GRE Math Things that I Forget too Easily

[Lines in coordinate geography. ALWAYS get the equation of the line and the slope.] How many points (x, y) lie on the line segment between (22, 12 2/3) and (7, 17 2/3) such that x and y are both integers?

-Here we get y=-1/3x+20 -So to be integers our x's must be a multiple of 3 -So x=9,12,15,18,21 for a total of 5

[We can only take RECIPROCALS of things that are in fractionally equivalent forms.] The reciprocal of a/1=1/a

-What we CANNOT do is this: (1/2)+(1/3) = 5/6 -The reciprocal is 6/5 but we cannot arrive to this by doing the following, which is illegal: (2/1)+(3/1)=5

36-25 =

11

[Special Triangle Rules] 1-1-sqrt(2) [sqrt(2)/2]-[sqrt(2)/2]-1

5-5-5sqrt(2) [5(sqrt(2)/2)]-[5(sqrt(2)/2)]-5

[Roots CAN be MULTIPLIED] [sqrt(5+u)][sqrt(5-u)] =

sqrt(25-u^2)

[Dividing by a decimal] 1/0.91 > 1

-Also notice that the close the denominator is to 1, the close the fraction is to 1 as well. So the small the denominator the larger the value.

[WE can take PERCENTS of 0; CORNER CASES] Column A 22 percent of x Column B 2/9 of x

-B is obviously bigger UNLESS we take x to be 0 since THERE ARE NO RESTRICTIONS on values, then the answers are = -So the answer is D

[COMPARING FRACTIONS] 4A = 9B

-Which is bigger A or B? -This can be written as 4/9A=B. Which means that whatever A is (which is the whole amount), B is only a fraction of it. So B is smaller. -Another way of looking at it is: A=9/4B. Which means that one portion of A is equal to 9/4 (over 1) portion of B. So A by itself must be bigger.

[Reciprocal] 1/x = 2/5 and I want x. Just use the reciprocal.

x=5/2

[Std] 1 unit of std can equal anything for example 1.2. What would 1.5 be?

-You can either set up ratio or do metal math (second is what is preferred is you can see this is a ratio problem).

[Just knowing the angles of a triangle without any idea into what the size of the sides is going to be D]

-A good way to check this is to elongate sides and keep angles fixed

[NUMBERS vs. INTEGERS] For POSITIVE "numbers" p & q (p-q)/(p+q) = 2/3 Column A p + q Column B 5

-ALWAYS pay attention to the limits of the question. Here it is numbers, so that is very broad. -Also notice that the numbers are POSITIVE, so we can manipulate the denominator -Also notice that 2/3 can be anything: 0.0002/0.0003 in which case p+q is less than 5. OR 2000/3000 in which case p+q greater than 5.

[HOW to approach things that look like they need calculations but don't.] Set X:{5,6,9} Set Y:{0,1,4} Column A Standard deviation of set X Column B Standard deviation of set Y

-Based off this we think we might need to calculate but the GRE doesn't want us to necessarily do that -The answer can't be D -What do we know about two sets of data that we are comparing? -The std is the SAME IF the same number is subtracted from each one. Is this the case here? -So no calculations needed. Best way to approach this is to see if D and C could be eliminated then you would for sure have to do calculations but ONLY as a last resort

[ALWAYS remember the a^2-b^2 equation] 25,002^2 - 24,998^2=

-Can this be expressed as (a+b)(a-b)?

[How to REWRITE CONSECUTIVE NUMBERS.] A, B, and C are consecutive odd integers such that A < B < C. If A + B + C = 81, then A + C =

-Consecutive odd's would be just 2 bigger than the previous number so another way to write A+B+C=A+(A+2)+(A+4)=3A+6=81 -From this we can figure out that A = 25 so C must be 29 -So A+C=54

[Two questions to ALWAYS ask regarding combinations/probabilities] Set A: {1, 3, 4, 6, 9, 12, 15} If three numbers are randomly selected from set A without replacement, what is the probability that the sum of the three numbers is divisible by 3? 3/14 2/7 9/14 5/7 11/14 _________ If three primes are randomly selected from the prime numbers less than 30 and no prime can be chosen more than once, what is the probability that the sum of the three prime numbers selected will be even? 10% 27% 30% 36.5% 42%

-First thing to ALWAYS ask is: how many stages are there? -Second thing to ALWAYS ask is: does order matter? -3 stages -No, so it is a combination -Remember also the integer properties of divisibility. 3 and any of its multiples added together will give a number still divisible by 3. 5c3/7c3 ________ -How many stages? 3 -Does order matter? Well the numbers we are dealing with are: 2,3,5,7,11,13,17,19,23,29. Would 2+3+5 be different than 3+5+2? NO, so this is a combination -What combinations work? We have all ODD and one EVEN O+O+O=O O+O+E=E only this scenario works... Since E must be even 2 is there we are now choosing 9c2=(9)(8)/(2)=36 -Total options? 10c3=(10)(9)(8)/(3!)=120 Our answer is 36/120=30% -I did this wrong and used FCP and got 10% only

[LIMITATIONS on WHOLE number objects] In a group of 200 workers, 10 percent of the males smoke, and 49 percent of the females smoke. Column A Total number of workers who smoke Column B 59

-First thing you should think about is what happens if there is a 50/50 split since we are only told the percent of the two groups we don't know their actual representation -If that is the case then the number that smoke will be 59 -Now we can assume that we can get different compositions such that we give more weight to the female but notice that .49 times something is very restrictive and will give a decimal answer below 200. We can only ever double it but that would be impossible since our max is 200 and we have a positive number of Males as well

[CAREFUL what PART of the POPULATION the questions is referring to.] At a certain university, 60% of the professors are women, and 70% of the professors are tenured. If 90% of the professors are women, tenured, or both, then what percent of the men are tenured? 25 37.5 50 62.5 75 Previous

-Here I did everything right but couldn't find the answer which is got at first 30% -This is the percent of the ENTIRE population that is MEN & TENURED -The question is asking us for "percent of MEN that are tenured, which is 3/4=75%

[Are we given PART-to-PART or PART-to-WHOLE?] A container holds 4 quarts of alcohol and 4 quarts of water. How many quarts of water must be added to the container to create a mixture that is 3 parts alcohol to 5 parts water by volume?

-Here I was setting up the equation equal to 3/5 = 0.6 -This however is FALSE! -Since I want the total concentration, I need PART-to-WHOLE, which would be 3/8

[*MEDIAN* DIVIDES the list in HALF] [X and Y] The median of x , y , 8 and 11 is 19. Column A x Column B 23

-Here we are given that the median is 19. That means that two of the numbers on the list MUST be greater than 19 and two of the numbers MUST be smaller than 19. -So we can see that the first two numbers are 8,11__,__ -The variables are interchangeable and can be anything so for A to be the answer, the answer must ALWAYS be greater than 23.

[SQUARES on BOTH SIDES] If square ABCD has area 25, and the area of the larger shaded square is 9 times the area of the smaller shaded square, what is the length of one side of the smaller shaded square?

-Here we get: (5-x)^2 = 9(x^2) -We can square root both SIDES! -Another way to think about this is to write to equations -So, x+y=5, and y^2=9x^2 -Here you still have to square root BOTH sides

[If there is a -0, the negative DOES NOT CARRY OVER.] 2 - [1 - (1 - [2 - 3] - 2) + 3] =

-I can't believe I missed this -The middle bracket of [2-3]=-1, which gets turned to +1 -Now we have 1+1-2 = 0 to get... 2-[1-0+3]=2-4=-2 -Notice that 1-0+3=1+3 NOT 1-3

[PLACEHOLDER for numbers that are multiples of 10.] How many integers between 1 and 10^21 are such that the sum of their digits is 2?

-I keep messing this up -We have a total of 22 digits with 21 of them being 0s -The 22nd digit has to be 0 for this purpose since we can't have anything bigger than that -This means that we are working with 21 digits, any combination of which will be less than 10^21 -SO... we can two by just having 2 or 1+1, which are two scenarios: 1) Scenarios with 2 are 21 2) Scenarios with 1s are 21c2= (21)(20)/2=210 Our answer is 231

[READ the questions CAREFULLY. "NUMBER" of positive NUMBERS...] N equals the number of positive 3-digit numbers that contain odd digits only. Column A N Column B 125

-I made the mistake here in thinking that we are looking for combination of odd numbers whereby we compare the number they form to 125 for example: 1,3,5,7,9. Any combination is going to give a number greater than 125 -BUT the question is asking us how many numbers are possible using 3 digits that are only odd. Well that is (5)(5)(5)=125

[Assign VARIABLE and give it a NAME.] A certain barrel is 1/5 full. When k liters of liquid are added to the barrel, it becomes 2/3 full. In terms of k, what is the capacity of the barrel, in liters? 3/8 k 7/15 k 15/7 k 7/3 k 8/3 k

-I made the mistake of making to variables but let's think this through: -Right now a barrel's full capacity is F -Only 1/5 of that is full so... F/5 -If we add k to F/5 then this would be equal to = (F)(2/3) -The answer is 15/7k

[BEWARE of what the variables represent] Yesterday, a certain school had an equal number of boys and girls. Today, 18 boys left the school, and the ratio of the number of boys to girls is now 3 to 4. Column A Number of boys in the school now. Column B 72 ___________________ [Another Example] If f(x) = 5 - 2x and f(3k) = f(k + 1), then f(k) = 0.5 1 3 4 6 __________________ [Another Example] If A is the initial amount put into an account, R is the annual percentage of interest written as a decimal, and the interest compounds annually, then which of the following would be an expression, in terms of A and R, for the interest accrued in three years? A(R)^3 A(R+R^3) A(3R+3R^2+R^3) 3A(R)^3 3A(R+R^2+R^3) _____ [Another that I missed] For the first 5 hours of a trip, a plane averaged 120 kilometers per hour. For the remainder of the trip, the plane travelled an average speed of 180 kilometers per hour. If the average speed for the entire trip was 170 kilometers per hour, how many hours long was the entire trip? 15 20 25 30 35

-I missed this because I used B:G as the variables but did not realize that B is representative of the original scenario and Column A is asking for the number of boys now, which is B-18 ____________________ -Notice that the original equation has the x in it and we are solving for k. We find out that it is 1/2. BUT THIS IS NOT THE ANSWER. We need to find out f(k)=f(1/2). This is equal to 4. -This is a huge trick for the GRE. Those who only solve part way and do not understand what is being asked will be tricked. _______________________ -Notice here that A is the initial amount -When we multiply by the multiple (1+R) that A is part of it. -Taking that into account we should have A+[A(3R+3R^2+R^3)]. The part after the first + is our answer ______ -Here I got the equation to be: [600+180x]/(5+x)=170 -So x has to be be 25 -THIS IS ONLY PART OF THE ANSWER. 25 is the TRAP ANSWER!!!!!! -The question is asking for the duration of the ENTIRE trip. So the answer is 5+25=30

[Inequalities] If y - 3x > 12 and x - y > 38, which of the following are possible values of x? -60 -30 -6 4 20 40 80

-If the inequalities are in the same direction, you can ADD them. -If they are in opposite directions, you can SUBTRACT them.

[In GEOMETRY, just because we are given multiple variables doesn't mean we have to use them separately.]

-If we are given A, B, C doesn't mean we have to use B by itself if we can write an equation that has 180-B. This means we are still including B! -Another way to think about this is that we have to pick 3 children to receive shirts of same color. The rest have to just get whatever is left. So this is a combination problem 6c3=20

[Quartiles and Distribution] Min 1st Q = 25% of the Pop 2nd Q = 50% of the Pop 3rd Q = 75% of the Pop Max

-If we need to find out how many people are between percentile subtract the percentiles and multiply by the total number of people B. J. Upton, who played on the Tampa Bay Rays that season, hit 78 RBIs in 2012; this is the 90th percentile value on this chart. How many players hit more than 56 (75th) and less than or equal to 78 (90th) RBIs? -Total people: 280 90-75=15%(280)=28+14=42

[SIMPLIFY the EQUATION FIRST.] [(a/b)+1]/(c/b) In the expression above, a, b and c are different numbers and each is one of the numbers 2, 3 or 5. What is the greatest possible value of the expression? 8/3 4 9/2 5 6

-If you don't simplify you have to test multiple cases, but if you simplify then get: (a+b)/c -It is clear from this that c should be 2 and a+b should be 8 for the greatest value of 4

[The word "DIFFERENT" is usually a case for COMBINATIONS. But to ensure...] Given that n = 10^a + 10^b + 10^c, where a, b, and c are distinct positive integers, how many different positive values of n result if n is less than 1 billion (1,000,000,000) ?

-In this case we have 8 choices: 1,2,3,4,5,6,7,8 since 9 would make the number a billion -a,b,c have to be all different. We can achieve this by the FCP. BUT NOTICE that a=1,b=2,c=3 is equal to a=3,b=2,a=1 since they are all being tied to the factor of 10 -So we HAVE to USE COMBINATIONS

[Ratios and How They Can be Arranged] z/y=35/6 z/x=35/8

-Look at the commonality. These have to be all related. So we can rewrite these as: x:y:z=8:6:35

[Ratios] Part-to-Whole The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k?

-Notice here that 4/5=(a+k)/(b+k). -The total parts are 9. a+b+2k=117. -117/9=13. So each part if worth 13. -So... 4/5=52/65 -Then... (52-k)/(65-k)=3/4 -k=13

[YOU MUST keep TRACK of SCENARIOS. Here is a tricky one.] If points A and B are randomly placed on the circumference of a circle with radius 2, what is the probability that the length of chord AB is greater than 2? 1/4 1/3 1/2 2/3 3/4

-Notice that a chord of 2 will mean that the angle is 60. But if we pick one point and there are two places we can place the other point. To the left or right of the first point. -This gives us a degree of 120 -So 240/360=2/3

[Take things ONE at a TIME. Use the given information to your advantage.] a1+a11+a21=99 In the above sequence, each term after the first is equal to the proceeding term plus c. Column A a3+a19 Column B 66

-Notice that the answer here CANNOT be D. We can solve for something. -Our sequence follows the below pattern: a1=A a2=a1+c=A+c a3=a2+c=A+c+c=A+2c a4=a3+c=A+2c+c=A+2c an=A+(n-1)c -Our given equation becomes: A+A+10c+A+20c=3A+30c=3(A+10c)=99 So... A+10c=99/3=33=a11 -Column A is: A+2c+A+18c=2A+20c=2(A+10c)=2(a11)=2(33)=66 -So our answer is C

[How to WRITE equation when working with TOTAL.] A sum of money was distributed among Lyle, Bob and Chloe. First, Lyle received 4 dollars plus one-half of what remained. Next, Bob received 4 dollars plus one-third of what remained. Finally, Chloe received the remaining $32. How many dollars did Bob receive? 10 20 26 40 52

-Notice that we are dealing with "REMAINING". This can be interpreted as Total (T) - Chloe = R -Using this we can create all our equations (substitute T-32 for R) -The total is equal to the sum of the below three: 32 4+(R/2) 4+(R/3)

[NO PROBABILITY can be GREATER than ONE] Events A and B are independent. The probability that events A and B both occur is 0.6 Column A The probability that event A occurs Column B 0.3

-Notice that we are essentially multiplying the two numbers together either of these numbers can be A -The largest that a number (probability) can be is 1, if that's the case then the other number has to be 0.6. These are both bigger than 0.3 -So if we lower one number from 1 then the probability of the other must increase from 1. Thus the lowest the numbers can be is 0.6

[When Distance is the Same] Andy drove from Townville to Villageton at an average speed of 40 miles per hour. He then drove the same route back from Villageton to Townville at an average speed of 60 miles per hour. Column A 50 Column B The average speed of Andy's entire trip in miles per hour.

-Remember if the distance is the same then the total distance is 2D. -Also you can make up a number for D. Use that if nothing else works.

[Exponent Rule] (a^n)(b^n)=(ab)^n

-So (2^-n)(7^-n)=(14)^-n

[*MEAN* is the middle number if the list has an ODD number of things in it.] If the average (arithmetic mean) of seven consecutive integers is k + 2, then the product of the greatest and least integer is k^2 - 9 k^2 - 2k + 1 k^2 + 4k - 12 k^2 + 6k + 9 k^2 + 4k - 5

-So we have a list of 7 things meaning the middle number of the list is equal to the MEAN which is k+2 __,__,__,(k+2),__,__,__, -We can figure out the rest of the numbers to be: (k-1),(k),(k+1),(k+2),(k+3),(k+4),(k+5) -The product of the largest and the smallest is... E

[The Quotient] -When you think of quotient we think it must be an integer. THIS IS NOT TRUE

-Take for example -1.5/1.1=-1.4 -Here the quotient is -1.4

[DO NOT make the mistake of not checking each number by plugging it in] For which values of x is (x^2 - 10x + 25)(x^2 - 13x + 40) < 10? 3 4 5 6 7 8 9 10

-The equation simplifies to (x-5)^3 (x-8) < 10 -I missed the number 4 using just the logic or negative vs. positive numbers. Since the number is of course positive it must be checked

[Never forget the word INCLUSIVE in math] Column A The product of integers from -87 to -36 inclusive Column B The product of integers from -58 to -34 inclusive

-The first one has 87-36=51 + 1 = 52 numbers inclusive. This MUST be a positive product -The second has 58-34=24 + 1 = 25 numbers inclusive. This MUST be a negative product

[Area of a Triangle] Point A (-4, 2) and Point B (2, 4) lie in the xy-coordinate plane. If point C lies in the first quadrant and contains the coordinates (p, q), where p < 2 and q < 4, which of the following could be the area of triangle ABC? 1.1 3.9 11.9

-The lowest the area of the triangle regardless of the size can be 0+. -In this case the max can be 12 and the min can be 0+. Using that what's possible?

[FIGURE out which to change and use that to plug into ORIGINAL equation.] If x + |x| + y = 7 and x + |y| - y = 6 , then x + y = -1 1 3 5 13

-The mistake I kept making was combining different scenarios, which is a nightmare -Let's start with the second equation: 1) If y is positive than x=6, now let's plug 6 into the other original equation. 2) We get 6+6+y=7 which means that y is -5. Well this is a CONTRADICTION. This means that y MUST be NEGATIVE. 3) Now let's look at the first equation and make x negative. If that's the case then we get y=7. Well this is now a CONTRADICTION, since we determined that y HAS to be NEGATIVE. So... x must be positive. 4) Our equations now are: 2x+y=7 x-2y=6=x-y-y From this we get that x=4 and y=-1 and our answer is 3

[REMAINDERS -- The remainder you are looking for is itself one of the numbers that can be divided to get that remainder.] How many positive integers less than 100 have a remainder of 2 when divided by 13? 6 7 8 9 10

-The trick answer is 7 -The first number that should run through you HEAD is 2. This IS ONE OF THE numbers that when divided by 13 will give you a remainder of 2 The rest are: 13+2=15 26+2=28 39+2=31 52+2=54 65+2=67 78+2=80 91+2=93

[ABSOLUTE VALUES & Inequalities] 6abs[(-k/3)+4]>12

-To figure out the values of k, we need two equations but we need to reverse the sign AND the sign of the number -So... abs[(-k/3)+4]>2 AND abs[(-k/3)+4]<-2

[ANGLES on a LINE]

-Two angles on a straight line are always 180 in total -The lines don't even have to be parallel to draw conclusions about supplementary angles

[Identical variables in NUMERATOR & DENOMINATOR]

-We CANNOT eliminate them for example: [(x+12)(x+1)]/[(x+12)(x+3)] -Now you may be tempted to cross of (x+12) but that would mean that -12 could work as an answer choice, but clearly it DOES NOT

[Tables for testing integer properties] If x and y are integers, and w=(x^2)y+x+3y, which of the following statements must be true? If w is even, then x must be even. If x is odd, then w must be odd. If y is odd, then w must be odd. If w is odd, then y must be odd.

-We are essentially testing four scenarios: two stages composed of x and y either of which can be even or odd. -EE -EO -OE -OO -For even pick 0 -For odd pick 1

[How to be SMART with answer choices] x is a positive integer. When x is divided by 2, 4, 6 or 8, the remainder is 1. Column A x Column B 24

-We are trying to find values of x such that the remained is 0 -We know that x could be 1 -Now we need to see if x could be another value such that is it BIGGER than 24. Because this would make the answer D -NOTICE that I don't care about any other values LESS THAN 24 since it doesn't give me anything new -So, let's test 25 and we find out that it WORKS -I got this wrong because i gave up. I should've gone through multiples of 8 up till at least 24 -Notice that any LCM of all the numbers +1 will give you the answer too

[Go for simplest solution. Notice that we are already give the MODE of 16. We can't have another number is a Mode too.] List S = {16, 9, 23, X, 13, 16} In list S, the mean and median and mode are all equal to one another. What is the value of X?

-We are trying to find x and we know the mean to be 16 and we know there are 6 numbers. So solving for x we get 19

[DO NOT be stuck with the preconception that your x's HAVE TO BE POSITIVE] A right triangle has legs of 6 and x, and a hypotenuse of r. If 5r = 5x + 9, what is the value of r + x?

-We are working with two equations: x^2 + 6^2 = r^2 (r-x)=9/5 -We know that with right triangles the quadratic equations always come in handy -I was tempted to have my x's be positive and didn't want to put( r^2 - x^2) = (r+x)(r-x) -DO NOT MAKE this mistake -ALSO notice that 36 CANNOT be negative since it is a side of a triangle and object

[MAKING up things for the RATIOS] Jack has 5 cats and 1 dog. If the dog's weight is 3 times the average (arithmetic mean) weight of the cats, then the dog's weight is what fraction of the total weight of all 6 animals? 1/4 1/3 3/8 3/7 3/5

-We can just say that each cat weighs xlbs so the average is xlbs (Total is 5x) -The dog is 3x lbs -The ratio must be 3x/8x

[How CONVERSIONS work. Keep TRACK of what the OUTCOME is.] A computer can perform c calculations in s seconds. How many minutes will it take the computer to perform k calculations?

-We have c/s and need to convert is to c/minutes c/s=x/60 => 60c/s = Calculations per minute -So... 60c/s=k/x where x is the number of minutes it takes to perform k calculations. This gets us to ks/60c

[STAGES DETERMINE how MANY options we have.] From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

-We have to account for the OO scenario: OO__, The blank is any one of 5 letters now and the OOs can be arranges 3c2 ways = (3)(2)/2=3. So we have (3)(5)=15 options here -The other scenario is with MAGOSH: How many stages? 3. Does order matter? Yes. __,__,__=(6)(5)(4)=120 -Our answer is 120+15=135 -The MISTAKE that I made was thinking we had 6 stages and instead did 6! Which is WRONG since we only have 3 stages

[KNOW what points are EQUAL to each other on the COORDINATE PLANE.] [USE ALL information given.] The points A(0, 0), B(0, 4a - 5) and C(2a + 1, 2a + 6) form a triangle. If angle ABC = 90, what is the area of triangle ABC? 102 120 132 144 156

-We know that one point is the origin, we know that point C is in the I quadrant, that means point B MUST be on the x-axis and on the positive part since the angle is 90 degrees -Now we have variable and we need their values. We need to set equations equal to each other when we have multiple variables -Knowing the 90 degrees configuration we know that the y-points for points B and C are the SAME so... 4a-5=2a+6 => 2a = 11 => 11/2 = a -We can use this to solve this problem now

[QUADRILATERAL RULE] Given that the length of each side of a quadrilateral is a distinct integer and that the longest side is not greater than 7, how many different possible combinations of side lengths are there? 21 24 32 34 35

-We know that that lengths can be 1,2,3,4,5,6,7, so that's 7c4 to get 35 -BUT... we can't have the following combinations: 1,2,3,7 1,2,3,6 1,2,4,7 Since three of the first side added together is NOT greater than the last side. Because of this we can't have a quadrilateral . So our answer is 35-3=32

[RATIOS can be solved by putting TWO VARIABLES on OPPOSITE sides of the equal sign.] Choose the option that best answers the question. In a group of children, the average (arithmetic mean) weight of the boys is 60 pounds, and the average weight of the girls is 48 pounds. If the average weight of all of the children in the group is 50 pounds, what is the ratio of the number of boys to the number of girls? 1/12 1/6 1/5 1/4 1/3

-We should get the following equation pretty easily: (60B+48G)/(B+G)=50 -This simplifies to: 10B=2G B/G=1/5

[HOW are the ANSWERS FORMATTED?] sqrt(0.00001) =

-We should notice that all of the answers are in fractions. -This is a HUGE giveaway that we need to format this into a fraction -So... sqrt(.00001)=sqrt(1/100,000) BECAUSE there are 5 spots to the right of the decimal including the 1 so there MUST be 5 0's in the denominator -This becomes 1/sqrt[(100)(100)(10)]=1/[100sqrt(10)] -We can rationalize this by dividing the top and bottom by sqrt(10) to get: sqrt(10)/1000

[PERCENT CHANGE When the Whole also Changes] The Bakery makes $200k per year and 6% of that is baked goods. If the backed goods triple but everything else stays the same, then what is the baked goods going to be as a percentage of the entire revenue?

-Well 6% is 12K. This is going to triple to be 36K in baked goods in total. -The whole is 200K and part of it has INCREASED BY 36-12=24K. So the TOTAL revenue is 224k -Then 36K/224K = 16.1%

[How to estimate using NORMAL interest.] Cindy invests $10000 in an account that pays an annual rate of 3.96%, compounding semi-annually. Approximately how much does she have in her account after two years? $10079.44 $10815.83 $12652.61 $14232.14 $20598.11 ------ [ANOTHER Example] Sarah invested $38,700 in an account that paid 6.2% annual interest, compounding monthly. She left the money in this account, collecting interest for three full years. Approximately how much interest did she earn in the last month of this period? $239.47 $714.73 $2793.80 $7,888.83 $15,529.61

-Well let's first round up to 4%. Half of this is 2% -Since this is semi-annually we are going to get roughly 4(2%)=8% -So 8% of 10,000 = $800 -The answer has be to B ------- -Notice we are only asked for the interest in the very last month! -Well let's estimate 40,000(6/12)=40,000(0.5%)= $200 -Our answer can only be slightly bigger than this, so we have to go with A

[Make sense of what the question is asking.] A weighted coin has a probability p of showing heads. If successive flips are independent, and the probability of getting at least one head in two flips is greater than 0.5, then what could p be? 0.1 0.2 0.3 0.4 0.6 0.7

-What could p be? -Well we know that p is the probability of getting "heads" and it is asking for the probability "AT LEAST ONCE" -The formula is p(at least once) = 1 - p(not even once) -p(not even once) = (1-p)(1-p) -Since we are given the p's we need to be able to plug them in to get values that are greater than .5 -So 0.1 and 0.2 will give us .9 and .8 respectively and squaring them will be .81 and .64, which subtracted from 1 will be a value less than .5 -This means the rest of the numbers work

["AT LEAST" = TWO OPTIONS] A weighted die, numbered one through six, has a probability of 1/4 of rolling a six. If this die is rolled three times, and each roll is independent, what is the probability of rolling at least two sixes?

-When we get an "AT LEAST" question we have two options: 1) With an "at least one" we can use the complement rule 2) If we have an "at least" of MORE than 1, then we have to take into account MULTIPLE SCENARIOS where we meet the "at least" AND the scenario where we have MORE THAN THE at least -Here I made the mistake of only doing the first one Scenario one "at least two": (3c2)(.25^2)(.75^1)= 9/64 Scenario two "all of them are 6's" and we still meet our requirement: (1/4)(1/4)(1/4)=1/64 (9/64)+(1/64)=10/64=5/32

[Geometry - If there are multiple ways of equations set to 180, the simplest one is the one you need.]

-You can assign one variable and get two equations set to 180 -Or you can assign two variables -Clearly the first is easier to deal with

[Difficult BUT necessary way to evaluate this problem.] In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?

1) How many stages are there? 4 Main stages (each of which is divided into further 4 stages). 2) Does order matter? No, this is a combination problem. -There are 16 different presents and each can be divided into four sets. Let's start with the first set of four: -We have 16 presents to pick from for a total of 4 to get 16c4 -Then for our next stage we have 12c4 -Then 8c4 -Then 4c4 -Which equals 16!/(4!^4)

[NO REASON TO MISS THESE.] Options: 1) Multiply both equations as if by reciprocals 2) Multiply one to eliminate one 3) Look at what questions is asking. Like this one. If 3 apples and 4 bananas costs $1.37, and 5 apples and 7 bananas costs $2.36, what is the total cost of 1 apple and 1 banana?

3A+4B=1.37 5A+7B=2.36 -Notice that we want to 1-to-1. To get this one of our equation's variables must be 1 greater. Well this is EASY! -Multiple the top by 2 and then subtract

[DO NOT be confused. If they WANT us to deal with NOW numbers they will use the word "NOW". Otherwise solve the NORMAL way.] Ten years ago, Josh was three times as old as Tim. In five years, Josh will be 10 years more than twice as old as Tim. How old is Tim right now?

Now: J & T -10: J-10 * T-10 so J-10 = 3(T-10) +5: J+5 = 2(T+5) + 10 -This requires a careful answer, so solve with diligence and get 35

[When going for percents, we go for the RETAIL price in the denominator.] If the retail price of a shirt is R dollars, and the price including sales tax is T dollars then the sales tax, as a percent, is

[(T-R)/R]100

[The GOAL of every GEOMETRICAL equality is to set things equal to the same thing. As long as we keep the variables in the equation NO MATTER what the equation we should get the right answer.] If AC = BC and CD = DE then, in terms of x, the value of y is GRE linesandtriangles_figure

b=180-2x b=(180-y)/2