Many times it is quicker to pick an If you do not ﬁnd a solution after 30 seconds, ■ answer and check to see if it is a solution. When move on. You will be given 25 minutes to answer you do this, use the c response. It will be the mid- questions for two of the Math sections, and 20 dle number and you can adjust the outcome to minutes to answer questions in the other section. the problem as needed by choosing b or d next, In all, you will be answering 54 questions in 70 depending on whether you need a larger or minutes! That means you have slightly more than smaller answer. This is also a good strategy when one minute per problem. Your time allotted per you are unfamiliar with the information the question decreases once you realize that you will problem is asking. want some time for checking your answers and When solving word problems, look at each extra time for working on the more difﬁcult prob- ■ phrase individually and write it in math lan- lems. The SAT is designed to be too complex to ﬁn- guage. This is very similar to creating and assign- ish. Therefore, do not waste time on a difﬁcult ing variables, as addressed earlier in the word problem until you have completed the problems problem section. In addition to identifying what you know how to do. The SAT Math problems can is known and unknown, also take time to trans- be rated from 1–5 in levels of difﬁculty, with 1 late operation words into the actual symbols. It is being the easiest and 5 being the most difﬁcult. The best when working with a word problem to repre- following is an example of how questions of vary- sent every part of it, phrase by phrase, in mathe- ing difﬁculty have been distributed throughout a matical language. 147