■ If given a percentage, write it in the numerator position of the number column. If you are not given a percentage, then the variable should be placed there. ■ The denominator of the number column repre- sents the number that is equal to the whole, or 100%. This number always follows the word “of” in a word problem. ■ The numerator of the number column represents the number that is the percent. ■ In the formula, the equal sign can be inter- changed with the word “is.” Examples: Finding a percentage of a given number: What number is equal to 40% of 50? Solve by cross multiplying. 100(x) = (40)(50) 100x = 2,000 ᎏ 1 1 0 0 0 0 x ᎏ = ᎏ 2 1 ,0 0 0 0 0 ᎏ x = 20 Therefore, 20 is 40% of 50. Finding a number when a percentage is given: 40% of what number is 24? Cross multiply: (24)(100) = (40)(x) 2,400 = 40x ᎏ 2, 4 4 0 00 ᎏ = ᎏ 4 4 0 0 x ᎏ 60 = x Therefore, 40% of 60 is 24. Finding what percentage one number is of another: What percentage of 75 is 15? Cross multiply: 15(100) = (75)(x) 1,500 = 75x ᎏ 1, 7 5 5 00 ᎏ = ᎏ 7 7 5 5 x ᎏ 20 = x Therefore, 20% of 75 is 15. Ratio and Variation A ratio is a comparison of two quantities measured in the same units. It is symbolized by the use of a colon—x:y. Ratio problems are solved using the concept of multiples. Example: A bag contains 60 red and green candies. The ratio of the number of green to red candies is 7:8. How many of each color are there in the bag? From the problem, it is known that 7 and 8 share a multiple and that the sum of their prod- uct is 60. Therefore, you can write and solve the following equation: 7x + 8x = 60 15x = 60 ᎏ 1 1 5 5 x ᎏ = ᎏ 6 1 0 5 ᎏ x = 4 Therefore, there are (7)(4) = 28 green candies and (8)(4) = 32 red candies. Variation Variation is a term referring to a constant ratio in the 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 # % __ = ___ 75 100 x15 # % __ = ___ x 100 40 24 # % __ = ___ 50 100 40 x –THE SAT MATH SECTION– 142 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 142 in one causes an increase in the other. This is also true if a decrease in one causes a decrease in the other. The ratio, however, must be the same. Example: Assuming each child eats the same amount, if 300 children eat a total of 58.5 pizzas, how many pizzas will it take to feed 800 children? Since each child eats the same amount of pizza, you know that they vary directly. Therefore, you can set the problem up the following way: ᎏ Ch P i i l z d z r a en ᎏ = ᎏ 5 3 8 0 . 0 5 ᎏ = ᎏ 80 x 0 ᎏ Cross multiply to solve: (800)(58.5) = 300x 46,800 = 300x ᎏ 46 3 , 0 8 0 00 ᎏ = ᎏ 3 3 0 0 0 0 x ᎏ 156 = x Therefore, it would take 156 pizzas to feed 800 children. ■ If two quantities change in opposite directions, they are said to vary inversely. This means that as one quantity increases, the other decreases, or as one decreases, the other increases. Example: If two people plant a field in six days, how may days will it take six people to plant the same field? (Assume each person is working at the same rate.) As the number of people planting increases, the days needed to plant decreases. Therefore, the relationship between the number of people and days varies inversely. Because the field remains constant, the two expressions can be set equal to each other. 2 people × 6 days = 6 people × x days 2 × 6= 6x ᎏ 1 6 2 ᎏ = ᎏ 6 6 x ᎏ 2= x Thus, it would take six people two days to plant the same field. Rate Problems You will encounter three different types of rate prob- lems on the SAT: cost, movement, and work-output. Rate is defined as a comparison of two quantities with different unites of measure. Rate = Examples: ᎏ m ho il u e r s ᎏ , ᎏ d h o o ll u a r rs ᎏ , ᎏ po co u s n t d ᎏ Cost Per Unit Some problems on the SAT will require you to calcu- late the cost of a quantity of items. Example: If 60 pens cost $117.00, what will the cost of four pens be? ᎏ t # o o ta f l p c e o n s s t ᎏ = ᎏ 1 6 1 0 7 ᎏ = To find the cost of 4 pens, simply multiply $1.95 × 4 = $7.80. Movement When working with movement problems, it is impor- tant to use the following formula: (Rate)(Time) = Distance Example: A scooter traveling at 15 mph traveled the length of a road in ᎏ 1 4 ᎏ of an hour less than it took when the scooter traveled 12 mph. What was the length of the road? First, write what is known and unknown. Unknown = time for scooter traveling 12 mph = x Known = time for scooter traveling 15 mph = x – ᎏ 1 4 ᎏ Then, use the formula, (Rate)(Time) = Distance to make an equation. The distance of $1.95 ᎏ pen x units ᎏ y units –THE SAT MATH SECTION– 143 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 143 the road does not change; therefore, you know to make the two expressions equal to each other: 12x = 15(x – ᎏ 1 4 ᎏ ) 12x =15x – ᎏ 1 4 5 ᎏ –15x –15x ᎏ – – 3 3 x ᎏ = x = ᎏ 5 4 ᎏ , or 1 ᎏ 1 4 ᎏ hours Be careful, 1 ᎏ 1 4 ᎏ is not the distance; it is the time. Now you must plug the time into the formula: (Rate)(Time) = Distance. Either rate can be used. 12x = distance 12( ᎏ 5 4 ᎏ ) = distance 15 miles = distance Work-Output Problems Work-output problems are word problems that deal with the rate of work. The following formula can be used of these problems: (rate of work)(time worked) = job or part of job completed Example: Danette can wash and wax two cars in six hours, and Judy can wash and wax the same two cars in four hours. If Danette and Judy work together, how long will it take to wash and wax one car? Since Danette can wash and wax two cars in six hours, her rate of work is , or one car every three hours. Judy’s rate of work is there- fore , or one car every two hours. In this problem, making a chart will help: Rate Time = Part of Job Completed Danette ᎏ 1 3 ᎏ x = 1 car Judy ᎏ 1 2 ᎏ x = 1 car Since they are both working on only one car, you can set the equation equal to one: ᎏ 1 3 ᎏ x + ᎏ 1 2 ᎏ x = 1 Solve by using 6 as the LCD for 3 and 2: 6( ᎏ 1 3 ᎏ x) + 6( ᎏ 1 2 ᎏ x) = 6(1) 2x + 3x = 6 ᎏ 5 5 x ᎏ = ᎏ 6 5 ᎏ x = 1 ᎏ 1 5 ᎏ Thus, it will take Judy and Danette 1 ᎏ 1 5 ᎏ hours to wash and wax one car. Special Symbols Problems The SAT will sometimes invent a new arithmetic oper- ation symbol. Don’t let this confuse you. These prob- lems are generally very easy. Just pay attention to the placement of the variables and operations being performed. Example: Given a ∇ b = (a × b + 3) 2 , find the value of 1 ∇ 2. Fill in the formula with 1 being equal to a and 2 being equal to b. (1 × 2 + 3) 2 = (2 + 3) 2 = (5) 2 = 25 So, 1 ∇ 2 = 25. Example: b ca 2 31 If = _____ + _____ + _____ a − b a − c b − c c b a Then what is the value of . . . 2 cars ᎏ 4 hours 2 cars ᎏ 6 hours ᎏ – 4 15 ᎏ ᎏ –3 –THE SAT MATH SECTION– 144 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 144 Fill in variables according to the placement of number in the triangular figure; a = 1, b = 2, and c = 3. ᎏ 1– 3 2 ᎏ + ᎏ 1– 2 3 ᎏ + ᎏ 2– 1 3 ᎏ = – ᎏ 1 3 ᎏ + –1 + –1 = –2 ᎏ 1 3 ᎏ Counting Principle Some word problems may describe a possibilities for one thing and b possibilities for another. To quickly solve, simply multiply a × b. For example, if a student has to choose one of 8 different sports to join and one of five different com- munity service groups to join, we would find the total number of possibilities by multiplying 8 × 5, which gives us the answer: 40 possibilities. Permutations Some word problems may describe n objects taken r at a time. In these questions, the order of the objects matters. To solve, you will perform a special type of calcu- lation known as a permutation. The formula to use is: n P r = For example, if there are six students (A, B, C, D, E, and F), and three will be receiving a ribbon (First Place, Second Place, and Third Place), we can calcu- late the number of possible ribbon winners with: n P r = Here, n = 6, and r = 3. n P r = = 6 P 3 = = = = 6 × 5 × 4 = 120 Combinations Some word problems may describe the selection of r objects from a group of n. In these questions, the order of the objects does NOT matter. To solve, you will perform a special type of calcu- lation known as a combination. The formula to use is: n C r = ᎏ n r P ! r ᎏ For example, if there are six students (A, B, C, D, E, and F), and three will be chosen to represent the school in a nationwide competition, we calculate the number of possible combinations with: n C r = ᎏ n r P ! r ᎏ Note that here order does NOT matter. Here, n = 6 and r = 3. n C r = ᎏ n r P ! r ᎏ = 6 C 3 = ᎏ 6 3 P ! 3 ᎏ = ᎏ 3 × 12 2 0 × 1 ᎏ = ᎏ 12 6 0 ᎏ = 20 Probability Probability is expressed as a fraction and measures the likelihood that a specific event will occur. To find the probability of a specific outcome, use this formula: Probability of an event = Example: If a bag contains 5 blue marbles, 3 red marbles, and 6 green marbles, find the probability of selecting a red marble. Probability of an event = = ᎏ 5+ 3 3+6 ᎏ Therefore, the probability of selecting a red marble is ᎏ 1 3 4 ᎏ . Multiple Probabilities To find the probability that two or more events will occur, add the probabilities of each. For example, in the problem above, if we wanted to find the probability of drawing either a red or blue marble, we would add the probabilities together. Number of specific outcomes ᎏᎏᎏᎏ Total number of possible outcomes Number of specific outcomes ᎏᎏᎏᎏ Total number of possible outcomes 6 × 5 × 4 × 3 × 2 × 1 ᎏᎏᎏ 3 × 2 × 1 6! ᎏ (3)! 6! ᎏ (6 – 3)! n! ᎏ (n – r)! n! ᎏ (n – r)! n! ᎏ (n – r)! –THE SAT MATH SECTION– 145 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 145 The probability of drawing a red marble = ᎏ 1 3 4 ᎏ and the probability of drawing a blue marble = ᎏ 1 5 4 ᎏ .So, the probability for selecting either a blue or a red = ᎏ 1 3 4 ᎏ + ᎏ 1 5 4 ᎏ = ᎏ 1 8 4 ᎏ . Helpful Hints about Probability ■ If an event is certain to occur, the probability is 1. ■ If an event is certain not to occur, the probability is 0. ■ 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. –THE SAT MATH SECTION– 146 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 146  Part 1: Five-Choice Questions The five-choice questions in the Math section of the SAT will comprise about 80% of your total math score. Five-choice questions test your mathematical reason- ing skills. This means that you will be required to apply several basic math techniques for each problem. In the math sections, the problems will be easy at the begin- ning and will become increasingly difficult as you progress. Here are some helpful strategies to help you improve your math score on the five-choice questions: ■ Read the questions carefully and know the answer being sought. In many problems, you will be asked to solve an equation and then perform an operation with that variable to get an answer. In this situation, it is easy to solve the equation and feel like you have the answer. Paying special attention to what each question is asking, and then double-checking that your solution answers the question, is an important technique for per- forming well on the SAT. ■ If you do not find a solution after 30 seconds, move on. You will be given 25 minutes to answer questions for two of the Math sections, and 20 minutes to answer questions in the other section. In all, you will be answering 54 questions in 70 minutes! That means you have slightly more than one minute per problem. Your time allotted per question decreases once you realize that you will want some time for checking your answers and extra time for working on the more difficult prob- lems. The SAT is designed to be too complex to fin- ish. Therefore, do not waste time on a difficult problem until you have completed the problems you know how to do. The SAT Math problems can be rated from 1–5 in levels of difficulty, with 1 being the easiest and 5 being the most difficult. The following is an example of how questions of vary- ing difficulty have been distributed throughout a math section on a past SAT. The distribution of questions on your test will vary. 1. 1 8. 2 15. 3 22. 3 2. 1 9. 3 16. 5 23. 5 3. 1 10. 2 17. 4 24. 5 4. 1 11. 3 18. 4 25. 5 5. 2 12. 3 19. 4 6. 2 13. 3 20. 4 7. 1 14. 3 21. 4 From this list, you can see how important it is to complete the first fifteen questions before get- ting bogged down in the complex problems that follow. After you are satisfied with the first fifteen questions, skip around the last ten, spending the most time on the problems you find to be easier. ■ Don’t be afraid to write in your test booklet. That is what it is for. Mark each question that you don’t answer so that you can easily go back to it later. This is a simple strategy that can make a lot of difference. It is also helpful to cross out the answer choices that you have eliminated. ■ Sometimes, it may be best to substitute in an answer. Many times it is quicker to pick an answer and check to see if it is a solution. When you do this, use the c response. It will be the mid- dle number and you can adjust the outcome to the problem as needed by choosing b or d next, depending on whether you need a larger or smaller answer. This is also a good strategy when you are unfamiliar with the information the problem is asking. ■ When solving word problems, look at each phrase individually and write it in math lan- guage. This is very similar to creating and assign- ing variables, as addressed earlier in the word problem section. In addition to identifying what is known and unknown, also take time to trans- late operation words into the actual symbols. It is best when working with a word problem to repre- sent every part of it, phrase by phrase, in mathe- matical language. –THE SAT MATH SECTION– 147 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 147 . throughout a math section on a past SAT. The distribution of questions on your test will vary. 1. 1 8. 2 15 . 3 22. 3 2. 1 9. 3 16 . 5 23. 5 3. 1 10. 2 17 . 4 24. 5 4. 1 11. 3 18 . 4 25. 5 5. 2 12 . 3 19 SAT MATH SECTION 14 3 5658 SAT2 006[04](fin).qx 11 / 21/ 05 6:44 PM Page 14 3 the road does not change; therefore, you know to make the two expressions equal to each other: 12 x = 15 (x – ᎏ 1 4 ᎏ ) 12 x. event by adding the known probabili- ties together and subtracting from 1. THE SAT MATH SECTION 14 6 5658 SAT2 006[04](fin).qx 11 / 21/ 05 6:44 PM Page 14 6  Part 1: Five-Choice Questions The five-choice