A little lesson in trickery

a) permutation - order matters 336 b) computation - order does not matter 56

(Day 5 #10) A talent contest has 8 contestants. Judges must award prizes for first, second, and third places, with no ties.(a) In how many different ways can the judges award the 3 prizes? (b) How many different groups of 3 people can get prizes?

a) 1/9 b) 1/5 c)4/5 (easier to subtract probability of opposite outcome from 1)

(Day 5 #11) If an integer is randomly selected from all positive 2-digit integers, what is the probability that the integer chosen has (a) a 4 in the tens place? (b) at least one 4 in the tens place or the units place? (c) no 4 in either place?

- order DOESNT matter because DN = ND - use combination n! / r!(n-r)! 1/45 - without replacement = total # of outcomes decreases in second case

(Day 5 #12) In a box of 10 parts, 2 are defective. If you choose 2 parts without replacement, what is the probability of choosing 2 defective parts?

3048/4581 = 67% - total is not total students in school - ON campus

(Day 5 #13c) If one full-time student who is a freshman or sophomore is selected at random, what is the probability that the student will be a student who lives on campus?

a) 0.4 b) 0.2

(Day 5 #14)

a) 0.56 b) 0.94 c) 0.6 - could just find AND, then use probability of neither to find b

(Day 5 #15) Lin and Mark each attempt independently to decode a message. If the probability that Lin will decode the message is 0.80 and the probability that Mark will decode the message is 0.70, find the probability that (a) both will decode the message (b) at least one of them will decode the message (c) neither of them will decode the message

- mean, median, mode increase by 7 - range, IQR, SD are unchanged if multiplied by 7, ALL change: mean, median, mode, SD, range & IQR are x7

(Day 5 #2)

a) pretend all numbers are mean 85x20 = 1700 75x30 = 2250 3250/50 = 79 (combined average) b) cant be determined (can imagine multiple scenarios with different answers)

(Day 5 #3) A group of 20 values has a mean of 85 and a median of 80. A different group of 30 values has a mean of 75 and a median of 72. (a) What is the mean of the 50 values?(b) What is the median of the 50 values?

- relative frequencies are a % - imagine it is out of 100 - mean = 2.03 - median = 1 (average of 50th and 51st number because 100 is even)

(Day 5 #4)

40

(Day 5 #5)

5! = (5 possible letters for first slot)(4 for second slot)(3...)(2...)(1...) = 120 - can also be solved as a permutation (n!/(n-r)!)

(Day 5 #6) In how many different ways can the letters in the word STUDY be ordered?

Rules: no odd in last digit, no 5 in any digit Hidden rule: no 0 in hundreds place _ _ _ = 8(no 0, no 5) x 9(no 5) x4(no odd, no 5) = 288 - start from restrictions - can also be solved as a permutation (n!/(n-r)!)

(Day 5 #8) How many 3 digit odd positive integers do not contain the digit 5?

210

(Day 5 #9) From a box of 10 lightbulbs, you are to remove 4. How many different sets of 4 lightbulbs could you remove?

30; 0! = 1

(Day 5 #special) How many ways to arrange TOOTH?

17.5 Height doesn't need to be a straight line/right triangle as long as base is flat and height is given - height doesnt need to be inside triangle = measurement perpendicular to base

(Day 3 #10d) Find area of triangle AEF

3√17

(Day 3 #14b) Length of diagonal

378

(Day 3 #9)

0.000025

1/4 of 0.01 percent

0.16

0.16 to nearest 0.05

Independent events: can happen at same time a) AND: 2/3 x 1/4 = 1/6 b) OR: - Because they overlap, the area of overlap is counted twice. Therefore, it must be subtracted out. 2/3 + 1/4 - (2/3 x 1/4) = 3/4

2/3 chance of shirt, 1/4 chance of fedora. What is the probability of: a) One person wearing the shirt, and the other person wearing the fedora? b) One or the other?

3

25th digit to the right of decimal

a) 36 b) 12 c) 24

4500 a) how many factors b) how many odd factors c) how many even factors

"factors" implies when all answer choices are multiplied together A) 11 - taking away from a smaller factor will have less impact than in a larger factor of the product: 2*3 = 6 2*2 = 4 1*3 = 3

A decrease of 1 in which of the factors below would result in the greatest decrease in the product? A) 11 B) 13 C) 17 D) 19 E) 23

4400

A developer has land that has x feet of lake frontage. The land is to be subdivided into lots, each of which is to have either 80 feet or 100 feet of lake frontage. If 1/9 of the lots are to have 80 feet of frontage each and the remaining 40 lots are to have 100 feet of frontage each, what is the value of x ?

pick and test numbers for each option - finding ANY example that doesn't work disproves the category - implied; it does not explicitly say all numbers of the category must satisfy the condition - include the number itself as a factor D

A positive integer with exactly two different divisors greater than 1 must be A) prime B) an even integer C) a multiple of 3 D) the square of a prime E) the square of an odd integer

6 * sqrt 2

A rectangular rug covers half of a rectangular floor that is 9 feet wide and 12 feet long. If the dimensions of the rug are in the same ratio as those of the floor, how many feet long is the rug?

A) 10 - can also plug in distances until you get 40 minutes

At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm? a)10 b)11 c)12 d)13 e)14

average > median when skew is towards higher numbers In a normal distribution, the mean and the median are the same number while the mean and median in a skewed distribution become different numbers: A left-skewed, negative distribution will have the mean to the left of the median. A right-skewed distribution will have the mean to the right of the median. - mean (average) stays closer to skew, median stays with bump - skew is where tail is

right skew

C) 30 relative speed x time = 40 - relative speed of x from the perspective off y: pretend y is stationary, x-y = relative speed of x - 40 because we're acting like y is not moving

Car X is 40 miles west of Car Y. Both cars are traveling east, and Car X is going 50% faster than Car Y. If both cars travel at a constant rate and it takes Car X 2 hours and 40 minutes to catch up to Car Y, how fast is Car Y going? A. 20 miles per hour B. 25 miles per hour C. 30 miles per hour D. 35 miles per hour E. 40 miles per hour

quotient

the answer to a division problem

dividend

the number that is being divided

15.8%

From 2011 to 2012, Jack's annual salary increased by 10 percent and Arnie's annual salary decreased by 5 percent. If their annual salaries were equal in 2012, then Arnie's annual salary in 2011 was what percent greater than Jack's annual salary in 2011?Give your answer to the nearest 0.1 percent.

17 2(^4)^2 = 2^8 = 2^4 * 2^4 = 16*16

Greatest prime factor of 2(^4)^2 -1

6

How many 0s at end of 25!

- eliminate prime numbers (only have themselves and 1 as factors) - take first couple of prime numbers, see which work, and find pattern - if # of factors =4, then exponents +1 x each other must = 4 - x^1y^1 and X^3 are both possible

How many integers less than 100 have exactly 4 odd factors but no even factors?

10

How many positive integers less than 20 are equal to the sum of a positive multiple of 3 and a positive multiple of 4?

Remainder problems

If N/D has remainder R, then N can = R(minimum), R+D, R+2D, R+3D....

E - algebra or pick numbers (any numbers will show the relationship)

If a taxicab charges x cents for the first 1/9 mile and x/5 cents for each additional 1/9 mile or fraction thereof, what is the charge, in cents, for a ride of y miles, where y is a whole number? (A) x + (xy-x)/45 (B) x - (xy-x)/45 (C) 2x+9y/5 (D) x + (9x-y)/5 (E) x + (9xy-x)/5

even (8) * anything (p) = even B (only one that can be odd) - look at question as evens and odds, because only odds can be primes

If p is a positive integer, which of the following could be a prime number?

A - pick numbers

If p is a prime number greater than 11, and p is the sum of the two prime numbers x and y, then x could be (any answer that works) which of the following? A) 2 B) 5 C) 7 D) 9 E) 13

676,000

In State X, all vehicle license plates have 2 letters from the 26 letters of the alphabet followed by 3 one-digit numbers. How many different license plates can State X have if repetition of letters and numbers is allowed?

41

In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners? A) 49 B) 47 c) 45 D) 43 E) 41

A

Is A or B greater

C, D - prime factorization indicates that n and n^2 will be some combination of 2s and 3s - must have AT LEAST LCM (2^3*3^3) - try these primes to different exponents for n and n^2 until n^2 is divisible by 24 and 108

Let S be the set of all positive integers n such that n^2 is a multiple of both 24 and 108. Which of the following integers are divisors of every integer n in S?

C) 12 - convert relative rate to minutes, answer should be in minutes

Marla starts running around a circular track at the same time Nick starts walking around the same circular track. Marla completes 32 laps around the track per hour and Nick completes 12 laps around the track per hour. How many minutes after Marla and Nick begin moving will Marla have completed 4 more laps around the track than Nick? (A) 5 (B) 8 (C) 12 (D) 15 (E) 20

1/2 - use ((total-divisor)/divisor) + 1 to identify how many multiples of 30 - list some multiples of 30 and identify pattern - every other is divisible by 12

Of the positive integers that are multiples of 30 and are less than or equal to 360, what fraction are multiples of 12?

75% - % of students IN SCIENCE CLUB

Of the students in a school, 20 percent are in the science club and 30 percent are in the band. If 25 percent of the students in the school are in the band but are not in the science club, what percent of the students who are in the science club are not in the band?A : 5%B : 20C : 25D : 60E : 75%

0

Remainder when 9^1 + 9^2 + ... 9^8 is divided by 6?

40

The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto A. 20 B. 30 C. 40 D. 60 E. 90

14

The remainder when positive integer N is divided by 2 is 1, when divided by 3 is 2, when divided by 4 is 3, and when divided by 5 is 4. What is the sum of the digits of the least possible value of N? (A) 11 (B) 13 (C) 14 (D) 16 (E) 17

e) 120 - don't forget the first hour and 15 miles train A travelled

Trains A and B are 190 miles apart. Train A leaves one hour before train B does, traveling at 15mph directly toward train B.Train B travels at 10mph directly toward train A. When the trains meet, how many miles has train A traveled? a) 70 b) 85 c) 95 d) 105 e) 120

1:1

Two alloys A and B are composed if two basic elements. The ratios of the composition of the two basic elements in the two alloys are 5:3 and 1:2 respectively. A new alloy X is formed by mixing the two alloys A and B in the ratio 4:3. What is the ratioof the composition of the two basic elements in alloy X?

A) 36

Two horses begin running on an oval course at the same time. One runs each lap in 9 minutes; the other takes 12 minutes to run each lap. How Many minutes after the start will the faster horse have a one lap lead?

20

What is the greatest positive integer n such that 2^n is a factor of 12^10? 10, 12, 16, 20, or 60

4/36 = 1/9

What is the probability of rolling a total of 5 with a single roll of two fair 6-sided dice, each with the distinct numbers 1 through 6 on each side?

3 - closest 4th in pattern is 32, one more has remainder of 3 (last digit of number is always remainder when dividing by 10)

What is the remainder of 3^33 / 10

0 - all prime factors of 26 can also be found in 25!, so the numbers will divide evenly with no remainder

What is the remainder when 25! is divided by 26?

9 DIVIDEND=(DIVISOR*QUOTIENT)+REMAINDER

When a positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n? 11, 9, 7, 6, 4

11

When the positive integer n is divided by 3, the remainder is 2, and when n is divided by 5, the remainder is 1. What is the least possible value of n?

greatest even factor of 180 that is less than 90 - odd = only odd - even = anything (including odds) with at least one even

Which is greater: the greatest even factor of 180 that is less than 90, or the greatest odd factor of 180

A) 18

Working alone, R can complete a certain kind of job in 9 hours. R and S, working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can S, working alone, complete one of these jobs? (A) 18(B) 12(C) 9(D) 6(E) 3

divisor

factor

28

multiples of 4 from 13 to 125

in all possible cases

must be