SATstu#2Problems involving remainders are popular on the SAT test.REMAINDERS: Suppose you have 20 apples and want to make piles of size 7 . You canmake 2 piles, with 6 left over. The number left over (remaining) is called theremainder.Numbers have lots of dierent names! The name 20 = 2 7+6 shows that you can get 2 pilesof 7 , with 6 left over.Your calculator can help you nd remainders. To nd the remainder when 20 is dividedby 7 , do this: How many piles of 7 can I get from 20 ?Key in 20/7 , giving something like 2.857142857 .You can get2 full piles. (Throw away the stu after the decimal point.) Then,20 2 7 = 20 14 = 6 gives the remainder.The key to remainders is making as many piles of a given size as possible, and seeing whatis left over.TRY THESE:(a) What is the remainder when20 is divided by 6 ?(Do this without your calculator.)(b) What is the remainder when217 is divided by11 ?(Use your calculator)(c) If a number goes inton evenly, then what is the remainder?Heres a typical problem.Try it yourself rst (on this page).Youll get 1.25 minutes75 secondswhich is the average time for a multiple-choice prob-lem (if you do them all) on the SAT.Then, three dierent solution approaches are discussed.Finally, youll apply the concept to several practice problems.Whenn is divided by 4 , the remainder is 2 . What is the remainder when 6n is dividedby4 ?(A) 0 (B) 1 (C) 2 (D) 3 (E) 4Note: When you divide by 4 , youll never get a remainder of 4because if you have 4 leftover, thenyoucanmakeanotherpileof 4 ! So, theonlypossibleremaindersare0, 1, 2,and3 . You should eliminate (E) immediately.PICTURESOLUTION: (For people who like pictures.) Draw a picture representingn :piles (of size4) and whats left over. Repeat this picture6 times. (Of course, once you dothis a couple times, you wont really have to draw the whole picture.)Focus on whats leftover; make piles of size4 . Whats left over?None! The answer is (a).PICKANUMBER: (For people who like working with actual numbers.) We know thatwhenn is divided by4 , the remainder is2 . What numbers couldn be?Start with the remainder only (no full piles) and then keep adding on piles of size4 :n = 2, 6, 10, 14, 18, . . . .Pick one of the simplest numbers thatn could be. For example, choosen = 10 .Now, whatis6n ? Well, 6n=6 10=60 , whichis15pilesof 4withnoneleftover. Theanswer is (a).(Pick other values ofn and see that you still get the same answer.)ALGEBRAICSOLUTION: (For algebra lovers, like Dr. Fisher, this is the best solution.)Whenn is divided by4 , the remainder is2 . So, n = 4p + 2 . (n can be made intop piles of4 , with2 left over.)How many piles of4 in6n ?6n = 6(4p + 2)= 6 4p + 6 2= 4 6p + 12= (piles of size4) + 12= (piles of size4) + (4 + 4 + 4)So,6n can be completely broken into piles of size4 . None left over! The answer is (a).Note that we really only needed to deal with the6 2 = 12 partwhy?PRACTICEPROBLEMS:1. Whenxisdividedby4 , theremainderis3 . Whatistheremainderwhen7xisdivided by4 ?(A) 0 (B) 1 (C) 2 (D) 3 (E) 42. Whennisdividedby11 , theremainderis7 . Whatistheremainderwhen5nisdivided by11 ?(A) 0 (B) 2 (C) 4 (D) 7 (E) 93. Whenn is divided by5 the remainder is2 . Whenm is divided by5 the remainderis1 . What is the remainder whenm + n is divided by5 ?(A) 0 (B) 1 (C) 2 (D) 3 (E) 44. If a=4b + 26, andbisapositiveinteger, thenacouldbedivisiblebyall of thefollowing EXCEPT(A) 2 (B) 4 (C) 5 (D) 6 (E) 75. If the remainder whenx is divided by5 equals the remainder whenx is divided by4, thenx could be any of the following EXCEPT(A) 20 (B) 21 (C) 22 (D) 23 (E) 24Problems involving averages are popular on the SAT test.AVERAGES: To averagen numbers, you add them all up and then divide byn . That is,the average of then numbersx1, x2, . . ., xn is given by the formulax1 + x2 + + xnn.Whenyoureworkingwithaverages,youshouldkeepasee-sawinmind! Theaveragegivesthebalancingpoint, asshownbelow: imagineasinglepebble(all thesamesize)placedat eachof thenumbersbeingaveraged. Whereshouldthesupport gosothateverything is in balance? avThereareseveral resultsaboutaveragesthatfollowimmediatelyfromthisbalancingviewpoint. The average of two numbers is exactly halfway between the numbers. When you averagenumbers,theaverageisalwaysbetween thegreatestandleastnumber. Forexample, supposeyoureaveragingthenumbers1, 1, 3, and5 .Then, the average must be between 3 and5 . (Check that the average is0.5 .) If youhaveanaverage, andyouaddinanothernumberthatsgreaterthantheaverage, thentheaveragemustgoup. Similarly, if youaddinanothernumberthats less than the average, then the average must go down.Note: An AVERAGE is also called an ARITHMETIC MEAN.TRY THESE:(a) What is the average of 5 and 9 ?(Dont do any arithmeticthink of balancing points.)(b) What is the average of 101, 103, and 105 ?(Dont do any arithmeticthink of balancingpoints.)(c) What is the average ofx b, x, andx + b ?(d) What is the average ofx 2, x 1, x + 1, andx + 2 ?Heres a typical problem.Try it yourself rst (on this page).If youre shooting for a 550 on the Math SAT, then you should only attempt about 14 outof 20 questions in a 25-minute multiple-choice section; thats about 1 minute 47 secondsper problem. [Cracking the SAT, 2009 edition, the Princeton Review, page 27]Then, three dierent solution approaches are discussed.Finally, youll apply the concept to several practice problems.The average weight of ve packages is 20 pounds. Four of the packages weigh 5, 10, 12,and22 pounds. What is the weight of the fth package (in pounds)?(A) 15 (B) 25 (C) 45 (D) 51 (E) 100INTUITION/GUESS-AND-CHECK: A quick sketch and some good intuition about thebalancing point will tell you that the answer CANT BE (A), (B), or even (E). Try theremainingtwonumbers. Averaging5, 10, 12, 22, and45gives(useyourcalculator)18.8,which is a bit too small. The correct answer is (D).ALGEBRAICSOLUTION: Letx be the unknown number. Then,5 + 10 + 12 + 22 + x5= 2049 + x = 100x = 51The correct answer is (D).ALTERNATEALGEBRAICSOLUTION: Thesumof thedistancesfromtheaveragemust be the same on both sides. Look at the sketch below: the pebbles to the left givea combined distance of15 +10 +8 = 33 . So, the pebbles on the right must also sum to33.Then,2 + x = 33 means that the remaining distance from the average must be31 . So, thelast number is20 + 31 = 51 .PRACTICE PROBLEMS:1. The average weight of ve packages is 30 pounds. Four of the packages weigh 10, 20,40, and45 pounds. What is the weight of the fth package (in pounds)?(A) 15 (B) 25 (C) 35 (D) 39 (E) 482. A is a number. The numbersA2,A1,A+1, andA+2 are averaged. What is theaverage (arithmetic mean) of these four numbers?(A)A + 0.5 (B)A 0.5 (C)A (D)0 (E)13. CarolneedstobuysixgiftsforChristmas. Shehas$100tospendonthesegifts.Shehasalreadyboughtfourgifts, costing$15, $20, $12, and$13 . Whatistheaverage amount that she can spend on the remaining gifts?(A) $40 (B) $20 (C) $15 (D) $17 (E) $304. A survey of Town X found an average (arithmetic mean) of 3.2 persons per householdand a mean of 1.2 televisions per household. If 48,000 people live in Town X, howmany televisions are in Town X?(A) 15,000 (B) 16,000 (C) 18,000 (D) 40,000 (E) 57,6005. The average (arithmetic mean) of the rst 3 numbers in a certain series is 10. If theaverage of the rst2 numbers is10, what is the third number?(A) 0(B) 5 (C) 10 (D) 20 (E) It cannot be determined from the information given.6. If x is the average (arithmetic mean) of a andb, then in terms of x andc, what isthe average ofa, b, andc ?(A)x + c (B)x + c2(C)x + c3(D)2x + c2(E)2x + c3