Quantitative Problems

Basic idea: equation for working together

(1/a) + (1/b) = 1/h where a is hours it takes a to do job alone and b is hours it takes b to do job alone

Arithmetic: Properties of Numbers If p is the product of integers 1-30 inclusive, what's the greatest integer k for which 3^k is a factor of p

1. Find which numbers 1-30 are multiples of 3 (have 3^x as a factor). 2. Under each of those numbers check how many factos they have. Etc. 27 has both 3 & 9 -- so 2 factors 3. Total up the number factors, that how many 3's will be in the final factorial A: 14 factors

Basic Idea: area of trapezoid

1/2(sum of bases * height)

Basic Idea: y % =

y/100

Data Sufficiency: Joanna bought only $0.15 stamps and $0.29 stamps. How many $0.15 stamps did she buy? (1) She bought $4.40 worth of stamps.(2) She bought an equal number of $0.15 stamps and $0.29 stamps.

Answer= A

Basic Idea: a^2 - b^2 can be factored into...

(a-b) * (a+b)

Basic Idea: Equation for Arc Length

= radius * internal/central angle

Arithmetic: ProbabilityEight women and two men are available to serve on a committee. If three people are picked, what is the probability that the committee includes at least one man?

Because this is an "at least" question, use the 1 - x shortcut: (The probability of picking at least one man) + (The probability of picking no men) = 1 The probability of picking no men is an and setup: woman and woman and woman. For the first choice, there are 8 women out of 10 people: . For the second choice, there are (because one woman has already been chosen): 7/9 For the third choice, there are: 6/8 = 3/4 Multiply = 7/15 Subtract from 1 = 8/15

Data Sufficiency: If Y = 2xz what is the value of (2xz + yz)/(2xz - y) 1. 2x + y = 3 2. z = 2

Break it out into: (2xz + yz)/(2xz - y) = z ( 2x + y) / (2xz - y) Neither equation gives enough information when plugged in. Combining isn't enough either

Basic Idea: Combination

Order of selection does not matter; (n/k) = n!/[k!(n-k)!] Basically can use (n/k) = n/n-k

Arithmetic: Properties of Numbers What is the average (arithmetic mean) of 11 consecutive integers 1) Average of 1st 9 integers is 7 2) Average of last 9 integers is 9 GMAT Practice 1

Pay attention to the word CONSECUTIVE Either piece of information tells you where that set numbers is centered on. Can find mean w/ either 1 or 2 Answer: Either is sufficient

Arithmetic: Probability From a group of 3 boys and 3 girls, 4 children are to be randomly selected. What is the probability that equal numbers of boys and girls will be selected?

Probability of selecting 1st boy = 3/6 Probability of selecting 2nd boy = 2/5 (as we have already selected one boy above) Probability of selecting 1st Girl = 3/4 Probability of selecting 2nd Girl = 2/3 Multiplied together = 3/5 or [3!/2! * 3!/2! ]/ 6!/4!2! 9/15= 3/5

Basic idea: equation for rate problems

rate x time = distance

Data Sufficiency: If xy doesn't equal 0, is x^3 + y^3 > 0? 1) x + y > 0 2) xy > 0

1) factor x^3 + y^3 as (x + y) * (x^2 - xy + y^2) For this to work, both components have to be either positive or negative. Since we know that one of them is, both have to be positive. Sufficient 2) Not sufficient to prove that both components are positive

Basic Idea: sum of interior angles of polygon with n sides

(n-2)*180

Basic Idea: Quadratic equation

negative b plus or minus the square root of b squared minus 4 ac ,all divided by 2a

Data Sufficiency: At the beginning of the year, the F committee and P committee each had n members. No one was a member of both. At end of year, 5 members left F and 3 members left P. How many members did F have at beginning of year? 1) Ratio of total members who left at EOY to number of members at beginning of year was 1:6 2) At end of year, 21 members remained on planning committee

1) 8/2n = 1/6 can solve for N 2) n-3 = 21 can easily solve for N because same number of members on F and P Answer: D

Arithmetic: Properties of Numbers See Picture for Question GMAT Practice 1

1. Key is to create a butterfly equation (5^-x/4^18) * 1/2x10^35 2. Then once multiplied out, turn 4^18 into 2 base 2^36 = 5^-x * 10^35 3. Then break out 10^35 into 2^35 * 5^35 Find that x = 35

Algebra: Equations For a certain race, 3 teams were allowed to enter 3 members each. A team earned 6 - n points whenever one of its members finished in nth place, where 1 ≤ n ≤ 5. There were no ties, disqualifications, or withdrawals. If no team earned more than 6 points, what is the least possible score a team could have earned? GMAT Practice 1

9 total players Table is as follows: 1st = 5 points 2nd = 4 3rd = 3 4th = 2 5th = 1 No team earned more than 6 points so... Let's say 1st team wins the most possible. 1st + 5th + 9th= 6 2nd Also Does: 2nd + 4th + 8th= 6 3rd does everything else: 3rd + 6th + 7th = 3 Answer = 3

Data Sufficiency: In the figure above, is the area of the triangular region ABC = area of region DBA? 1) (AC)^2 = 2(AD)^2 2) ABC is isosceles

B) Need both Set equivalency (AC)(AB)= (AD)(AB) From 2) Know if its isoceles AD*AB= x^2 AB= x*sqrt2 AD = x/sqrt 2 from part 1, so also = x^2

Algebra: Interest If money is invested a r interest rate, compounded annually, it will double in 70/r years. If 5000 was initially invested at 8%, how much will there be in 18 years?

Concept: Investments double in 70/r 70/8= 9 roughly which roughly half of 18. ($5000)*2*2= roughly $20,000

Arithmetic: Properties of Numbers If n= 3^8-2^8, which is not a factor of N?

Concept: Can break bigger exponents into perfect squares. Can also treat exponents like they're x^2 functions 1. 3^8 = (3^4)^2 and 2^8 = (2^4)^2 2. So, 3^8-2^8 = (3^4+2^4)(3^4-2^4) 3. Can be further broken down into (3^4+2^4)(3^2-2^2)(3^2+2^2) 4. Just like an algebraic expression one can see what each of these equations equals to. We get the "answers" of 5, 13, & 97.

Arithmetic: Properties of Numbers Calculate: (0.99999999999/1.001) - 0.9999999999991/1.000003)

Concept: Can write 0.99999999 as 10- (10^-8) 0.999999991 = 1-9(10^-8) 1.0001 = 1+10^-4 1.0003 = 1 + 3(10-4) A=2(10-4)

Arithmetic: Properties of Numbers If 35% of consumers were influenced by TV, 22% by coupons, 18% by store displays and 15% by samples-- what fraction indicated wither coupons or store displays?

Concept: Check in every table if percentages total to 100

Algebra: Rates If Francine traveled x % of a total distance at an average speed of 40 miles/hour and the rest of the distance at 60mph, what was Francine's average speed?

Concept: Make a table where Time = Distance/Rate for each of the two legs of the journey 1. Get equations (x/40) and (100-x/60) 2. Add them together to get x+200/120 3. To find rate divide total distance (100) by total time Answer= 12000/(x+200)

Arithmetic: Properties of Numbers For every positive integer m, f(m) represents the product of all even integers from 2 to m inclusive. What is the greatest prime factor of f(24)?

F(24) = 2 x 4 x 6 x 8 x 10 x 12 x 14 x 16 x 18 x 20 x 22 x 24 6 = 2 x 3 10 = 2 x 5 ... 22 = 2 x 11 Answer = 11

Geometry: Areas In the figure above, points P and Q lie on the circle with center O. What is the value of s ? GMAT Practice 1

Have to realize that we don't know which way the right triangle is angled. The y-axis doesn't perfectly bisect it. Based on the coordinates of P, we know that it forms a 30 60 90 triangle. Key is to know 30:60:90 corresponds to x: x*sqrt3 : 2x So we know that the left side takes up 30 degrees and the right takes up 60 degrees (must add up to 180). Using the correspondence, we know s must be 1

Algebra: Second Degree Equations If Sqrt(3-2x) = Sqrt(2x) + 1 then 4x^2 =? 1, 3, (2-2x), (4x-2), (6x-1)

Idea: Scramble equation until you reach desired answer Work with equation until you can create 4x^2 A= (6x-1)

Algebra: Equations How many integers that satisfy inequality (x+2) ( x +3) / (x-2) > 0 are less than 5?

Inequality holds when x > 2 but less than 5, so 3 numbers But also when x is equal to -2 or -3 So 5 total

Data Sufficiency: Is the number of seconds required to travel d1 feet at r1 feet per second greater than the number of seconds required to travel d2 feet at r2 feet per second? (1) d1 is 30 greater than d2 (2) r1 is 30 greater than r2.

It is known from 1 & 2 that the distance travelled in the first instance was farther and that the average speed was faster in the first. However, it is still unknown whether time taken to travel greater distance at at higher speed was greater than lesser distance at lesser speed.

Data Sufficiency: If m, p, and t are positive integers and m<p<t, is the product mpt an even integer? (1) t - p = p - m (2) t - m = 16

Odd: odd x odd x odd odd + odd + odd Even: odd x even x even even x even x even even + even + even odd + even + even Even spacing can mean either odd odd or even even Neither is enough data

Basic Idea: Permutation

Order of selection DOES MATTER P(K) = n!/(n-k)!

Geometry: Angles In the figure shown, what is the value of angles a + b + c + d + e?

Rule: sum of interior angles of any polygon is 180(n-2) Thus, interior angles of pentagon = 540 Need to write 5 equations for each of the angles based on 5 triangles in the figure. Substitute middle angles for 540. Scramble equation until you reach desired answer. A= 180 degrees

Arithmetic: Properties of Numbers If the average of numbers: 3, 15, 32 and N+1 is 18, what is N?

Simple Q, but watch out for N+1 Concept: Simple questions may have a small trick. Look for what unknown is defined as!

Algebra: EquationsIf x, y, and k are positive numbers such that [x/x+y]∗10 + [y/(x+y)]∗20 =k and if x < y, which of the following could be the value of k? 10, 12, 15, 18, 30

We have: 10x+20y/x+y=k, if you look at this equation you'll notice that it's a weighted average. There are x red boxes and y blue boxes. Red box weight is 10kg and blue box weight 20kg, what is the average weight of x red boxes and y blue boxes? k represents the weighted average. As y>x, then the weighted average k, must be closer to 20 than to 10. 18 is the only choice satisfying this condition.

Basic Idea: Factorial (!) =

multiply all integers below or = to that number

Data Sufficiency: If 2^(x+y) = 4^8 what is value of Y? 1) x^2 = 81 2) x - y = 2

4^8 = (2^2)^8 = 2^16 We don't know if x is -9 or 9 based on the first one B) Only 2 is sufficient

Arithmetic: Properties of Numbers If s/t = 64.12 which could be a remainder when s is divided by t? 2, 4, 8, 20, 45

Idea: Write equation and test answers So, 64t + 0.12t = S With the 0.12t remainder, test answers to see if its an integer A= 45 bc 45/.012=375 Aka dividing answers by remainder % will tell if could be the integer remainder

Arithmetic: ProbabilityHow many different 6-letter sequences are there that consist of 1 A 2B's and 3C's?

720 Total Choices but need to remove arrangements of 2B's and 3C's bevause the letter sequences would look exactly the same. 2 B's can be arranged in 2 ways 3 C's can be arranged in 6 ways --> 720/12 = 60 different sequences

Algebra: Equations A seed mixture X is 40% R and 60% B by weight. Seed mixture Y is 25% R and 75%. If a mix of Mixture X & Y is 30% R, what % of the mixture is X?

0.4X+0.25Y = 0.3 (X+Y) X=0.5Y Or the ratio is 2:1 (Y:X) So answer is 33%

Algebra: Simplifying Equations If p doesn't = 0 and p- [(1-p^2)/p] = r/p then r=?

1. Multiply both sides by p 2. p^2 - (1-p^2) 3. 2p^2-1 = r Concept: Don't forget to carry the negative

Algebra: Equations[SUPER HARD] At his regular hourly rate, Don's labor cost of a repair job is $336. However, he took 4 hours longer than he'd estimated and earned $2 less than his hourly rate. What was the time he'd estimated for the job?

1. Write system of equations: (r-t)(t+4)=336; rt=336 2. FOIL: rt-2t+4r-8=336 3. Get rid of rt by plugging in: -2t+4r-8=0 4. Multiply both sides by t: -2t^2+4rt-8=0 5. Swap out rt again and divide by 2: t^2+4t-672=0 6. Factor quadratic into (t-24)(t+28) 7. T can't be negative, so answer is 24

Algebra: Equations Of the 150 houses, 60% have A, 50% have B, 30% have C. 5 have all 3, and 5 have none. How many have 2?

A: 55 195 have at 1 or 2 amenities 195- (150-10) = 55 10 comes from 150 -5 -5 for houses with 3 or none

Arithmetic: ProbabilityThe 5 letters in the list G, H, I, J, K are to be rearranged so that G is the 3rd letter in the list and H is not next to G. How many such rearrangements are there? 60 36 24 12 6

Answer = 12 Only 2 configurations possible: H _ G _ _ or _ _ G _ H Looks like 3 choices for first blank, then 2 choices, then 1, same for 2nd config So only 12 options FYI: Need to use 3! (3 factorial or 3 x 2 x 1) Adding just happens to get same thing but actually wrong kk?

Arithmetic: Properties of Numbers Of 300 subjects, 40% had sweaty pals, 30% vomited, and 75% were dizzy. If 35% of subjects had two of these effects, how many experienced only one effect?

Concept: In venn diagram questions, people who experience 2 conditions are "counted twice". 1. .35*300- 105. So 195 experienced 1 or 2 effects. 2. 120+90+225 (300 * % of people who experienced each condition) = 435 a + 2b + 3c = 435 a + 2(105) + 3c= 435 a + 3c = 225 a + c = 195 a + 3x = 225 -3a-3x= -585 a+3c=225 -2a=-360 or a = 180

Data Sufficiency: Each month Jim receives a base salary plus a 10% commission on the price of each car that he sells that month. Jim sold 15 cars last month, what's the total amount of base salary and commissions. 1) Last month Jim's base salary was $3000 2) Last month Jim sold 3 cars totaling 60K and 5 cars totaling 120K

Concept: READ READ READ. 3+5 does not give you the full 15 Answer: E

Algebra: Second Degree Equations Which of the following equations has 1+sqrt(20 as one of its roots?

Concept: When finding an equation from its roots, first get rid of radical. Then make the equation =0! x-1+sqrt(2) (x-1)^2=2 x^2-2x-1=0

Arithmetic: Probability There are 5 cars to be displayed in 5 parking spaces with all the cars facing the same direction. Of the 5 cars, 3 are red, 1 is blue and 1 is yellow. If the cars are identical except for color, how many different display arrangements of the 5 cars are possible?(A) 20 (B) 25 (C) 40 (D) 60 (E) 125

First consider the red cars. There are 5 spaces from which 3 must be chosen to display the red cars --> 5!/3!*2! = 10 Then there are 2 remaining spaces for the blue and yellow car so (2) * (1) possibilities Thus 20 is the answer (See combination formula!)

Geometry: Areas In the figure above, triangle is inscribed in a circle. The measure of angle is greater than 90°, and the area of the circle is 25π. Quantity A: The length of line segment Quantity B: 10

Key is knowing that a 90 degree angle means that a diameter is formed. Less or more than 90 degrees forms a chord that is not the diameter. More than 90= less than diameter B is greater

Geometry: Areas A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r ? GMAT Practice 1

Key is that we don't know if the two pieces are equal ((40 - 2πr)/4)^2 is the area of the square --> (10-1/2πr)^2 Then just add the area of the circle, πr^2 Answer = πr^2 + (10-1/2πr)^2

Algebra: Simplifying [SUPER HARD] Aaron will jog from home a x miles/hour and then walk back at y miles/hour. How many miles can he jog so that he spends a total of t hours jogging and walking?

Let j be the number of hours Aaron spends jogging. Let t-j be total number of hours he spends walking. So, he jogs a distance of xj miles Walks a distance of y (t-j) miles Miles must be equal because its the same route. xj = y (t-j) xj = yt - jy Move all terms with a j to one side and factor out j (you want to get rid of/isolate the variable you made up) j (x+y) = yt j = yt/x+y Number of miles he can jog = xj remember So x * (yt/x+y) = xyt/ x + y

Basic Idea: Probability of this "or" that

Mutually Exclusive Events: Add probability of A & B Not ME: P(A) + P(B) - P(A & B)

Data Sufficiency: If n is an integer, is (0.1)^n greater than 10^n n>-10 n<10

Statement will only be true if n is < 0 (negative exponent means 10^n will get infinitely small, and 0.1^n will approach infinity) Answer: E neither

On a certain day, a bakery produced a batch of rolls at a total production cost of $ 300. On that day, 4545 of the rolls in the batch were sold, each at a price that was 50 percent greater than the average (arithmetic mean) production cost per roll. The remaining rolls in the batch were sold the next day, each at a price that was 20 percent less than the price of the day before. What was the bakery's profit on this batch of rolls? A. $ 150 B. $ 144 C. $ 132 D. $ 108 E. $ 90

Profit as (Total Revenue - Total Cost).The profit is the same for 300 rolls that cost $300 as it is for 30 rolls that cost $300.Let # of rolls = 30Cost per roll = $10 Day 1: 4545 of 30, or 24 rolls, are sold at 50 percent above cost -->Day 1, PRICE: 50% above cost is (1.5 * $10) = $15 each Day 1, Total Revenue: (24 * $15) = $360Day 2: remaining 6 rolls are sold at 80 percent of Day 1's price (of $15) -->Day 2's price is (.8 * $15)= $12 eachDay 2, Total Revenue: (6 * $12) = $72Total Revenue for both days: ($360 + $72) = $432(TR - TC) = PROFIT PROFIT = ($432 - $300) = $132

Data Sufficiency: Seven different numbers are selected from the integers 1 to 100, and each number is divided by 7. What is the sum of the remainders? (1) The range of the seven remainders is 6. (2) The seven numbers selected are consecutive integers.

Range of numbers doesn't necessarily mean that you know what the remainders are... could be 1,1,1,7,7,7 2 is enough.

Arithmetic: ProbabilityA couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys? (A) 3/8 (B) 1/4 (C) 3/16 (D) 1/8 (E) 1/16

Use binomial function: 4!/2!*2! * (1/2)^2 * (1/2)^2 = 6/16= 3/8

Algebra: Equations What was the percent increase in the population of City K from 1980 to 1990 ? (1) In 1970 the population of City K was 160,000. (2) In 1980 the population of City K was 20 percent greater than it was in 1970, and in 1990 the population of City K was 30 percent greater than it was in 1970. GMAT Practice 1

Write out the equations! You'd know that the x in the equation already cancels out in the 2nd equation. No need for the first piece of info. Answer: B

Arithmetic: Properties of Numbers 1, −3, 4, 1, −3, 4, 1, −3, 4, ... In the sequence above, the first 3 terms repeat without end. What is the sum of the terms of the sequence from the 150th term to the 154th term?

You see that the 3rd term is 4, and every 3rd term after that, in other words, the 6th, 9th, 12th, 15th, and so on, is 4. Since 150 is a multiple of 3, the 150th term is 4. Therefore the 150th to the 154th terms are 4, 1, −3, 4, 1. The sum of these 5 terms is 7, so the correct answer is 7.

Algebra: Equations A store sold copies of Newspaper A for 1.00 and Newspaper B for 1.25. if r% of store's revenue was from newspaper sales from Newspaper A and p% of newspapers sold were Newspaper A, which of the following expresses r in terms of P?

p% of N= (p/100)*N Revenue from Newspaper A: (p/100)*N * $1 Revenue from Newspaper B: [(100-p%) /100 ]* $1.25 Add those two together to get: 4p/500-p

Basic idea: volume of sphere

4/3 pi r^3

Data Sufficiency: If x is a positive integer greater than 1, what's the value of x? 1) 2x is a common factor of 18 & 24 2) x is a factor of 6

Concept: Factor is less than number, don't get it confused with multiple Answer: A 1 only

Geometry: Volumes A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What's the ratio of the height of the cone to radius of the hemisphere?

Concept: Height of the cone is the radius of the hemisphere. A: 1 to 1 ratio! Pay attention to what is inscribed in what!

Geometry: Angles If 2<k<7 how many different values of k is there a triangle with sides of lengths 2, 7, & k?

Concept: In a triangle sum of smaller two sides must be larger than largest side. 2+x > 7 x can't be greater than 7 x can only be 6 A: only 1 value of k

Arithmetic: Probability The letters G, D, I , I & T can be used in a 5 letter sequence. How many strings can be formed such that the I's are seperated by one other letter?

Concept: Think of this as two different problems. First, how many ways can the I's be placed and then how many unique combinations of the remaining letters? 1. For I's 5!/(3!*2!) - 4 = 6 or draw it out 2. D, G, T : 3*2*1= 6 6*6=36

Geometry: Triangles What is the floor area in square feet of the bedroom?

Concept: When a parallel line cuts through a triangle, the sides it intersects have the same ratio of intersected parts. So, here 30ft:30ft corresponds to 15ft:15ft. This tells us that the "bedroom" is an equilateral triangle, which allows us to apply the x:sqrt(3)x:2x property of 30-60-90 triangles. This gives us an area of 225*sqrt(3)

Arithmetic: Probability If Team A has 3 males and 3 females and decides to line up M, F, M, F, M, F how many different possible lineups are there?

Concept: When calculating a permutation just multiply... but when determining how many "options" there are for each position, split between categories. 1. For the first spot, it can be any of 3 males. 2. For 2nd, any of 3 females. Eventuall you get: 3x3x2x2x1x1=36

Arithmetic: Properties of Numbers If there are 15 white flowers and 85 red flowers, what's the greatest number of bouquets that can be made if there's the same ratio is the same in each bouquet?If

Concept: When making equal sets of items, look for the Greatest Common Denominator as the max number of sets

Arithmetic: Properties of Numbers What is the smallest integer n for which 25^n > 5^12

Concept: When one exponential expression's base is the square, etc. of the other's base-- you can assign a common base. 25^n = 5^2n 5^2n > 5^12 So 2n>12

Algebra: Interest Alex deposited x dollars into a new account that earned 8% annual interest, compounded annually. 1 year later she deposited additional x dollars. If there were no other transactions, and the account contained w dollars after two years, which expresses x in terms of y

Concept: With interest over multiple years, calculate the state of interest after Y1, at the beginning of Y2, at the end of Y2 etc. with as many breaks as needed. After Y1: x*1.08 Begin Y2: x*1.08 + x End of Y2: (x*1.08 + x)*1.08 w=x(1.08)^2+x(1.08) So x= w/ ((1.08)^2+(1.08))

Data Sufficiency: A paint mixture was formed by mixing exactly mixing exactly 3 colors of paint. By volume, mixture was x% blue, y% red, and z% green. If 1 gallon of blue paint and 3 gallons of red paint were used, how many gallons of green paint were used? 1) x=y 2) z=60

Concept: break down percentages y% --> y/100*T = 1 Answer: D either

Arithmetic: Properties of Numbers If y is the smallest positive number such that 3150 multiplied by y is the square of an integer, then y must be? 2, 5, 6, 7, 14

Concept: when finding smallest integer such that a number is the square of the integer, first find the prime factorization. Also, to be a perfect square, all factors must be squared or paired. 3150= 10*315= 2*3^2*5^2*7 2 and 7 are unpaired factors. So we need 14!

Arithmetic: Properties of Numbers If d= (1)/(2^3 x 5^7), how many nonzero digits will d have?

Concept; A factor with an integer power of 10 has no effect on the number of non-zero digits 1. Pair together equal powers of 2 and 5 so that you can create 10 and take it out of the equation. = 1/(2^3 x 5^3 x 5^4) = 10^-3 x 1/5^4 2. Boils Down to: 0.2^4 = .0000...16`

Geometry: Volumes Cylindrical tank contains 36π cubic feet of water and is filled to half capacity. When tank is placed upright, height of water is 4ft. When tank is placed on side, what is height in feet of surface of water?

Logic: When a volume is half full, it will be filled to half its height no matter what side its on. When a cylinder is on its side, 1/2 height = radius 36π = πr^2*h 36 = r^2 (4) r=3

Data Sufficiency: A conveyor belt moves bottles at a constant speed of 120 CM per second. If the belt moves a bottle from a loading dock to an unloading dock, is the distance the conveyor belt moves the bottle less than 90 meters? 1) Takes conveyor belt less than 1.2 minutes to move the bottle 2) Takes conveyor belt more than 1.1 minutes to move the bottle

Rate = 1.2 M per second Conveyor belt will move less than 90 meters if if moves less than 90/1.2 = 75 seconds 1) Indicates length of time that the conveyor belt moves is less than 1.2 minutes --> most it could've moved is 1.2 minutes or 72 seconds, which is less than 75 seconds 2) Too open ended, we don't know how much more than 1.1 minutes the belt runs Answer: A 1 alone

Arithmetic: Probability A certain club has 10 members including 10 members. 1/10 is selected to be president, one of remaining is secretary, then one of remaining is treasurer. What's the probability that Harry will be chosen to be secretary or treasurer?

Rule: probability--> (#of desired outcomes/total # of outcomes) "Or" signals two independent events, so need to add together two probabilities. So first calculate the probability of secretary, then probability of treasurer. (9/10)*(`1/9)*(1)= 1/10 (9/10)*(8/9)*(1/8)=1/10 1/10 + 1/10= 1/5

Arithmetic: Probability How many ways can the letters in MISSISSIPPI be rearranged?

This is a combinatorics problem, and the WOW example is intended to make it clear that any W is considered identical to any other W—switching one W with another would not result in a different combination, just as switching one S with another in MISSISSIPPI would not result in a different combination. Therefore, solve this problem using the classic combinatorics formula for accounting for subgroups among which order does not matter: Because MISSISSIPPI has 11 letters, including 1 M, 4 S's, 4 I's, and 2 P's:

Geometry: Areas If the shaded region (outside donut shape) is 3x the are of smaller region (interior, donut hole), then the circumference of the larger circle is how many times the circumference of the smaller circle?

piR^2-pir^2=3pir^2; R^2-r^2=3pir^2 When you solve the equations you get R=2r A: 2 Concept: Sometimes it's better to break info into two discreet equations to create a system of equations rather than combining everything altogerher.