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### 34 Cards in this Set

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 What is a way of helping you list your variables? Use organization charts. Age problems. 8 years ago, Geore was half as old as Sarah. Sarah is now 20 years older han George. How old will George be 10 years from now? -8 Now +10 George G-8 G G+10 Sarah S-8 S S+10 True or false given two speeds the average of those two velocities would be (v1+v2)/2. False. The average velocities is not the average of the speeds! The average speed is: Total distance = (average speed)* (total time) What is the best way to solve simulataneous motion problems? Use clicking charts. Example Sacey and Heather are 20 miles apart and walk towards each other along the same route Stacey walks at a constant rate that is 1 mile per hour faster than Heather's constant rate of 5 miles/hour. If Heathe starts her journey 24 minutes after Stacey, how far from her original destination has Heather walked when the two meet? A. 7 miles B. 8 miles. C. 9 miles. D. 10 miles. E. 12 miles. Stacey Heather Total V 6mph 5mph T t+24 t D 20-d d 20 miles Looking for d Fill in the chart using answer choices. Plugging in C yields Stacey Heather Total V 6mph 5mph T t+24 t D 11 9 20 miles Well 11+9 don't equal 20 Plugging in B yields Stacey Heather Total V 6mph 5mph T t+24 t D 12 8 20 miles The correct answer! True of false given two rates the total rate is the sum of the two rates. True. The total working rate is the sum of all the individual working rates. Example. Larry can wash a car in 1 hour, Moe can wash a car in 2 hours, and Curly can wash a car in 4 hours. How long will it take them to wash a car together? 1+1/2+1/4=7/4 cars/hr What is the best way to solve population problems? Use a population chart. Example. The population of a certain bacteria triples every 10 minutes. If the population of a coloney 20 minutes ago was 100, in approximately how long from now will the bacteria population reach 24000? Time elapsed Population -20 100 -10 300 Now 900 +10 2700 +20 8100 +30 24300 True or false if you're given a ratio then there's enough information to determine the exact quantity? False. Ratio ONLY express the relationship between two or more items; they do not provide enough information, on their own, to determine the exact quantity. For more complicated ratio problems what is good way to think of ratios? As an unknown multiplier. Example: A recipe calls for amounts of lemon juice, wine, and water in the ratio of 2:5:7. If all three combined yield 42 milliliters of liquid, how much wine was included? Juice+Wine+Water=Total 2x+5x+7x=42 x=3 Wine = 5x = 5*3 = 12 milliliters. If there are 4 people and 4 chairs in a room, how many different seating arragnements are possible? 4!= 120 n! In this problem, you have 4 choices for the person who will sit in the first chair. Once one personhas been assigned to the first chair, you have 3 choices for the person who will sit n the second chair. Then you have two choices for the person in the third chair and one choice for the person in the last chair. If there are 7 people in a room, but only 4 chairs avaialbe, how many different seating arranagements are possible? 7!/3! = 840 n!/(n-k)! In this problem, each of the 7 people i the room (A, B, C, D,E, F, and G) could be in seat #1, #2, #3, #4 - or each could be one of the 3 people not seated in the chair. Let's assign these 3 people the designation N, to show that they are not seated. We can use N's to represent all 3 of the standing positions, since the order in which they are standing is not important to the problem. A B C D E F G 1 2 3 4 N N N Then the question is, in how many ways can we arrange these letters and numbers? This problem can be modeled with anagrams of the world 1234NNN Using the factorial logical from the previous section, it would appear that there are 7! anagrams for this word. owever, 1234N1N2N3 represent the same seat assignements as 1234N3N2N1. Although the two arrangements represent differential mathematical permutations, they both mean that person A is seated in seat 1, person B is seated in 2 and so on. There are 3! ways in which you can arrange the 3 N's. Calculating 7! will give you 3! times as many anagrams as there really are. Since the order in which the non-seated people are standing is not important to the problem, we actually do not wish to count all the different anagrams of the 3 N's. In order to do this, we divided by 3!. 7!/3!=840 If a team of 4 people is to be chosen from 7 people in a room, how many different teams are possible? 7!/(4!*3!) = 35 n!(k!*(n-k)!) If the order does not matter, use the lettter Y and N to represent Yes and No situation. A B C D E F G Y Y Y Y N N N Each person is either on the team or not on the team. The problem can be modeled with the anagram of the word YYYYNNN. There are 7! ways to arrage 7 letters. However, since 3 of the letters are N's and four of the letters are Y's you must divided by 3! and 4!. When the order does not matter in a problem what kind of problem is it? A combination problem. Whe the order does matter in a problem what kind of problem is it? A permutation problem. Greg, Marcia, Peter, Jan, bobby and Cindy go to a movie and sit next to each other in 6 adjacent seats in the front rown of the theater. If Marcia and Jan will not sit next to each other, in how may different arrangemnt can the sit people sit. Draw a diagram to the show the 6 chairs. 1 2 3 4 5 6 Jan as 6 possible choices. If Jan is the first to sit, she can sit in any of the 6 seats. Marcia has 3 or 4 possble choices. If Jan has chosent to sit in one of the two end seats (1 or 6), only one seat is off-limits to Marcia- the seat immediately adjacent to Jan. This leaves 4 remaining seat options for Marcia. If Jan has chosen to sit in one of the 4 middle seats (2, 3, 4 or 5), exactly two seats are off limits to Marcia-the two seats on either side of Jan. This leaves 3 remaining seats options for Marcia. Thus, in 1/3 (i.e 2/6) of the cses, Marcia will have 4 seat options and, in 2/3 (i.e 4/6) of the cases, Marcia will have only 3 seat options. After seating Jan and Marcia, seat the non-constrained people. Greg has 4 possible choices. Peters has 3 possible choices. Bobby has 2 pssible choices. Cindy has 1 possible choice. To compute the total number of permutation, find the product of the number of choices for each of the six people: 6*(1/3*4+2/3*3)*4*3*2 = 480 What is the the mathematical definition of probability? (# of winning outcomes)/(total # of possible outcomes) True or false since the GMAT probability problems often involved addition and multiplication, it is better to work in fractions. True. What is the proobability of event X AND event Y will occur, given that X and Y are indepent events? P(X)*P(Y) What is the probability that a fair coin flipped twice will land on heads both times? This is an "and" problem, because it is asking for the probability that the choicn will land on head son both the first flip AND the second flip. The probability that the coin will land on heads on the first flip is 1/2. The probability that the coin will land on the heads on the second flip is 1/2. Therefore the probabiity that the coin will land on heads on both flips is 1/2*1/2=1/4. What is the probability that X OR Y will occur, given that X and Y are independent and mutually exclusive? P(X)+P(Y) What is the probability that a fair die rolled once will land on either 4 or 5? This is an "or" problem, because it is asking for the probability that the die will land on either 4 or 5. The probability that the die will land on 4 is 1/6. The probability that the die will land on 5 is 1/6. 1/6+1/6=1/3. What is the meaning of mutually execlusive in terms of probabilities? It means the two events cannot both occur. What is the probability that X OR Y will occur, given that X and Y are independent but not mutually exclusive? P(X)+P(Y)-P(X)*P(Y). A fair die is rolled once and a fair coin is flipped once. What is the probability that either the die will land on 3 or that the coin will land on heads? These events are not mutually exclusive, since both can occur. The probability that the die will land on 3 is 1/6. The probability that the coin will land on heads is 1/2. The probabiliyt of both these events occuring is 1/6*1/2=1/12. Therefore, the probability of either event occuring is 1/6+1/2-1/12=7/12. What is a good time saving probability trick on the GMAT? 1-X trick. A simple trick to save time is to find the probability that an event will not happen. What is the probability that, on 3 rolls of a single fair die, at least one of the rolls will be a six? Rephrase the question. What is the probability that of 3 rolls of a fair die none will yield a 6? 5/6*5/6*5/6=125/216. 1-125/216=91/216. What is the dominino effect? Is occurs when the probability of the first event effects subsequent events. IN a box with 10 blocks, 3 of which are red, what is the probability of picking out a red block on your first two tries? 3/10*2/9=1/15 What is the best way to solve the problem below? Renee has a bag of candy. The bag has 1 candy bar, 2 lollopops, 3 jelly beans, and 4 truffles. Jack takes one piece of candy out of the bag at random. If he picks a jellybean, he chooses one additional piece of candy and then stops. If he picks any non-jellybean candy, he stops picking immediately. After Jack picks his candy, Renee will pick a piece of candy. What is the probability that Renee picks a jelly bean? List the winning scenarios. Use this for tough probability problems. Renee has a bag of candy. The bag has 1 candy bar, 2 lollopops, 3 jelly beans, and 4 truffles. Jack takes one piece of candy out of the bag at random. If he picks a jellybean, he chooses one additional piece of candy and then stops. If he picks any non-jellybean candy, he stops picking immediately. After Jack picks his candy, Renee will pick a piece of candy. What is the probability that Renee picks a jelly bean? Jack1 Jack2 Renee jb no jb jb jb jb jb no jb - jb Jack1 Jack2 Renee Total 3/10 7/9 2/8 42/720 3/10 2/0 1/8 6/720 7/10 - 3/9 168/720 The probability of either event chain 1 OR 2 OR 3 happening is the sum of the individual probabilities. 42/720+6/720+168/720=216/720 =3/10 What is the best way to solve the probability problem below? Kate and her twin sister Amy want to be on the same relay-race team. There are 6 girls in the group and only 4 of them will be placed on the team. What is the probability that ate and Amy will both be on the same team? Use the Counting Method. Use this method for tough probability problems. A B C D E F Y Y Y Y N N There are 6!/4!*2! = 15 different teams. All the winning scenarios will include Kate, Amy and 2 other people. Therefore, we can create a combinatoric grid to figure out how many ways there are tow select the 2 other people from the remaining 4 girls. A B C D Y Y N N 4!/(2!*2!) = 6 WINNING OUTCOMES Therefore 6/15. True or false to find the average of a set you need to know each individual term. False. To find the average in a problem set you do not need to know each individual term. RECALL: SUM = AVERAGE* (NUMBER OF TERMS) Therefore AVERAGE = SUM/(# OF TERMS) True or false the quickest way determine the average of the set {5, 10, 15, 20, 25, 30} you need to sum up all the terms and divided by the number of terms? False. Because the set {5, 10, 15, 20, 25, 30} is evenly spaced the average will be the two middles. 15+20/2=17.5. This is also the average of the set. True or false the average of a set of N consecutive integers is the middle number? True. This applies to both even number of terms in a sequence and an odd number of terms in a sequence. This only applies to a series with consecutive number in a series. Example {3,5, 7, 9, 11} = average = 7 {5, 10, 15, 20, 25, 30} average = average of the two middle terms. (15+20)/2= 17.5 What is the definition of medium? It is the middle number in a set ordered from least to greatest. In a set with two middle numbers what is the medium? The average of the two middle numbers. For example: {3, 4, 9, 9} = medium = (4+9)/2 = 6.5 True or false in a set with an unknown number you can always find the medium? False. It depends on the set. Sometimes you can find the medium {x, 2, 5, 11, 11, 12, 33} the medium is 11 regardless of x. And sometimes you can't {x,x, 2,4,5} in this case you can't determine the medium without knowing the value of x. True or false given the sequence {x, x, y, y, y, y} arraged in ascending order the average and the medium is y. False. The only time that the average and the medium are equal are for evenly spaced sets. In this case the average is (2x+4y)/6 And the medium is average of the two middle terms. y. When is the average and medium equal? For evenly spaced sets. What is the definition of standard deviation? Distance from the average. What is the best way to solve problems which involve 2 sets that partially intersect? Use the double set matrix table organization. Of 30 integers, 15 are in set A, 22 are in set B, and 8 are in set A and B. How many integers are in neither set A or B. A NOTA TOTAL B 8 22 NOT B TOTAL 15 30 Fill in chart. A NOTA TOTAL B 8 14 22 NOT B 7 1 8 TOTAL 15 15 30 When confronted with a overallaping set problem with percents do you still use the double set matrix table organization? Yes. Pick 100 to represent the total. 70% of the guest at Company X's annual holiday party are employees of Company X. 10% of the guest are women who are not employees of Company X. If half the guest at the party are men, what percent of the guest are female employees of Company X. MEN WOMEN TOTAL X 70 NOT X 10 TOTAL 50 100 Fill in the chart. MEN WOMEN TOTAL X 30 40 70 NOT X 20 10 30 TOTAL 50 50 100 40% is the answer.