SATstu#3This weekwell reviewfractions, decimals, percentages, order of operations, andra-tios/proportions.FRACTIONS:ADDING/SUBTRACTING FRACTIONS: you must have a common denominator multiply by1 in an appropriate form to get a desired denominator23 + 15=23 55 + 15 33=1015 +315=1315MULTIPLYING FRACTIONS: just multiply across cancel BEFORE multiplying; cancelling is getting rid of extra factors of143 314=2 2332 7=27DIVIDING FRACTIONS: dividing by3 is the same as multiplying by13 dividing by25is the same as multiplying by52 dividing byabis the same as multiplying byba23 15=23 51=103WRITING FRACTIONS AS DECIMALS: the only fractions that can be written as nite decimals are ones with ONLY factors of2s and 5s in the denominator (when theyre in simplest form) all other fractions will be innite repeating decimals; do a long division320=32 2 5=32 2 5 55=1510 10=15100= 0.15TRY THESE:(a)13 27(b)x2y 5a2b(c) Suppose thatn is a power of2 andm is a power of5 . Is the fraction8034nma nite orinnite repeating decimal?(d) Write as a decimal:56PERCENTAGES:The word percentmeans per one hundred.The symbol% is used for percent.Whenever you see the symbol%, you can trade it in for a factor of1100 .Whenever you see a factor of1100 , it can be traded in for a% symbol.EXAMPLE: 2% = 2 1100=2100= 0.02(to go from a percent to a decimal, just move the decimal point two places to the left)EXAMPLE: 5 = 5 100100= 500 1100= 500%(to go from a decimal to a percent, just move the decimal point two places to the right)Use this memory device: PuDdLe DiPpeR(Percent to Decimal, Left; Decimal to Percent, Right)To do arithmetic with percents, change to decimals rst.PERCENT INCREASE AND DECREASE:To nd a30% increase, multiply by1.3 : x + 0.3x = 1.3xTo nd a7% increase, multiply by1.07 : x + 0.07x = 1.07xIf you have a20% decrease, only80% remains, so multiply by0.8 : x 0.2x = 0.8xIf you have a95% decrease, only5% remains, so multiply by0.05 : x 0.95x = 0.05xEXAMPLE:A pricex rst increases by30%, then decreases by25%.What is the resulting percent increase/decrease?new price =x(1.3)(0.75) = 0.975x1 0.975 = 0.025it is an overall2.5% decreaseTRY THESE:(a) nd5% of200(b) convert7% to a decimal(c) convert0.0037 to a percent(d) Suppose an item decreases by 20%, then increases by 20%. What is the overall percentincrease/decrease?ORDEROFOPERATIONS:The order that operations are to be performed (when not clearly identied) is summarizedwith the following memory device:PleaseExcuseMyDearAuntSally (PEMDAS)Do things insideParentheses rst (using PEMDAS, if needed, inside the parentheses).Then do allExponents, in order as they occur, going from left to right.Thendoall Multiplications/Divisions(theyhaveequal weight)inorderastheyoccur,going from left to right.Finally, doall Additions/Subtractions(theyhaveequal weight)inorderastheyoccur,going from left to right.In horizontal fractions, there are implied parentheses in both the numerator and denomi-nator.EXAMPLES:1 + 3 5 2 = 1 + (3 5) 2 = 1 + 15 2 = 122 10 5 + 3 = 2 105+ 3 = 32 37 4=63= 2 (Key this into your calculatorcorrectly!)TRY THESE:(a) 3 6 2 4 7(b)3 + 511 9(do both by hand, and on your calculator)ARATIO is a comparison of two things, that gives information about how the quantitiesof each relate to each other.For example, suppose youre told:the ratio ofgirls to boys in a given group is3 to 4This information is often written as:girls : boys = 3: 4This means that for every3 girls in the group, there are4 boys.For example, there might be only3 girls and4 boys in the group (a group of size7).Or, there might be2 3 = 6 girls and2 4 = 8 boys (a group of size14).Or, there might be3 3 = 9 girls and3 4 = 12 boys (a group of size21).Fractions are often used to display and work with ratio information:3 girls4 boys=6 girls8 boys=9 girls12 boys= Notice that the ratio information can be expressed using lots of dierent names!Notice that a ratio does not give any absolute information about the size of a group: doesour group of girls and boys have size7, or14, or21, or some other multiple of7 ?Quick sketches are often very useful when working with ratio problems:TRANSLATINGRATIOINFORMATIONINTOFRACTIONS: Here are some samplequestions that might arise from the previous ratio information, and their answers:Question: Whatpart(fraction)ofthegroupisgirls?Answer: There are3 girls in a group of size7 , so37of the group is girls .Or, there are6 girls in a group of size14 , so614=37of the group is girls.Notice that it is easiest to use the smallest numbers to answer this question.Question: Whatpart(fraction)ofthegroupisboys?Answer: Since there are4 boys in a group of size7 ,47of the group is boys.TRY THESE:(a) A recipe calls for3 cups of our for every2 cups of sugar.If these are the only ingredients, what fraction of the mixture is our?(b) What is the ratio of sugar to our in this mixture?THECROSS-MULTIPLYINGTECHNIQUE: Supposethatab=cd . Multiplyingbothsides bybd and cancelling gives:ab=cdab bd =cd bdad = bcThis observation is often called the cross-multiplying technique:Cross-multiplying is often useful in solving ratio problems.Heres a sample problem.Try it on your own; then, several solution approaches will be discussed.SAMPLEPROBLEM:When manufacturing widgets, the ratio of defective to nondefective widgets is 2 to 17 .How many defective widgets would you expect ifNwidgets are being produced?(A)219N(B)2N17(C)2N19(D)N + 2 (E)17N2SOLUTION#1: Make a quick sketch, noting the total group size:So,2 of every19 widgets is defective.That is,219of the widgets are defective.So, in a group of sizeN , we expect219Nto be defective.Notice that219Ncan also be written as219N=219 N1=2N19. The correct answer is (C).SOLUTION#2: Letx denote the number of defective widgets in a group of sizeN .The question involves comparing defective widgets to the total number in the group. So,we need a ratio that expresses this information:2 defective19 widgets=x defectiveNwidgetsCross-multiply and solve forx :19x = 2Nx =2N19The correct answer is (C).SOLUTION#3: Choose a number forN , and check the possible answers.Take the simplest numbers to work with. Suppose there are2 defective, 17 nondefective,and a total group size of2 + 17 = 19 . So, letN= 19 . (The number of defective in a groupthis size should be2 .)SubstituteN= 19 into (A), giving219 19which is clearly not correct!SubstituteN= 19 into (B), giving2 1917, which is not correct.SubstituteN= 19 into (C), giving2 1919, which is correct!Stop and record answer (C).Now, you try these:1. When manufacturing widgets, the ratio of defective to nondefective widgets is2 to17 . If there areNdefective widgets, how many nondefective widgets do you expectto have?(A)19N17(B)17N2(C)2N17(D)17N19(E)17N22. When manufacturing widgets, the ratio of defective to nondefective widgets is2 to17 . If there areNnondefective widgets, how many defective widgets do you expectto have?(A)19N17(B)17N2(C)2N17(D)17N19(E)17N23. When manufacturing widgets, the ratio of defective to nondefective widgets is2 to17 . If there areNwidgets, how many do you expect to be nondefective?(A)19N17(B)17N2(C)2N17(D)17N19(E)17N24. When manufacturing widgets, the ratio of defective to nondefective widgets isD toN . If500 widgets are produced, how many do you expect to be defective?(A)500DD + N(B)D + N500(C)500DN(D)N500D(E)500ND + N5. When manufacturing widgets, the ratio of defective to nondefective widgets isD toN . If500 widgets are produced, how many do you expect to be nondefective?(A)500DD + N(B)D + N500(C)500DN(D)N500D(E)500ND + N6. When manufacturing widgets, the ratio of defective to nondefective widgets isD toN . If a given group has70 defective widgets, what is the total size of the group?(A)70(D + N)D(B)D + N70D(C)70ND (D)70 + D + N (E)70DD + N7. In a certain game, 12 players form a team. If a team must have at least one maleplayer for every 3 female players, what is the minimum number of male players ona team?(A) 1 (B) 2 (C) 3 (D) 4 (E) 58. Ifx + 1 = y , what is the ratio ofx toy ?(A)12(B)23(C)34(D)45(E) It cannot be determinedfrom the informationgiven.9. Acertainrecipefora6-poundnutmixrequires4poundsof peanuts,12poundofcashews, and112poundsofalmonds. Howmanypoundsofalmondswouldberequired to make 16 pounds of this mixture?(A)113(B) 4 (C) 6 (D)1023(E)111210. Inacertainclass,23ofthestudentstakePhysicsand23ofthosestudentstakingPhysics alsotakeChemistry. What fractionof theclass takes Physics but notChemistry?(A)19(B)29(C)13(D)49(E)2311. On a certain diagram, 1 inch represents 10 feet. On this diagram, 1.2 inches repre-sents how many feet?(A) 120 (B) 12 (C) 10.2 (D) 1.2 (E) 0.1212. Ifab=34andbc=25 , thenac=(A)310(B)35(C)23(D)45(E)5313. Five bags contain the following proportions of marbles:Red Marbles White MarblesBag 1 1 2Bag 2 5 7Bag 3 3 6Bag 4 2 3Bag 5 3 4In which bag is the ratio of red marbles to white marbles the greatest?(A) Bag 1 (B) Bag 2 (C) Bag 3 (D) Bag 4 (E) Bag 514. Acertaininsectcrawlsattherateof0.005inchespersecond. Atthisrate, howmany seconds will it take for the insect to travel an inch?(A) 200 (B) 500 (C) 1000 (D) 2000 (E) 500015. George is one-half as old as Tom who is one-half as old as Bill. What is the ratioof Bills age to Georges age?(A)14(B)12(C)11(D)21(E)4116. A line 120 meters long is divided into two portions in a ratio of1: 3 . The longerportion is how many meters longer than the shorter portion?(A) 30 (B) 40 (C) 60 (D) 80 (E) 9017. In a certain school there are 100 boys and 200 girls. How many students ride thebus if 20 percent of the boys and 30 percent of the girls ride the school bus?(A) 60 (B) 70 (C) 80 (D) 90 (E) 10018. Ateacherboughtsome$4booksandsome$2booksforatotal of $42. If theteacher purchasedthree$4books for every$2book, howmanybooks didtheteacher purchase?(A) 3 (B) 7 (C) 9 (D) 12 (E) 14