Magoosh - Quantitative Section [Hard], Quantitative Section

*Learn to recognize patterns from two distinct patterns!* *For hard you need to make connections between the information given. Do not go off on a tangent. Try to connect two pieces of information that you've solved to the max.* (algebra + geometry/triangles) If a right triangle has area 28 and hypotenuse 12, what is its perimeter?

ab=56 a^2+b^2=144 so... if we do 2(ab)=112 we get the square of sums!

*Remember that we cannot start with 0 as the even since 0 is neither positive nor negative.* X = sum of the first 31 positive odd integers Y = sum of the first 30 positive even integers Column A X - Y Column B 30

-Check out saved image

*Remember your rules. You CANNOT assume that the numbers are arranged already. You have to use the information given to figure our the order first.* List A: {x, x, x, y, y, y, 3x+y, x-y } If the median of list A is 10 and 0 < x < y, what is the range of list A?

-Given that 0<x<y how can we arrange our variable first?

*You can factor out something and then the sum of the inside of the parenthesis is something easier to deal with.* K = sum of the integers from 1 to 500 inclusive that are divisible by 5.

-Here a five can be factored out... Check it out to see a different approach

Check out saved image 94 If 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 = 2^x, then x =

-Here it's easier to deal with this in the format that If we divide something with something else the exponents subtract top-bottom -The answer should be -3

*Remember if not told explicitly, all of these integers could be the same exact number.* If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?

-The answer here is 5

*Remember with quadratics you cannot cancel out variable unless you know what the variable is.* If sqrt(sqrt(sqrt(3x)))) = (2x)^1/4, what is the greatest possible value of x?

3x = 4x^2 -> 4x^2 - 3x = 0 factor and set equal to 0 So we have 2 solutions: 0 and .75

*Another interesting coordinate plane problem.* (coordinate + fraction comparison) Which of the following lines intersects the vertical line x = 3 between (3, 1) and (3, 2)?

Figure out the limitations and then approach

*Look out for shortcuts when available.* If (8-x)/(x+1)= x, then x2 + 2x - 3 =

Here we get 8 = x^2 + 2x -If we subtract -3 from both sides we get 5

*Remember the Mississippi rule.* A librarian has three identical copies of a single cookbook and four different novels that he wants to display. Assuming all seven books will be in a single row, how many different arrangements can he make?

Here we have (7!)/(3!)

*Another example where assigning numbers for variables gets you a quicker answer.* 2x + y years ago, Roberto was 3x + y years old. How many years old was Roberto x years ago? x 4x + 2y 5x + 2y 6x + 2y 6x + y

I used x=2 and y=3 and came up with the number 14 for Robert's age x years ago -This gives us B as our answer

*We could set this equal to the 180 equations. But in this case we can solve for x and y and see what works.* Check out saved image 45 Column A x Column B y

x=180-w+1 y=180-w-1 now compare

*Remember to write out the equations neatly.* *If we do not know the exact number and let a variable equal to the actual number then we have to add or subtract from the variables.* Every person in a certain group is either a Dodgers fan or a Yankees fan, but not both. The ratio of Yankees fans to Dodgers fans is 5 to 3. If 22 Yankees fans change teams to become Dodgers fans, the ratio of Dodgers fans to Yankees fans will be 1 to 1. How many people are in the group?

y/d=5/3 y-22/d+22=1/1

*Remember that average times the number of numbers is total.* The average (arithmetic mean) of y numbers is x. If 30 is added to the set of numbers, then the average will be x - 5. What is the value of y in terms of x ?

xy is total so... xy+30/y+1 = x-5

*For something as small as this, write out neatly all of the possibilities.* Set K consists of all fractions of the form x/(x+2) where x is a positive even integer less than 20. What is the product of all the fractions in Set K ?

(2/4)(4/6)(6/8)...(18/20)

*Divide into stages of which there are 2. Next realize that this is a combination problem since order does not matter.* From a total of 5 boys and 4 girls, how many 4-person committees can be selected if the committee must have exactly 2 boys and 2 girls?

(5C2)(4C2)=60

Check out saved image 62 Column A AB + AD Column B DC + BC

*Follow your intuition. Remember that angles do not dictate how long or short the sides of a triangle are! They can only tell us which side is shorter or bigger but the shape could be changed dramatically and the angles can still hold.*

*This is a counting question of possibilities.* A knockoff website requires users to create a password using letters from the word MAGOSH. If each password must have at least 4 letters and no repeated letters are allowed, how many different passwords are possible?

-At the least we must have a scenario with 4 letters: 4! -Another scenario that we must add to this is the 5 letter password: 5! -Lastly, we need to consider the 6 letter scenario: 6! S1: (6)(5)(4)(3)=360 S2: (6)(5)(4)(3)(2)=720 S3: (6)(5)(4)(3)(2)(1)=720 The sum is 1800

*You can group multiplication to make sense of things.* (p)(q)(r)<0 (pq)^2/r<0 Column A pq Column B 0

-Based off the second equation we know that r must be negative -From the first equation we can make two groups (pq)(r)<0. If we know that r is negative that means that p*q MUST be positive. And any positive is greater than 0. So A is the correct answer

*Use triangles for this!* In the standard x,y plane, a circle has a radius 6 and center (7, 3). The circle intersects the x-axis at (a, 0) and (b, 0). What is the value of a + b?

-But notice the shortcut, since the two triangles are congruent it will be 7+x+7-x, where x is the length of the bottom leg. This just equals 14

*Percentiles go left to right and essentially give a percentile mark to each person.* Check out saved image 80 Suppose all students surveyed answered in integer number of hours only. Suppose, of 82 surveyed, only one respondent answered "16 hours." Within this group, the approximate percentile of this person would be: 32nd percentile 51st percentile 67th percentile 75th percentile 80th percentile

-Here we want the person that is at 16 which means the first three columns added together divided by 82 will give us our percentile because that person will be above all the first three columns 65/82 ~ 64/80 ~ 8/10 ~ 80%

*Careful with these questions. You need to figure out if this is proportional or inversely proportion.* Working together, 7 identical pumps can empty a pool in 6 hours. How many hours will it take 4 pumps to empty the same pool?

-If it take 7pumps 6 hrs then it must take 1 pump 42 hours

*This is a division problem. First eliminate the decimal because this number is misleading saying that we can have 5^10 hidden in there but that is not possible since this number multiplied out is not a whole number.* *When trying to figure out if a number can be divided, write out the complete prime factorization.* Which of the following are divisors of 1.2x10^10?

-Notice that when you factor this out and group everything you get: (2^11)(3)(5^9) Now you can clearly see what numbers this number can be divided by

*Once you reach a point where you can't move further. Plug numbers to see what makes sense. We are comparing 8 so plug than into the matrix to see if that works or not.* In a group of 45 children, 60 percent of the children are boys, and 60 percent of the children are left-handed. Column A Number of boys who are left-handed Column B 8

-Plug and chug when you can't move forward with anything else

*Notice the small things like this. We can simply flip number of people to number of televisions to get what we want.* In 1955, the ratio of the number of televisions to the number of people was approximately A) 1 to 13 B) 1 to 23 C) 1 to 26 D) 1 to 50 E) 1 to 90

-Population per television is 13 (or 13:1) -televisions to the number of people then is (1:13)

*When dealing with percent increases we can directly deal with the multiplier first. The answer will be in 1._,_,_ format from which we can easily see what the percent increase of decrease is.* Suppose China's production remains more or less constant from 2004 to 2006. Suppose India is able to sustain the same percent increase in both of those years. By what percent would India's rice production have to increase from 2004 to 2005 and again from 2005 to 2006 so that it equaled China in rice production in 2006? A) 10.0% B) 12.8% C) 16.3% D) 21.5% E) 27.4% India: 88 China: 112

-So here it would be 88(r)(r)=112

*This may look like a recursive sequence but we have to think of it in terms of arithmetic sequence.* The nth term (tn) of a certain sequence is defined as tn = tn-1 + 4. If t1= -7 then t71 =

-Solve this using arithmetic sequence and recognize that

*The largest integer value that a remainder can be has to be one less than the divisor. Otherwise the divisor would go into the dividend.* n is a positive integer. n is not divisible by 4. n is not divisible by 5. Column A The remainder when n is divided by 4 Column B The remainder when n is divided by 5

-Test multiple scenarios values less than both the divisors and value above

*Be able to spot the meaning of parallel lines. Here that is key to see that parallel lines allow to equate two angles to be the same.* Check out saved image 93 In the figure below, ∠ADE = 60°, ∠EFC = 40°, and ∠DAE = 55°. If AB || CD, what is the value of x?

-The angle at the top right is equal to the top angle of the bottom right triangle

*Become confident in dealing with equalities and converting things to decimals to make the easier to deal with. Also be very careful in dealing with additions or subtractions.* *Something which is flatly applied to the individual parts can be applied to the whole.* The sum of the pre-tax costs of Item A and Item B is $300. In Alumba, each item would be charged a flat 7%. In Aplandia, Item A is subject to 5% tax and Item B is subject to 10% tax. If the tax in Aplandia on the purchase of both items is exactly $3 more than it is in Alumba, then what is the pre-tax price of Item A?

-Write out the equations carefully

*Do not forget how the line looks like. In this case it is downward sloping so the answer will be negative!* In the xy-coordinate system, line k has y-intercept 12 and an x-intercept greater than zero. If the area of the triangular region enclosed by line k and the two axes is 30, what is the slope of line k?

Answer: -12/5

*GRE question tip.*

Be careful about questions asking about the "difference" of something. For ex: "how much shorter would it be to go from x to y directly rather than x to z to y?" -This was a triangle question and the trick answer was 5 (x to z is 4; z to y is 4) -The right answer is 7-5=2

*Prime factorization. If k is a multiple of 1440 then it must be divisible by 1440 and for that to happen, all the prime factors of 1440 must be available in 1 to n.* n is a positive integer, and k is the product of all integers from 1 to n inclusive. If k is a multiple of 1440, then the smallest possible value of n is

First, take the prime factorization of 1440 ----- 1440 = 144 * 10 = 12 * 12 * 10 = (2 * 2 * 3) * (2 * 2 * 3) * (2 * 5) 1440 = 2 * 2 * 2 * 2 * 2 * 3 * 3 * 5 1440 has five factors of 2, two factors of 3, and one factor of 5. Any number has to have at least that many of each one of those factors in order for it to be divisible by 1440. Well, we see a 5, so we know n has to be at least as big as 5 5! = 5 * 4 * 3 * 2 * 1 = 5 * (2 * 2) * 3 * 2 So far, that's three factors of 2 and one factor of 3 --- neither of those is enough 6! = 6 * 5! = (2 * 3) * 5 * (2 * 2) * 3 * 2 Now, we have the two factors of 3, which is enough, but we have only four factors of 2, not enough yet. The next factor, 7, doesn't help us at all. Let's go up to 8 ---- 8! = 8 * 7 * 6! = (2 * 2 * 2) * 7 * (2 * 3) * 5 * (2 * 2) * 3 * 2 Now, we have enough factors to cover all the prime factors in 1440. Therefore the smallest value of n is 8.

*We can either work with our own numbers of equations.* Captown is the capital city of Maltania. If the population of Captown is 25 percent of the rest of the population of Maltania, then the population of Captown is what percent of the entire population of Maltania?

If P=100 then (100-C)(1/4)=C. What is C/P? -Another approach is C/(M-C)=25/100 and we need C/M

Check out saved image 77 *On the GRE when we see one of the algebraic equations separate it if it is in complex fraction form and a solution will emerge.* If x + y ≠ 0, which of the following is a solution to the inequality x=3 and y =7 x=-3 and y =7 x=-11 and y =-9 x=9 and y =-6 x=-20 and y =-24 x=12 and y =9 x=-2 and y =16

In this case we can separate to get [(x^2-y^2)/(x+y)]-[1/(x+y)]>-1/(x+y) -Here our -1/(x+y) get canceled out and we are left with [(x^2-y^2)/(x+y)]>0 and this simplifies to x-y>0 which equals x>y -This should allow us to see the solutions easily

*Percent trick. We don't have to deal with the number at all just the percent.* At the moment there are 54,210 tagged birds in a certain wildlife refuge. If exactly 20 percent of all birds in the refuge are tagged, what percent of the untagged birds must be tagged so that half of all birds in the refuge are tagged?

Let's work with an easy number of 10 2 of these are tagged and 8 aren't we need to have 5 tagged well 3/8 is 37.5% of the untagged birds

*Know how to set up the equations. A good place to start may be by asking how many unknowns are there?* -In this care there are two unknowns SD and M and they are used to set up and equal 12.5 and 8.9. *Notice that SD is just a given number. Is SD is 2 and we need to find three SDs above the mean. Then that is (2)(3)=6 in actual movement in score. Thus M+1 represents one standard deviation above the mean so that would be 2, M+2 = 4=2+2=2S, M+3=6=2+2+2=3S.* In a certain set of numbers, 12.5 is 1.5 units of standard deviation above the mean, and 8.9 is 0.5 units of standard deviation below the mean. What is the mean of the set?

M+1.5SD = 12.5 M-0.5SD = 8.9 -Now this should be pretty simple

*Best to use variables when dealing with comparisons and use arithmetic to narrow things down.* (advanced QC trick) Gerry is three times as old as Pat. Column A Gerry's age 20 years ago Column B Pat's age in 12 years

Now: G=3P 20yrs ago Gary must have been: 3P-20 12 yrs from now Pat will be: P+12

*Approximation: Meaning in this case we can use simple interest.* *Simple interest you just need the annual rate and don't even have to worry about dividing it in quarters because it's the equivalent of applying a base rate to everything and adding versus adding and then applying the base rate.* Diana invested $61,293 in an account with a fixed annual percent of interest, compounding quarterly. At the end of five full years, she had $76,662.25 in principal plus interest. Approximately what was the annual percent rate of interest for this account? 1.2% 4.5% 10% 18% 25.2%

(4.5)(400)(61,293)(4)(5) = ~75,000 ~60,000*~5%=3000/yr*5=15,000 60K+15K=75K

a^2b^2

(ab)^2 so... (a^-n) (b^-n) = (ab)^-n

*Keep in mind that even a variable divided by the same variable is 1.* If w/x = 2/3 and w/y = 8/15, then (x + y)/y =

(x+y)/y = (x/y)+(y/y) = (x/y) + 1 = ? We can easily use the relationships given to get what x/y must be

*Consecutive integers* x is a positive integer. k is the remainder when x3 - x is divided by 3. Column A k Column B 1

(x^3-x)=x(x^2-1)=(x-1)x(x+1) -There there are 2 consecutive integers one of their products must be divisible by 3 and if that is the case then their product must be divisible by 3 as well and so the remainder will always be 0 -I did this just by plugging in numbers to see what happened and missed it only because I accidentally picked A instead of B

So, we know the one for addition and subtraction, here is the one for multiplication: x>0 xy = 2 yz = 5 xz = 10

(xy)(yz)(xz) = (2)(5)(10) (x^2)(y^2)(z^2) = 100 xyz = 10

*Corner cases, 0s and 1s* Column A p/q Column (p/q)^2

*1) Always check corner cases (0, 1); and multiple options 2) just because we have two different variables p & q, doesn't mean that the number they represent has to be be different* Here it can be seen that any fraction lower than 1 will be lower in column 2. But if both p and q are equal then the two columns will equal. And if the fraction were greater than 1, then squaring it will lead to a larger number.*

Column A The average (arithmetic mean) of 3^2, 3^4, 3^6 Column B 3^4

*A number with an exponent represents ONE number. So the three numbers are divided by 3 not 12.*

If x is a positive integer and x+2 is divisible by 10, what is the remainder when x2+4x+9 is divided by 10?

*A quadratic can be broken down to help you solve. We cant split the 9.* *Another thing is to pay attention to what x is and use that to plug in and solve.*

Cam is 20 percent taller than Bea, and Bea is 20 percent taller than Ann. Column A Cam's height minus Bea's height. Column B Bea's height minus Ann's height.

*An easy way to solve this is to set up with variable and compare the variable* A=x, B=1.2x, C=1.2(1.2x) column A = 1.44x - 1.2x = .24x column B = 1.2x-x = .2x *Also when order is available, the order is preserved. So you don't have to check multiple options. But if you solve this way, you don't have to worry about checking multiple option.*

To create paint with a certain shade of gray, one must combine 2.016 liters of black paint with every one liter of white paint. Approximately how many liters of white paint must be combined with 350 liters of black paint to create the certain shade of gray?

*Be careful about which direction to go in approximations.* We can round 2.016 down to 2 350/2=175 so... 350/2.016 must be less then 175... *To avoid confusion, first set up the problem... 350/2.016=x. Now it's easier to see that dividing by 2+ will gives us something less than 175*

If x > 21/40 , then x could be which of the following?

*Break down fractions into components* 21/40 = 20/40 + 1/40 so... 1/2 + 1/40 19/36 would equal 18/36 + 1/36, well all we are comparing is 1/40 to 1/36 and we know that 1/36 is bigger so 19/36 would have to be bigger as well

2^n + 2^n + 2^n + 2^n = 4^(n + 3)

*Careful about how you group these together.* 4(2^n) = 4^(n+3) 2^2(2^n) = 4^(n+3) 2^(2+n) = 2^2(n+3) *You CANNOT add the 2^n's together because that would change the base.*

If k ≠ 0, k ≠ ±1, and (k^3 k k^4)^2/(k^n k) , then n =

*First combine all the like terms* (k^8)^2/(k^n+1)= k^14, now you can solve *Get the same base and use the fact the x^a = x^b, so a = b.*

Square ABCD has a side of 20. Circle O has a radius of r. If the circle has more area than does the square, then which of the following could be the value of r? Indicate all possible values of the radius. 8 9 10 11 12 13 14 15

*First solve what you know and write it down. Area of square is 400 and Area of a circle is 3.14(r^2). Start with C and start eliminating.*

The revenue generated by Company X is divided between Doug and Moira in a 6 to 5 ratio respectively. Column A Moira's share when the revenue generated by Company X is $15,700 Column B $7900

*Half and half* If it is split evenly then 7850, but Moria right now is receiving less than that so A has to be smaller so B is the right answer

In a group of 100 adults, 75 percent of the women are left-handed. If there are 12 right-handed women in the group, how many men are in the group?

*Here there are two people with two characteristics, left and right handed. We can use a double matrix method.*

Which of the following, when rounded to the nearest integer, are rounded to 3? 3.4 3.445 3.494 3.499 3.501 3.61

*Here we only look at the number directly to the right of the decimal point. Thus the first four are the right answers.*

*Must check for extraneous solutions!* *Notice for things that look really similar to the quadratic algebra equations.* What is the sum of all possible solutions of the equation |x + 4|^2 - 10|x + 4| = 24? -16 -14 -12 -8 -6

*I missed this because I did not check for extraneous solutions.* a=|x + 4| a^2-10a-24=0 (a-12)(a+2)=0 a=12 and a=-2 |x + 4|= 12 |x + 4|= -2 *Before we even solve these; plug in to see whether the values work.* -We see that 12 does work but -2 does not. So we only need to solve using 12 |x + 4|=+ or - 12 x=8 x=-16 -16+8=-8

*Pay attention to what the question is asking. What will be the greatest range?* Positive integers a, b, c, d and e are such that a < b < c < d < e. If the average (arithmetic mean) of the five numbers is 6 and d - b = 3, then what is the greatest possible range of the five numbers? 12 17 18 19 20

*I missed this because of a stupid reason* Well i figured out that the largest number must be 19 and the smallest 1 -The greatest range must then be 18

*Build a table with 0 as even and 1 as odd* *Always four scenarios with 2 numbers. EE;OO:EO;OE* 2 variables/stages with with 2 options (E or D): (2)(2)=4. Since there are only 4 possible scenarios then we must be able to evaluate each expression to be true or false. If x and y are integers, and w=(x^2)y+x+3^y, 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.

*I missed this* -Check out saved image 137 for the solution ABC are true

*ALWAYS, ALWAYS, ALWAYS, consider the case of the remainder.* *We should also consider the negative integers, in this case we don't need to.* 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

*I missed this* -Here there are on two possible values for x -Can can either be a multiple of 25 in which case A works OR -X can be 1 in which case B is true -That being the case the answer must be D

*Corner cases or 0s and 1s* If x is an odd negative integer and y is an even integer, which of the following statements must be true? I. (3x - 2y) is odd II. xy^2 is an even negative integer III. (y^2 - x) is an odd negative integer

*I missed this: Very Hard* -I is always ODD -II is a corner case where if y=0 then this equals 0 which is not a negative numbers -III is false since we can get a positive ODD Only I is true

*Corner cases where either variables or plugging in numbers = 0 or 1* If x and y are positive odd integers, then which of the following must also be an odd integer? I. xy+1 II. x(y+1) III. ((y+1)^x-1) + 1

*I missed this: Very Hard* -I is always ODD -O(E) = Even =Note ODD -The third is tricky! Usually this will be an ODD. But check for the corner case! If x=1 then this number will be EVEN So only I must be true

*I was on the right track with prime numbers. Only the squares of integers have an odd number of divisors. Here we only want to focus on the square of prime numbers which have a only 3 divisors.* How many integers from 1 to 900 inclusive have exactly 3 positive divisors? 10 14 15 29 30

*I missed this: Very Hard* 2,3,5,7,11,13,17,19,23,29 = 10 numbers whose squares have only 3 divisors meaning we only have 10 numbers from 1-99 that have 3 divisors 4,9,25,49,121,169,289,361,529,841 -Notice that we can't go above 29 since the square of 30 is 900 -Do not be confused by repetition: 2^2=4 is divisible by 1,2,2,4 for a total of 1,2,4 three divisors

A, B, and C are consecutive odd integers such that A < B < C. If A + B + C = 81, then A + C =

*I was right in that you have to set this up in terms of one variable. When the starting variable is not given in this case assume x is the odd and the next odd then would be x+2, and the next odd would be x+4.*

If k is an odd integer, which of the following must be an even integer?

*If you notice that logic is going to take some time. Use 1 for odd and 2 for even*

For positive numbers X,Y,Z we have Z/Y= 35/6 and Z/X=35/8. Column A X Column B Y

*It is a mistake to assume that Z = 35 and X = 8 and Y = 6* These are all ratios, so the best we can assume is that Z = 35k, X= 8k, and Y= 6k. And so we notice that 8k is always going to be bigger for any positive number than 6k.

84/400

*Notice that both of the numbers can be divided by 4* so.. 21/100 = 21%

If K is the least positive integer that is divisible by every integer from 1 to 8 inclusive, then K =

*Pay attention to the question. It is asking for the least positive integer. You can go through the multiplication and if all the ones work then the least is A.* Another way to solve is to figure out what the factors need to be to be divisible... (1)(2)(3)(2)(5)(7)(2) = 840. This is the lowest number that could be divisible by this number. Everything else is a multiple of it.

If xy ≠ 0 , and 75 percent of x equals 125 percent of y , which of the following is true?

*Realize that 75% is 3/4 and 125% is 5/4 and solve out that way.*

k is a positive number. If k is twice its reciprocal, and j is twice k, then jk =

*Solve for k immediately. Do not just leave it hanging.*

Column A 43 percent of 63 Column B 63 percent of 43

*Sometimes it is helpful to write out percents as 43/100 and 63/100.* So (43/100)*63 = (63/100)*43 through the associative property of multiplication.

The sum of 5 consecutive even integers is 0.

*The best way to solve this is to put it in terms of x* x+x+2+x+4+x+6+x+8=0 x=-4 so our numbers are -4, -2, 0, 2, 4

Column A 43 percent of 207 Column B 85

*To avoid calculator realize that 43% of 100 is 43, and so 43% of 200 must be 86. So 43% of 207 must be 86+. Thus A is the larger column.*

Nina has exactly enough money to purchase 6 widgets. If the cost of each widget were reduced by $1.25, then Nina would have exactly enough money to purchase 8 widgets. How much money does Nina have?

*To set this up you have to assign the right variables.* x=price of 1 widget n=amount of money nina has so... 6x = n is the price of each widget goes down by 1.25 and nina can still buy 8 then... 8(x-1.25) = n ... since both equations are equal to n 6x = 8(x-1.25)

*We don't know what sides are what and what angles are what. So, do not draw and triangle and label it and assume all triangles to be that way DEF could have different sides than DEF where DE from the first is different from DE of the second.* Two sides of triangle DEF are equal to 3. Which of the following, taken alone, would be sufficient in finding the area of triangle DEF? The ratio of DE to EF = 1 : √2 The sum of angles DEF and EFD is 135 degrees The sum of angles DEF and FDE is 90 degrees

*Very Hard - I missed this* With the information provided in [A], there are still two possible scenarios: DE: EF: DF = 1:√2:1 DE: EF: DF = 1:√2:√2 Notice that two of the sides are equal for each of the scenarios, since the question states two sides equal 3. Yet, the two different proportions above result in two different areas. Thus, [A] is not sufficient. With the information provided in [B], there are still two possible scenarios: 45 degrees, 45 degrees, and 90 degrees 67.5 degrees, 67.5 degrees, and 45 degrees [B] is not sufficient. For [C], DEF and FDE must be the two sides that equal Since the remaining angle, EFD, must equal 90. And you can't have two angles in a triangle that both equal 90 degrees. Therefore, given the information in [C], we have a 45:45:90 triangle with sides 3:3:3√2. Since, such a triangle has only one possible area, [C] is sufficient.

*If an integer is divisible by 2, the resulting number doesn't have to be be EVEN!* If n = 2×3×5×7×11×13×17, then which of the following statements must be true? I. n^2 is divisible by 600 II. n + 19 is divisible by 19 III. (n+4)/2 is even

*Very Hard, I missed this* -It's easy to see that will not be divisible by 600 since the prime factors of 600 do not cancel out -It's easy to see that n+19 will not be divisible by 19 since to be divisible by 19, n must be divisible by 19 and that is not the case -The last one is tricky (2×3×5×7×11×13×17+4) *is divisible by 2* but III is asking for the result of (2×3×5×7×11×13×17+4)/2 which is 255257, this is odd!

*This is a double matrix questions, but highlights how important is it to be able to read it correctly.* 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?

*Very Hard: I missed this* T ~T M 40 F 60 70 30 We should now recognize that based off the last piece of information, 90% or 90 professors fall in the FT,F~T, and MT. This means that our remaining box M~T will have 10 members we can use this to figure out the rest and get that 30MT 30/40=75%

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

*Very Hard: I missed this* -The first point can be placed anywhere -The probability depends solely on the second point which can be on the right or left of the point -The entire circle is 360; the space taken up by the three points (taking into account both scenarios) is equal to 120; this is the case since the radius is 2 and the chord length is 2 and the distance form the other point to the center is 2 So 240/360 is the answer = 2/3 -Check out Very Hard 151 for solution

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?

*We can create new equations, by subtracting from other equations.* *Multiplication is the route that GRE wants you to take usually.*

Cylindrical tank A has radius x and height y. Cylindrical tank B has radius y and height x. x = 2y Column A Volume of Tank A Column B Volume of Tank B

*When given something like x=2y, set everything in terms of y because then you don't have to deal with FRACTIONS.* You don't need to plug in numbers simply set up the same equations and solve

Ben is three times as old as Ron. Ed is 8 years younger than Ben. Ron is 7 years older than Ken. If the sum of the ages of all four people is 161, how many years old is Ron?

*When there are multiple items, set the problem in terms of one variable. Here it makes sense to do it according to Ken's age, since he is the base and his age isn't related to anything else.* K=ken's age K+7=Ron 3(K+7)=Ben 3(K+7)-8=Ed

Strange Operators trick

*You can simplify the operation first, and you can make this go even faster* Do this if it is possible. Usually works with algebraic equations common on the standardized tests.

Seven years ago Bob was k times as old as Ann. If Ann is now 11 years old, what is Bob's present age in terms of k? *Do not keep re-writing formulas. Think logically about what is happening when you plug one thing into another.*

*You cannot treat the past as now. 7 years ago was 7 years ago. Treating the past as now will lead to double the right answer or lead you to subtract years instead of adding them.* You can organize age in a double matrix format with the people being of the rows and two time periods on the columns.

*Great case of 0 (non-negative and non-positive integer). And 1 being a divisor of every single integer.* If k is a non-negative integer and 15^k is a divisor of 759,325 then 3^k - k^3 =

-15=3*5 since the number is not divisible by 3 the number cannot be divisible by any power of 15 -0 must = k

*When we see a sqrt, we the test only means the positive root*

-3 = sqrt(9) This says that -3 = +3 and that is not true. So that can't work.

*Absolute values can be just two values positive or negative.* If x + |x| + y = 7 and x + |y| - y = 6 , then x + y = -1 1 3 5 13

-Assume that x + |y| - y = 6, y is positive if that is the case then we get x=6 if we plug this into the other equation we get y=-5 but that's a contradiction! So y must be a NEGATIVE number -Assume that x + |x| + y = 7, x is negative and we get y=7. But this can't be right since we determined that y must be negative. So x MUST be POSITIVE -Using this info we get: 2x+y=7 x-2y=6 We solve and get that x=4 y=-1 4+-1=3

*This is very, very tricky! You must test corner cases of 0 & 1 and negative numbers.* Column A 22 percent of x Column B 2/9 of x

-B would be bigger if x is positive -If x were 0 then the two columns are the same So the answer is D

*Tip with square roots*

-Besides squaring all of these, you can also write everything as roots, double the outside and put a square root over it and multiply it out. For example 2sqrt(13)=sqrt(4*13)=sqrt(52)

If four numbers are randomly selected without replacement from set {1, 2, 3, 4}, what is the probability that the four numbers are selected in ascending order?

-Best way to do this is to think about how many possible ways are there to organize 1,2,3,4? Well 24. And only one of those is in ascending order so 1/24 -Another way to thing about this is (1/4)(1/3)(1/2)(1/1)

*Roots*

-Careful about how the operator is applied! Work from inside to outside -With roots higher than 1/2 work in fractions

*This is pretty easy to figure out using algebra, just need to pay attention to what is being asked.* Pump A can empty a pool in A minutes, and pump B can empty the same pool in B minutes. Pump A begins emptying the pool for 1 minute before pump B joins. Beginning from the time pump A starts, how many minutes will it take to empty the pool?

-Check out saved image 150 for solution, which is in variable form A(B+1)/(A+B)

*Areas of circle.* Check out saved image 119. A cow is tethered to the corner of a rectangular shed. If the length of the rope is 5, and the shed has length 4 and width 3, what is the maximum area that is accessible to the cow? (The cow cannot enter the shed).

-Check out solution but think in terms of circles first

*Mathematical mistake to avoid.* (p+q)/p = 10/7. What is p-q/q solve by realizing that p/p + q/p and break down the second equation too. p/q - q/q

-DO NOT separate numerator and denominators. The fraction is one number it cannot be broken down further! -Instead trying breaking up the fraction into smaller components and solve out. Of course you can use can use reciprocals of a number so a/b can be b/a (to represent reciprocals)

*GCF of 1 means that the two integers are the same or are prime numbers. So you have to check multiple cases.* If x and y are positive integers, and 1 is the greatest common divisor of x and y, what is the greatest common divisor of 2x and 3y? Cannot be determined 1 2 5 6

-Doing that should lead to A

Three friends are buying a gift for a friend. Declan contributes 4 dollars more than 1/4 the cost of the gift, Ed contributes 1 dollar less than 1/3 the cost of the gift, and Frank contributes the remaining 22 dollars. What is the cost of the gift?

-Good job on identifying what c was going to be *Notice that when you set this up: (c/4)+4 + (c/3)-1 + 22 = c *You do not have to combine the fractions with the whole numbers immediately. Think about making the process faster. Use the associative property of addition and add 1-3+22 to get 25 and then you are left with c/4+c/3+25. Much easier to deal with

*Realize that the only choices that don't work are the one's the make the denominator 0.* If , then which of the following are possible values of x? Check out saved image 127 Indicate all such values. -60 -12 -1 1 2 5

-Here both sides of the equation are equal except when...

Phil has three times as many DVDs as Sam as. Even after Phil gives Sam 6 DVDs, he still has 16 more DVDs than Sam has. What was the original number of DVDs that Phil had?

-Here make two equations but be careful about how to set up the second one... It is not like the age problem AND is a completely different equation than the first. 1) P = 3S 2) P - 6 = 16 + S + 6 S = 42

*Think about the whole number versus individual sums. Also "numbers" is not the same as "integers".* *Check out saved image 124* Column A p + q Column B 5

-Here the answer is D -Think about it like this let's say our *numbers* were 0.2/0.3=2/3; well in this case p+q is definitely less than 5 -On the other hand let's say our numbers were 2000/3000=2/3; Well 3000 is definitely bigger than 5

*How averages work.* Check out saved image 125 There are 10 employees in an office, not counting the office manager. The table shows how many employees have 0, 1, 2 or 3 pets. If the office manager also were included in the table, the average (arithmetic mean) number of pets per person would equal the median number of pets per person. How many pets does the office manager have? 3 4 5 6 7

-I solved this first by looking at the answer choices -The way averages here work is total animals/total employees -The middle numbers must be an integer by itself (given the answer choices) and the fact that we will have 11 people to consider -Use one of the answer choices beginning with 7 to get the mean and see if is equals that numbers as well -Only 6 works out

*Even number list so the median is the average of the 50 and 51st digit.* What is the median number of televisions per household? Check out saved image 128 Cannot be determined 1 2 2.5 3

-In this case both the 50 and 51st are 3

*Do not assume that x y and z have to be in ascending order.* 10^x + 10^y + 10^z = n, where x, y, and z are positive integers Which of the following could be the number of zeroes, to the left of the decimal point, contained in n? x + y y - z z

-It's easy to see why A and B work -The only way to test c is to do a descending order where x=3,y=2,z=1 -So all three work

*This is not tough but good to highlight that something at the SD 3.2 and 10.4 are inclusive so you would count those numbers if those numbers were part of the list.* If the mean of list A is 6.8 and the standard deviation is 3.6, then how many elements of list A are within 1 unit of standard deviation of the mean? A = {2, 9, 2, 6, 9, 10, 7, 4, 5, 14}

-Nothing to comment simply solve and count correctly

*Manipulation choices.* Which of the following equations is true for all positive values of x and y? Check out saved image 117

-Notice here everything is in equation form -One side of the equation must equal the other. How can you manipulate for make sure one of them is true?

73^2 + 74^2 =

-One approach is just to do it the normal way -However based on the answer choices realize that the last digit of 73^3 is 9 and the last digit of 74^2 is 6 and so the last digit of the two numbers is 5... If the answer choice has the last digit as a 5 then that is the correct answer

*These one's can be confusing so just start with one statement at the time from the top.* Anne pays 150 percent more for a wholesale widget than Bart pays. Anne's retail price per widget is 15 percent greater than the wholesale price she paid. Bart's retail price per widget is 185 percent greater than the wholesale price he paid. Column A Anne's retail price. Column B Bart's retail price.

-Pick numbers that satisfy the first statement and move on from there

*Number of places* How many integers between 1 and 10^21 are such that the sum of their digits is 2?

-Recognize that we have 22 spaces of which 21 are available since 1 takes the 22nd spot -Here the only way to get digit sum to 2 is if we just have 2 by itself or two 1's -21 places and we can pick 2 spots = 21C2 = 210 -21 places and we can pick 1 spot = 21= 21 -231

*Prime factorization is key* In the game of Dubblefud, red chips, blue chips and green chips are each worth 2, 4 and 5 points respectively. In a certain selection of chips, the product of the point values of the chips is 16,000. If the number of blue chips in this selection equals the number of green chips, how many red chips are in the selection? 1 2 3 4 5

-The prime factorization of 16,000 is: (2)(2)(2)(2)(2)(2)(2)(5)(5)(5) -We have Blue chips worth 4 so group the twos to get (2)(4)(4)(4)(5)(5)(5) -We can now see that we have 1 red chip -- *Below is the mistake that I made.* We need the product of the point value of each chip not the sum of the value of the total chips FAQ: According to the problem I have 16000 = (2 * red) (4 * blue) (5 * green), which is the same as 16000 = (y * 2) (x * 5) (x * 4). I can rewrite that as 400 = x^2 * y. So 400 can be written as (10 * 10 * 4) or (20 * 20 * 1). So the number of red chips could be either 4 or 1. Right? A: Unfortunately, no. The question states that "the product of the point values of the chips is 16,000." So we have to find the product of the point value of each chip individually—not the sum. Your equation is the equivalent of summing the point values within a given color and then multiplying, not multiplying the individual chip values. If there are x blue chips, that means 4^x, not 4x. For example, say there were three blue chips. They are worth 4 points. We want the product of these chips, so 4 * 4 * 4 = 4^3 = 64. Again, not adding three 4's together or 4 * 3 = 12. Thus, the expression: (4^x)(5^x)(2^y) = 16000 or (20^x)(2^y) = 16000. But that's a very complicated equation with which to wrestle. I actually think it's a mistake to introduce variables into this problem at all. In general, in prime factorizations problems, introducing variables is not helpful.

*Concept review: how percentiles work.* The following boxplot shows the 2012 season runs batted in (RBIs) of 280 American League (AL) batters (the top 280 batters in terms of number of plate appearances). *Check out saved image 118* Five-Number Summary for AL RBIs in 2012: Minimum = 0 First Quartile = 9 Median = 25 Third Quartile = 56 Maximum = 139

-There is a fast way to solve this by realize that the percentile represents the number of people so total times the percentile will give you the people at or between those percentiles -90-75=15% 15%(280)= 28+14=42

*Stages and independency* Hal has 4 girl friends and 5 boy friends. In how many different ways can Hal invite 2 girls and 2 boys to his birthday party?

-Think about this in two stages, which are independent so it will be a multiplication of the two (stage 1 = girls) (stage 2 = boys)

*Probability lesson: If on the first try Ann were supposed to pick a specific ball the probability would be 1/10 but since she can pick any her probability is 10/10.* A box contains 10 balls numbered from 1 to 10 inclusive. If Ann removes a ball at random and replaces it, and then Jane removes a ball at random, what is the probability that both women removed the same ball? 1/100 1/90 1/45 1/10 41/45

-This is an "and" scenario Ann: 10/10 Jane: 1/10 (1)(1/10)=1/10

*Check out saved image 122* What was the approximate population of Town X in 1945? 150 750 1500 3000 6000

-This isn't hard and I had the right approach but picked the wrong year to get the number of population per television B is the right answer

Eighteen students attained an average score of 70 on a test, and 12 students on the same test scored and average of 90. What is the average score for all 30 students on the test? A) 78 B) 80 C) 82 D) 85 E) Cannot be determined

-This problem although not hard can become tedious if not approached with a GRE mindset -Do not multiple out the numbers, just write out the equation and see if things simplify before attempting to do arithmetic Ex: [(70*18) + (90*12)] / 30 = can be simplified first by removing the zeros since 10 can be factored from the top. Also, both the multiplication numbers are multiples of 3 and can be simplified as well

The average length of 10 objects is 25 inches. If the average length of 2 of these objects is 20 inches, what is the average length of the remaining 8 objects? A) 22 1/2 B) 24 C) 26 1/4 D) 28 inches E) Cannot be determined

-This question is not hard but can be hard depending on the choices that the test makers can put down if you forget what you are looking for -We want the average of the 8 objects left so... [(20*2)+(8x)]/10 = 25 If you just get x and forget to to add the 8 then you will just get the sum of the remaining 8 scores not their averages

Strategy for Multiple Correct Answers

-Usually you have to go through and check each one to see which one is correct -Key is to identify an equation that you can quickly go through and plug each one into it to see if the answers work or not

*When numbers must be INTEGERS.* 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

-We are not given the number of F or M so we have to make them up -The only solutions that works is if we have 100 males and 100 females which gives us a total of 59 smokers so the answer is C

Number Property Rule:

-With fractions less than 1, let's say you have 3 of them, then there is absolutely no way to get a number greater than 3, since then the fractions would have to be greater than 1 -But it is possible to get a whole number just not greater than the number of fractions available - sum of value of fractions less than one < the number of fractions available

*"At least" means we have to check out cases in which we can have more digits. In this case since we are testing multiple scenarios that are mutually exclusive we have to add them.* A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible? 9! + 10! 2 x 10! 9! x 10! 19! 20!

0-9=10 digits S1: 9digits: (10)(9)(8)(7)(6)(5)(4)(3)(2) = 10! (this takes into account any combination and permutation of the digits!) S2: 10 digits: (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 10! So the answer is 2*10!

*How many times does each number appear in the list?* The sum of all the digits of the integers from 18 to 21 inclusive is 24 (1+8 + 1+9 + 2+0 + 2+1 = 24). What is the sum of all the digits of the integers from 0 to 99 inclusive? 450 810 900 1000 1100

0=20x 1=20x 2=2x so... our sum is: (0)+(20)+(40)+(60)+...(180)=900

In a certain town in Connecticut, the ratio of NY Yankees fans to NY Mets fans is 3:2, and the ratio of NY Mets fans to Boston Red Sox fans is 4:5. If there are 300 baseball fans in the town, each of whom is a fan of exactly one of those three teams, how many NY Mets fans are there in this town?

1) Always just like the given information as is, so you don't confound the information. Then proceed to answer the question 2) For ratios always write down in terms of n. 6n:4n:5n

Strategy: Rule for exponents

1) x^2 (or any other even number): We don't know if x is negative or positive 2) x^3 (or any other odd number): We know for a fact based on the outcome that x can only be positive or negative

*I solved this first by assuming 55 are the right answer. Another way would be to keep increasing the number of candies from 1 onwards till you get to the 10 child who gets the rest.* Joan has 100 candies to distribute among 10 children. If each child receives at least 1 candy and no two children receive the same number of candies, what is the maximum number of candies that a child can receive? 10 34 39 45 55

1+2+3+4+5+6+7+8+9+____ The sum is equal to [(9)(10)/2]=45 That means that the 10th child gets 55

*Division by the decimal or fraction*

1/decimal OR 1/fraction is going to be a BIGGER number than 1/whole number

*Great scenario where you can cancel off variables.* x and y are integers greater than 5. x is y percent of x2 Column A x Column B 10

10*10 are the only integers greater than 5 that give 100 so C is the answer

*Prime factorization if very important. I only went down to 21 and 10 and did not simplify more* How many positive integers less than 10,000 are such that the product of their digits is 210? 24 30 48 54 72

10,000 = 1,_,_,_,_ Notice that we can only deal with 4 spaces since we can't have anything bigger than 1 on the first space because that would violate our rule. That's why we don't consider the scenario of organizing 1,2,3,5,7 -The prime factorization of 210 is 2,3,5,7 S1: We can arrange these numbers as 4! S2: We can combine 2 and 3 for 6 (which is a one digit number) for a total of 3! S3: We have to account for 1,6,5,7 for a total of 4! so we get 24+6+24= 54

*Look out for problems in which the equations are equal to the same thing! Then you can set the part equal to each other.* If 100,000 microns = 1 decimeter, and 1,000,000,000 angstroms = 1 decimeter, how many angstroms equal 1 micron?

100,000m=1,000,000,000a

Check out the saved image 41 *Write down what equations you know.* *Big Rule: Something+x > x* *Try to use what is given first.*

180-a-b=x [top triangle] 180-c-d=y Well there is another way to get x... 180-b-c-d=x Set these equal to each other and you get that a=c+d Since we know that c+d=a and the column asks as to compare c+d vs a+b; it should be obvious that a+b is bigger -- Another way is: a+x=180 [x being in the bottom triangle] c+d+x=180

*Break into stages and see what happens.* In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts? 16^4 (4!)^4 16!/(4!)^4 16!/4! 4^16

1st: 16C4 2: 12C4 3: 8C4 4: 4C4 This equals to C

*Remember grouping things together and then multiply that by the ways that in groups can be arranged.* In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated? 24 36 72 144 288

2(6)(6)

*Tips for Roots*

2^n+2^n=4^n 2^3 = (2)(2)(2) = basically the root shows us that the base is 2 and we have 3 of them

*This is another ratio problem that we can cross-multiply to make things simpler.* If 3/4 of the number of women working at Company X is equal to 2/3 of the number of men, what fraction of the employees at Company X are women?

3/4W=2/3M -> 9W=8M -We want ratio of women to total. Well W/M would be 8/9 -In another words there would be 8 W for 17 (8+9) employees

*Total possible divided by 2* There are 10 people in a room. If each person shakes hands with exactly 3 other people, what is the total number of handshakes? 15 30 45 60 120

30 total possible but we are double counting so 30/2=15

*Your job should be to draw a connection between equations.* (algebra/triangles + geometry) 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?

36=r^2-x^2 5r-5x=9

*Draw out the figure! In this case the figure is drawn with the 90 degrees being formed at the y-axis. This means that the y-coordinates are equation to each other for points B and C!* *Here it doesn't even matter what quadrant the triangle is in. We know that B is 90 degrees so just pick the first quadrant and work with that since any triangle would be congruent.* The points A(0, 0), B(0, 4a - 5) and C(2a + 1, 2a + 6) form a triangle. If ABC=90degrees , what is the area of triangle ABC? 102 120 132 144 156

4a-5=2a+6 a=11/2=5.5 Plug this in to get the coordinates and solve for area to equal 102

*Realize that in questions like this, the weights of the cats could be anything. They could all weight the same!* 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?

5x/5=x Dog = 3x 3x/8x=3/8

*Be careful about how you organize this. Our first multiple of 5 is 85 NOT 80!* How many multiples of 5 are there between 81 and 358?

85/5= 17 355/5=71 [71-17]+1=55 -I started at 80 and got the answer to be 56 which is wrong since it includes 80 as a multiple but that's not part of our range

*Power rule.* If 8^n+1 + 8^n = 36, then n =

8^(n+1) = 8^n(8)

*Multiplier problem.* 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+R3) A(3R+3R2+R3) 3A(R)3 3A(R+R2+R3)

A 1st: A(1+R) 2nd: (A+AR)(1+R)=A+AR+AR+AR^2 (basically multiply everything by 1 and add it to everything multiplied by R) 3rd: (A+AR+AR+AR^2)(1+R)=A+AR+AR+AR^2+(AR+AR^2+AR^2+AR^3) C is the answer -Another way is A(1+R)^3

A is what percent greater than B. What does this translate to?

A = [(x/100)B] + B

*The change multiplied by 3 is equal to A. Pay attention to the key word "then".* Appleton's population is 400 greater than Berryville's population. If Berryville's population were reduced by 900 people, *then* Appleton's population would be 3 times as large as Berryville's population. What is Berryville's current population?

A=B+400 A=3(B-900)

*It is better to first just assign variables and the determine the ratio. Be careful about how you read these.* Sue planted 4 times as many apple seeds as she planted orange seeds. 15 percent of the apple seeds grew into trees, and 10 percent of the orange seeds grew into trees. If a total of 420 apple trees and orange trees grew from the seeds, how many orange seeds did Sue plant?

A=apple seeds O=orange seeds 4O=A

*Strategy: Geometry*

Always label everything you want to find with your own variables and then try to make an equation with what you know

*Do the formula that you come up with. Don't just sit there. Mess around with the formulas you do know.* 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

B/x=60 G/y=48 (B+G)/(x+y)=50 => (60x+48y)=50x+50y => 10x=2y => y=5x B:G = x:y => x:5x => 1:5

*Key: Units Comparisons* Machine A produces 11 nails in 14 secs Machine B produces 7 nails in 9 secs Quantity A: Time is takes machine A to produce a nail Quantity B: Time is takes machine B to produce a nail

Careful about how you set this up. We don't want nails by second but "seconds per nail"... so.. 14secs/11 nails ?? 9secs/7nails Now we can cross multiply to get the right answer "B" Otherwise is we had done 11/14 ?? 7/9 we would've have gotten A since machine A can produce more nails per second since it is faster. But this questions is asking about time and B takes more time!

*Be mindful of what you are labeling.* If ABCD is a square with area 625, and CEFD is a rhombus with area 500, then the area of the shaded region is 125 175 200 250 275

Check out saved image 168 for the figure

*If the exponents are different but the exponents are the same then we can do the same things to the base as with any other number.* Check out saved image 104 If [image], then k = 8 12 16 24 36

Check out the saved image solution

Strategy: When comparing fractions

Cross multiplication is usually the best way to go

A man rode a bicycle a straight distance at a speed of 10mph and came back the same distance at 20mph. What was the man's total number of miles for the trip back and forth, if his traveling time was 1hr?

D1 = D2 D1 = 10mph * T1 D2 = 20mph * T2 T1+T2 = 1 So be pretty easy to solve now. When you are given information regarding time see if it equals a certain number.

If y+2q=15, q+2p=5, and p+2y=7, then find the value of p+q+y

Do not approach this by solving for each variable, they want what p+q+y is so solve that way y + 2q + _ = 15 _ + q + 2p = 5 2y+ _ + p = 7 So, 27 = 3y+3q+3p The answer is 9

If 0 < x < 1, then which of the following must be true? I) 2x < 2 II) x-1 < 0 III) x^2 < x

Do this by splitting the original into two parts

Given that a/b is less than 1, a>0, b>0. Which of the following must be greater than 1?

Do this by taking the statement a/b < 1 and manipulating is around

a>1 Quantity A: (a-3) (2a+1) Quantity B: (a-3) (a+2)

Here is one of the fundamental rules of comparisons: The equations must hold in EVERY SINGLE SCENARIO -So if A is bigger with one set of numbers, but is smaller with another set of numbers then the relationship is indeterminable

sqrt(28) + sqrt(175) = a*sqrt(b) What is a+b?

Here to add radicals you have to get a common radical and we get 2sqrt(7) + 5sqrt(7) = 7sqrt(7) Now... Do not be confused a is obviously 7 b is also 7 NOT sqrt(7). sqrt is just an operator of sorts, the number inside is b. *Variables can only be numbers not the operators on them. They can be positive or negative though.*

4/250 = ?

Here we are multiplying by 4/4 not to cancel but to make the denominator easier to deal with... (4 * 4) / (250 * 4) = 16 / 1000

*Combinations in stages* Car X can come with any of these 5 additional features: sunroof, stereo, tinted windows, leather seats and cruise control.

Here we have five stages where we can decide to either have an option or not so the combinations then become (2)(2)(2)(2)(2)=32 Can this be solved using combinations? Yes, though it's a little more complicated. We have 5 options, so we need to choose 0, 1, 2, 3, 4, and 5 options and add them together for our total possibilities. That gives us: 5C0+5C1+5C2+5C3+5C4+5C5 1+5+10+10+5+1 32

Strategy: Dash Method for Counting

How many number are there below 20,000 which have prime numbers as each of their digits? Answer is 340; figure it out

If (p+1)/(r+1) = 1 then...

I just want to highlight here that you can keep going when a variable is equal to another.. p+1 = r+1 equals... p = r ... You can keep going p/r = r/r = 1

*Triangle inside a Square*

If the triangle's base is the entire side, then the ratio of area of triangle to area of rectangle will always be 1:2

Geometry Rule:

If two opposite sides of a quadrilateral are 90-degrees then that means that the other two angles must form 180-degrees and can be bisected so that two sides of that shape will be equal and the other two will be equal (try to draw a 90-140-90-40; it's kind of like a weird rhombus-like figure)

*When things are equal to each other, write out in terms of one variable.*

In 2003, the number of girls attending Jefferson High School was equal to the number of boys. In 2004, the population of girls and the population of boys both increased by 20 percent. Which of the following could be the total student population at Jefferson High School in 2004? G=B so G=G so total equals 2G. In 2004 1.2(2G) = total this is equal to 12(x/5) so we need a number divisible by 3 and 4

*How to approximate* 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

Let's round the deposit up to $40,000, and the percentage down to 6% annual. Compounding monthly means each month, Sarah will accrue 6/12 = 0.5% in interest. Well, 1% of $40,000 is $400. Divide by 2: then 0.5% of $40,000 would be $200. That would be the simple interest amount, as well as the interest in the first month. We expect the amount in the last month to be a little more than this, but certain not even as large as double this amount. The only possible answer is (A).

*Mean and Median are equal if the list is spread out evenly OR if the deviation from the mean is equal.*

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? the deviation that would make this work is +3 so 19 has to be the answer.

*Use prime factorization to your advantage. If you must check answers, start with E first.* If k is an integer, what is the smallest possible value of k such that 1040k is the square of an integer?

My first intuition was that squares of a number have an odd amount of total factors because all their individual prime factors must be symmetrical and even. This approach leads to (10)(104) = 2,5,2,2,2,13,k we need to group these evenly so all factors are symmetrical (2,2,5,13)(2,2,k) so... k must be 65

2k years ago Frank was 3k years old. In k years Frank's age, in years, will be 4k 5k 6k 7k 8k

Notice that Right now the age is F F-2k=3k so F=5k In k more years the age will be 5k+k=6k

*Concept knowledge about medians.* The median of x , y , 8 and 11 is 19. Column A x Column B 23

Notice that our median is 19 in which case we know that the average of the two middle numbers must be 19 *That means that two of our numbers are bigger than 19 and two of our numbers are smaller than 19* -Knowing this we can place 8 and 11 to the left and know have to determine the 3rd place number which has to be 27 which is bigger than 23

*Exponent rule.* If 4^n + 4^n + 4^n + 4^n = 4^16 , then n =

Notice that the left side simplifies to 4(4^n) = 4^16 -We can divided both sides. To do this correctly with exponents realize that (4^16)/(4^1) = 4^15

Strategy: When equation/inequality is set up

Once an equation or inequality is set up or given then first try to figure out how to manipulate (add or subtract or multiple or divide both sides by a variable or number) to see if you can get it to equal what you need. This should be especially obvious if the question is asking for a formula like answer rather than a straightforward number

*Area vs. size limitation. Area can range from 0+ to the maximum possible at right degree angle REGARDLESS of the size of the sides.* Two sides of a triangle have length 6 and 8. Which of the following are possible areas of the triangle? I. 2 II. 12 III. 24

Our last side must be 2<x<14 -But imagine starting with a straight line at 8 if we raise our other two sides just a tiny bit (6+2.01)=8.01 we area can be as small as 0 to just a bit above 0. So two could technically work

*Independent Probability: P(A and B) = [P(A)] [P(B)].* *Mutually Exclusive: P(A or B) = P(A)+P(B)-P(A and B)* 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

P(A and B) = [P(A)] [P(B)] -We use this formula when events are *independent* 0.3 = P(A) P(B) P(A) and P(B) must both be greater than 0.6; because if not than one of them will end up being greater than 1 and that is not possible since the maximum probability can only be 1 or 100%

Rule: Circumference to Diameter ratio

Pi or 3.14

*If you are comparing things and something pops up every time in the comparison. This thing can be left out to only focus on the remaining part.*

Population density is the population of a region divided by its geographic area. Of the seven continents, Asia has by far the largest population density. Which continent has the second largest? Africa North America South America Europe Australia -Here we have %of population/%of land. We can leave out the population/land and just focus on the percents.

The greatest prime factor of 40,002 is x The greatest prime factor of 80,004 is y

Prime Factorization of a number*2 so all the prime factors are the same multiplied by 2

*Thinking about what is left first.* 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

R=T-32, where R is the remaining money. This should make sense since R is the unit we need to deal with. Once we take Chloe out of the equation we are left with the money that is "remaining" L=4+(1/2)(T-32) B=4+(1/3)(T-32) C=32 Adding all these = T

*This question has 2 scenarios. When we deal with scenarios we "add" since they are mutually exclusive.* A box contains 4 red chips and 2 blue chips. If two chips are selected at random without replacement, what is the probability that the chips are different colors? 1/2 8/15 7/12 2/3 7/10

S1: B, R S2: R, B S1: (2/6)(4/5)=4/15 S2:(4/6)(2/5)=4/15 S1+S2=8/15 *I did this through POE, where I knew the denominator (30) had to be a divisible by the answer choices. C does not work. There is no way this is a 50/50 so A is wrong. 2/3 and 7/10 look too big so I went with B.*

*At least case means you have to test the next options available too.* A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

Scenario 1: (10)(9)(8)(7)(6)(5)(4)(3)(2)= 10! Scenario 2: (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 10! S1+S2= 2(10!)

*This deals with tricky scenarios with the double O's which are repetitive.* 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?

Scenario with OOs: XOO; OXO; OOX. In each of these cases we can pick 5 of the remaining letters for X for a total of 5+5+5=15 choices Then we have to consider just the 6 letter scenario of MAGOSH (6)(5)(4)=120 120+15=135

*Strategy: Set up correctly, and then start plugging and chugging if nothing else makes sense. Something will make itself apparent.* a, b and c are positive integers. If b equals the square root of a, and if c equals the sum of a and b, which of the following could be the value of c? A) 21 B) 30 C) 45 D) 72 E) 100 F) 331

Setup: b^2 = a c = a + b so... c = b^2 + b -> *b(b+1)* (DO NOT THINK that b^2+b=b^3) -Now notice that we are dealing with consecutive integers and c is the result of two consecutive integers. Notice further that if that is the case then b(b+1) must be an even number so we can narrow down to B, D, and E but only B and D can be results of two consecutive integers *Also, think about it this way. c = b^2 + b. So go down the list starting from 1. Notice that once you get to 5 you notice that 5*5=25 + 5 = 30 so B could work... Also notice that 72 can work (b = 8) *You are asked what C could be. That doesn't mean you have to find what a and b have to be. Set up the problem. Solve as far as you can. And then start plugging and chugging if nothing else makes sense.*

If 6(x^5) = y^2 and x = 1/y, then y = ?

Since we are given an equation we can multiply the original by BOTH side to the one we need so x[6(x^5) ] = [y^2](1/y) = 6(x^6) = y

If x+y=7 and xy=4, then find the value of x^2 + y^2

Strategy: -Whenever you see the x^2 + y^2 of any format think of the following three rules: 1) (x+y)^2 2) (x-y)^2 3) x^2 - y^2 In this case approach via solving for (x+y)^2 = 49 and you are given xy=4 so 2xy=8 Hence x^2 + y^2 = 49 - 8 = 41

*Think about it theoretically. Otherwise this could be solved using numbers.* 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 -Check out saved image 106 for options and answers

T=retail + tax R= Retail T-R=tax rate (T-R/T)100 = Percent tax rate

*Remember to count 2 as a number.* How many positive integers less than 100 have a remainder of 2 when divided by 13? 6 7 8 9 10

That should give us 8

*Opposite angles are 90 degrees.* Check out saved image 131 Column A AB Column B BC

The answer is D; you can have a triangle in which the two sides differ and a triangle in which the two sides are the same

*Read very carefully and consider what it means. Especially if a number is negative* If [x] is defined as the greatest integer less than or equal to x. [-3.4] + [21] =?

The answer is NOT 18 and is a trick answer -The key is to notice "less than or equal to. If the number were 3.4, then 3 would make sense. But the number is -3.4, so the integer less than this would be -4 not -3 which would be larger! -4 + 21 = 17

*Dealing with average questions*

The equation can be simplified to average*number of items = sum of total items -Now the sum of total items can be dispersed any which way. It can be evenly distributed, it can be all towards one side or all towards the other side. There is no way to compare. *Thus the answer is usually D*

*Remember that when multiplying by a negative you can choose what part of the term you want to apply it to!* If 3xm + 2ym - 2yn - 3xn = 0 and m ≠ n , then what is the value of y in terms of x?

The equation simplifies to 2y(m-n)=3mx(n-m) since we want to cancel the m-n, apply -1 to 2y and apply -1 to (m-n) on the right hand side

*Assumptions on the math part*

The only assumption you can make is that if you see a line it is straight and not bent (can't make assumptions about parallel, etc.)

*When asking to check rules that you know, spell out the rule and do not doubt it and just check each one against it.* *For triangles this has to hold up for all sides! a > b+c but you also have to check if b > a+c. If not you have a problem.*

The sides of a triangle are 1, x, and x^2. What are possible values of x? We know that the sum of two sides must be greater than the third side. Plug in the values given for x and solve these out.

What are all the values of x such that (x-7)(x+3) is positive?

There is only one way multiplication of two things is positive: -When both the values are positive (x-7) > 0 and (x+3) > 0 OR -When both the values are negative Use that to figure this out

Strategy: Plugging in 1s and 0s

These work really well for easy problems [(-3x)^3]/[-3(x^3)]

sqrt(12 * 32 * 54)

This can be broken down into: sqrt(12) * sqrt(32) * sqrt(54)

*Remainder question* When positive integer N is divided by 167, the remainder is 35, and when positive integer K is divided by 167, the remainder is 17. What is the remainder when 2N+K is divided by 167?

To get the original number that is divided we can *add the remainder + divisor* -This is not the smallest number that can be divided. The smallest that can be divided is the remainder itself!

*Total - disobeying restriction = Obeying restriction* In how many ways can Ann, Bob, Chuck, Don and Ed be seated in a row such that Ann and Bob are not seated next to each other? 24 48 56 72 96

Total = 5! = 120 -Scenario is which A&B are next to each other are: __,__,__,__,__ There are 4 possibilities with AB leaving 3 chairs left so (4)(3x2x1)=24 But there is another scenario of BA which would be (4) for BA and (3x2x1) for the rest for a total of 48 120-48=72

*Tricky! Must take into account negative integers.* If k is an integer and 121 < k2 < 225, then k can have at most how many values?

Total is: 6

What is the value of 2 + 2^2 + 2^3 + 2^ 4 + 2^5?

Trick is to not go through this and solve... 1) set the equation equal to x 2) the equation 2x would add one 2 to each 2 already there: 2^2 + 2^3 + 2^ 4 + 2^5 + 2^6 = 2x But we want x, so subtract x from 2x so all the common terms cancel out you are left with 2x - x = x = 2^6 - 2

In a group of 40 people, 15 have visited Iceland and 23 have visited Norway. If 11 people have visited both Iceland and Norway, how many people have visited neither country?

Two ways to solve this: Double matrix and Total minus common For the matrix put as Iceland and Not Iceland and Norway and Not Norway *Notice that 15+23=38 that takes into account Norway, Iceland and Both. We need to cancel both (11) to get 27. That means 13 people visited neither.

*Mississippi rule works best here too.* In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?

We have 6 shirts so 6! total combinations but the two items are repetitive 6!/(3!*3!)=20 gives the fastest answer -I did this another way where I figure out the combinations possible for just 3 identical shirts __,__,__,__,__,__ (not counting repeats) we have 4+3+2+1=10 ways to organize; multiply that by 2 to get 20 (to account for the green shirts)

1/(2 - .25) = ?

We know that .25 is 1/4 is there a way to convert 2 to where it has the same denominator? Of course...

John is now m years old and Sally is 4 years older than John. Which represents Sally's sage 6 years ago?

Well right now J = m and S = 4+m Six years ago: S - 6 = 4 + (J - 6) So our answer is to the right of the equation m-2. Do not try to keep going and move 6 from Sally's side! That is not what the question is asking!

*We can get averages from percentages. It just depends what kind of information is given to us.* Percent is out of 100 so 0 households have 0 televisions. 8 households have 1*8 televisions. 24 households have 2 each so a total of 48, etc... so it would be total television divided by 100 households. BUT we can't solve this because of the "5 or more". We need the exact number.

What is the average (arithmetic mean) number of televisions per household? 0=16% 1=8% 2=24% 3=20% 4=20% 5 or more=12%

*Tip for quicker calculation*

When two fractions are set equal to each other. We can cross multiple and then multiply by something to cancel the denominators. n (15+ (10/n)) = can be simplified to 15n+10

*Pay attention to the limits! Our max can only be 90 AND that no two numbers are the same. Begin by working in reverse instead of trying out the answer choices.* The average (arithmetic mean) of 4 different integers is 75. If the largest integer is 90, what is the least possible value of the smallest integer? 1 19 29 30 33

____+____+____+90=300 -if not two numbers can be the same and the numbers are INTEGERS, then the next largest number must be 89, and the one after that must be 88 -That leaves us with 33 being the smallest possible value

*How to write out the equation* Whenever Art Dealer sells a sculpture, he earns a 20 percent commission on the first $12,000 of the sale price plus 15 percent of the sale price in excess of $12,000. If Art earned a $3,900 commission on the sale of a certain sculpture, what was the sale price?

commission = .20(12000) + .15(x - 12000)

*Set the equations equal to each other.* If f(x) = 5 - 2x and f(3k) = f(k + 1), then f(k) = 0.5 1 3 4 6

f(3k) = f(k + 1) and we have f(x) = 5 - 2x That means that 5-2(3k) = 5-2(k+1) Solve for k=1/2 So f(k)=4

*Do not forget that squares can be of negative or positive numbers.* If a and b are integers and ((ab)^1/3 (b)^1/2)^6 = 500, then a + b could equal 2 3 4 5 6

make the left look like the right and you get a^2(b^3)=(a)^2(5^3) here a could be 2 or -2

If x^2 - y^2 = 12 and x - y = 4, then x =

recognize the that (x+y)(x-y)=12 so (x+y)=3 since we want x just subtract x+y from x-y to eliminate y and get 2x=7 so x=3.5

*I arranged this in the way it was needed by failed to realize what I was looking at.* If sqrt(17+sqrt(264)) can be written in the form sqrt(a)+sqrt(b) , where a and b are integers and b < a, then a - b =

sqrt(17+sqrt(264)) = sqrt(a)+sqrt(b) 17+sqrt(264) = a+2sqrt(ab)+b 17+sqrt(264) = a+b+2sqrt(ab) Realize how similar the two equations are! -a+b=17 -The left hand side sqrt can be simplified to 2sqrt(66)=2sqrt(ab). So ab=66 -This means that a and be must be 11 and 6 since a is larger than b -The difference between the two is 5

Recognize that if we are multiplying by the same percent and we are looking for the difference then...

x%(a-b)= x%a - x%b

*Equation of a circle in the coordinate plane.* In the standard x,y-plane, a circle has a center = (1, -2) and a radius r = 5. Which of the following points are on the circle? (-4, -2) (-3, 1) (-1, -6) (1, -7) (3, 2) (4, -6) (5, 1) (6, -2)

x^2+y^2=r^2 in this case (x-1)^2+(y+2)^2=25

*How to write this into an equation: A helicopter company charges $85 for the first kilometer of a trip and $5 for every kilometer after that. If the total cost of a trip was $365, how many kilometers were flown?*

y=85+5(x-1) since is x=1, y=85

Check out saved image 44 W, X, Y and Z each represent a different number. If the sum of each column is shown beneath that column, and the sum of each row is shown beside that row, then n =

-Remember that the sum of all the columns and sum of all the rows must be equal

*Sometimes you just have to pick numbers for variables and solve that way.* If 3x < 2y < 0, which of the following must be the greatest? 2y - 3x 3x - 2y -(3x - 2y) -(3x + 2y) 0

-I used -10=3x and -1=2y

*Make your own numbers and keep track of the variables and numbers neatly.* It takes 1 pound of flour to make y cakes. The price of flour is w dollars for x pounds. In terms of w, x and y, what is the dollar cost of the flour required to make 1 cake? xy/w y/wx w/xy wx/y wxy

-Let's say that 1 pound of flour makes 5 cakes. So y=5 -Let's say that it costs 6 dollars for 2 pounds. So w=6 and x=2 that's $6 for 2 pounds which can make 10 (2*5) cakes 6/?=10/1 That's $0.6 for 1 cake -Plug in to see which option gives us this 6/(2*5)=w/(xy)

*Learn to save time by examining the limiting factors.* *System of equation of rebuilding the remainder doesn't work because the quotient is different for both the equations!* x is a positive integer less than 100. When x is divided by 5, the remainder is 4, and when x is divided by 23, the remainder is 7. What is the value of x?

-Notice that we could list all the possibilities of 5 but that would be tedious and cumbersome -Notice that we need a positive integer value below 100, so... use 23 instead start with 7 and keep adding 23 to that 7, 30, 53, 76, 99 -Now we only need to check 5 values of which it is easy to see that 99 must be the answer

Which of the following are equal to (1/560)^-4? Indicate all correct answers.

(70^4)(1/8)^-4 -First write everything out and not skip steps... You are missing questions when you do this. Notice that this is the equivalent of (70^4)(8^4), which can be written out as (70 70 70 70 8 8 8 8) = 560 560 560 560 = 560^4 -Also notice that (a^m)(b^m) = (ab)^m. So... (70^4 8^4)= (70*8)^4 = 560^4

*When given the algebraic we have to factor and use the information given.* x2 - y2 < 8 x + y > 3 If x and y are integers in the above inequalities and 0 < y < x, what is the greatest possible value of x?

(x+y)(x-y)<8 x+y>3 It's clear that the maximum value that x+y can be is 7 and x-y has to be 1 we can use system of equations to figure out that the max value of x is 4

n is a positive integer Column A (0.99)^n Column B 0.01

*Key is to note that any fraction raised to a "n" power will approach 0.* But the relationship can vary for this problem since different values of n give different relationships

For all numbers a and b, the operation & is defined by a b = a^2 - ab. If xy ≠ 0, then which of the following can be equal to zero? I. x & y II. xy & y III. x & (x + y)

*Key is to work out the operation and then simplify by factoring and then seeing what possible numbers could make the situation zero given our limitation.* Notice for the last one that we get: x^2 - x(x+y). This simplifies to x^2 -x^2 *-* xy {NOT x^2-x^2+xy}

*GRE jargon*

*Must be true* = the statement or equation must hold *ALL of the time* or in all cases

-Check out saved image 90 Column A AB Column B BC

*Triangle rule: An isosceles triangle will have the angle bisector as the altitude. But just because we see an angle bisector that does not mean that we have an isosceles.* check out saved image 90 solution for image of why D is the right answer

*Work with a table and keep variable constant per leap. and add years to a developed relationship. Then set the new relationship equal to the one you have been developing by adding or subtracting.* *Another method is to treat J and T as now; T-10 and J-10 as ten years ago; and T+5 and J+5 as five years from now. Once this is established, use these to write out the relationships and solve for a common variable.* 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?

-10yr: J=3T; T = Tim So... Now: J+10 = 3T+10; Tim = T+10 Then... 5yrs from now: J+15 = *[3T+10]+5 = 2(T+15)+10*; Tim = T+15

*Remember to keep track. Remember to not double count.* *Another thing to remember is that positive and negative need to cancel out.* The sum of k consecutive integers is 41. If the least integer is -40, then k =

-Based off the second rule we get -40 to 0 and 1-41 which gives us 81, the 82nd number is 41

*Be careful to read the statements clearly. Here is it asking for "MORE THAN 50%". That would mean that we need 2 not 1 since 1 would make it exactly 50/50.* -There are 44 nations in total What is the least number of nations currently not members of NATO that would have to join NATO so that more than 50% of the nations in Europe would be members of NATO? 1 2 3 4 5

-Check out saved image 88

*Key here is to realize 1) that the x-intercepts are symmetrically located. We should notice that the height is (0,k). If we plug in 0 for x (to find the x-coordinates we get sqrt(k),0 and -sqrt(k),0. This means that the total base is 2sqrt(k). Plug this into the area equation to solve.* Check out saved image 57 The figure shows the graph of the equation y =k-x^2, where k is a constant. If the area of triangle ABC is 1/8, what is the value of k?

-See solution image 57

*Slope gives a lot of answers.* *Divisibility rules and coordinate geometry.* 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?

-Solve for the slope and then see what works

*Remember that when diluting, we do not need to change the numerator; just the denominator.* 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?

-The concentration we want to achieve is: 3/(5+3)=.375

*The trick here is to realize that the numerators are going to be identical since adding a 0 doesn't do anything. But notice that 0 is a number and the second one is being divided by 51 to get an average.* Column A Average (arithmetic mean) of integers from -50 to -1 inclusive. Column B Average (arithmetic mean) of integers from -50 to 0 inclusive.

-The first one's sum is 50(-51)/2 -The second's is 51(-50)/2 These are identical

*This is a dependent probability with no replacement. With pairs or whatever the first chance is not dependent on anything. We can pick anything or in other words we have p=1 for the first pick.* A box contains 10 pairs of shoes (20 shoes in total). If two shoes are selected at random, what is the probability that they are matching shoes? 1/190 1/20 1/19 1/10 1/9

-The first time we can select any shoe (20/20) -The second time our shoe has to match the first, since we already picked a show there are 19 left so our probability is (1/19) -Overall our probability is (1)(1/19)=1/19

Check out saved image 43 *Remember your triangle properties and use them effectively.* *Magoosh trick is to come up with two equations equal to 180*

-The outer angle is the sum of the opposite inner angles We are asked to compare the opposite top angle (y) vs the outer angle -Well y+something = x. This means that by itself x ALWAYS greater than y alone -- Using Magoosh we should get that y+a+b=180 b+x=180 Set these equal to each other to get that y+a=x Meaning that x is greater than y by itself

Suppose in the next year, 2007, College D's expenses and enrollment remain about the same, but in addition to their current revenues, they receive an additional $50,000,000 grant. This would allow them to reduce average tuition by how much? A) $1388.89 B) $3571.43 C) $5555.56 D) $9500.00 E) $25888.89 number of students = 36,000

-Think before answering. If nothing else changes, and we have more money, how much does our tuition go down by? Well it would be 50/(number of students) = how the money would be divvied up.

*Careful about arrangements! I missed this because I arranged smallest to greatest instead of greatest to smallest.* If A, B, C and D are positive integers such that 4A = 9B, 17C = 11D, and 5C = 12A, then the arrangement of the four numbers from greatest to least is

-To solve this start with A and get is alone for ex: A=9/4B meaning that A>B. Do this for all of them and see how things stack up Answer should be DCAB

*Special case where we can use percentage to get a RELATIONSHIP of the actual number of CDs. We don't know exactly know many CDs there are but the relationship will always hold.* Yesterday, Carl had 40 percent more CDs than Karen had. Today, Carl gave 20 percent of his CDs to Karen. Column A Number of CDs that Carl now has Column B Number of CDs that Karen now has

-Try 100 and 200

*Deal with equations rather than own numbers to make things easier to solve* The area of a circle is equal to the area of a square. Column A The circumference of the circle. Column B The perimeter of the square.

-Try it out 2(pi)(r) vs 4x -we know that area of square is x^2 and area of circle is (pi(r^2))

*You have to at least try plugging in numbers as a last resort before you make a random guess.* Triangle on a line a is at the top, b is the adjacent angle outside the left leg, c is the adjacent angle to the right of the triangle leg. A: 180+a B: b+c

-Try to use 70, 60, 50 -Try to use 90, 60, 30

*Do not assume unnecessary things. This is not asking us to multiply things out. Simply asking what do both these things have in common?* What is the Greatest Common Factor (GCF) of 25x2 and 16y4?

-We could multiply is out to get 400x^2y^4 but that is not what the questions is asking the question did not ask us to get the GCF of (25x^2)(16y^4). It asked us what the GCF is individually.

*I approached this right but missed it based off 5x+6* If x>0, and two sides of a certain triangle have lengths 2x+1 and 3x+4 respectively, which of the following could be the length of the third side of the triangle? 4x+5 x+2 6x+1 5x+6 2x+17

-notice that 5x+6 can never work (try plugging any number for x, it will be 1 greater than our required criterion) -6x+1 and 2x+17 could world

*Number and integer property.*

0 is an *INTEGER* that is *EVEN* [It is neither positive nor negative.]

*This was tricky. If using plug-and-chug start with E.* *Otherwise recognize that to solve this we need to add .2x top the top and x to the bottom and set that equal to (1/4=.25).* Solution Y is 40 percent sugar by volume, and solution X is 20 percent sugar by volume. How many gallons of solution X must be added to 150 gallons of solution Y to create a solution that is 25 percent sugar by volume?

1/4 = (60+.2x)/(150+x)

*Distinct means different.* What is the sum of each distinct prime factor of 9999?

11(3^2)(101) 11+3+101=115

*Write formulas and keep track.* For a certain event, 148 people attended. If all 148 had paid full admission price, the total revenue would be three times the cost of sponsoring the event. (Admission price was the only source of revenue.) As it happens, only 50 paid the full admission price, and the others paid nothing. Column A the total revenue Column B the cost of sponsoring the event

148P=3C. So C = 148/3P; that is the right hand side Total revenue is only going to be 50P So we are essentially comparing 148/3P to 50P and this means that 50P is always going to be greater

*How to set this in an equation format.* Helen mixes a first solution, 4 liters of 40% concentrated sulfuric acid solution, with a second solution, 5 liters of a sulfuric acid solution with a stronger concentration, and the resultant solution is 9 liters of 50% concentrated sulfuric acid solution. What was the concentration, as a percent, of the second solution?

4(.40) + 5x = 9(.50)

*First start by determining how many unique combos are possible and then move from there.* *Integer properties: If P and Q is a multiple of r then P+/-Q and P+-r will be multiples of r. We can also keep adding or subtracting Q and r multiple times.* 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?

7C3 = How many combos are possible that are unique that will be the denominator -Based off the integer rule we can see that 5 numbers are divisible by 3. So... 5C3 will be our numerator

*With quartiles our data must be organized. In this case it is from lowest score to highest. So all we have to do it count the number of that gets us to the middle of the bottom half of the data meaning between the 10th and 11th student will be where our Q1 score is.* *Box-and whisker takes population and spreads it over whatever metric is being measured. So the spread is the difference between the lowest and highest metric. The population itself determines the quartiles because it would show how many fall with what quartile of the metric being measured. As long as the data is organized all we have to do it count the population into even 4 parts.* Check out saved image 67 The first quartile SAT score among the forty scores at the Newboard Free School is in which score range? 1300 - 1400 1400 - 1500 1500 - 1600 1600 - 1700 1700 - 1800

40 students: 1-20 bottom half (half-war between this is 10-11) 21-40 upper half (30-31)

*Once you get a pattern notice that 47 is not divisible by 4 or less. So go to 46, and notice that the largest number that is divisible by is 2.* What is the units digit of 18^47?

46 divisible by 2 so... _4 47 then would have to be _2

*If everything is in ratio/percent we don't need to deal with the numbers.* If from 2004 to 2005, Vietnam's rice production increases by 25%, and all the other countries in the "Indochina" group maintain the same levels of production, then the rice production of the Indochina group would increase by what percent? Indochina: 80 million metric tons 28% = Vietname

A) 2.8% B) 4% C) 5.6% D) 7% E) 12.5% -Well right now we are 100% -If Vietnam increases out by 25% well that's a 28%/4= 7% increase -So if nothing else changes out total output is 107%. That's an increase of 7%

*Pick numbers and work with them. Remember is something is 20% less than someone else's score, that means that "something" is only 80% of someone else's score.* Ashley's score was 20% higher than Bert's score. Bert's score was 20% lower than Charles' score. Column A Ashley's score Column B Charles' score

A: 240 if B:200 Since B is 20% less than C that means that C=200/.8=250

*Be very careful about what is being asked. We are asked for the TOTAL TIME not just the second leg of the trip. I missed this despite solving everything correctly and picking 25 instead of 30.* 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?

Average speed = total distance/total time =(120*5)+(180T)/(5+T) -distance for second leg is 180T=D

*This is really tricky! We wants our answers in terms of k so do not forget to keep track of variables.* 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?

F=Capacity of barrel (1/5)F+k=(2/3)F k=7/15F [This is k in terms of F] *We are asked for capacity of barrel (F) to be represented in terms of k. So We need F=15/7k)*

*There are two ways to tackle this.* If 2^k = 3, then 2^(3k+2) =

For the right hand side you can recognize that 2^3k= (2^k)^3. *For multiple exponents we multiply out* We can pug in 3 for 2^k -Another way to recognize is that for the left equation we can simply raise both sides to the power of 3 and get that 2^3k=27 -Answer should be 108

Check out saved image 74 *This is a triangle rule: hypotenuse of a triangle is always bigger than the legs.*

Here we are essentially comparing leg/hypotenuse to hypotenuse/leg clearly the second must always be bigger since quantities are fixed and the ratios remain the property no matter what the numbers are (hypotenuse/leg 1+) vs (leg/hypotenuse 1-)

*Remember that when we add the same value to the numerator and denominator the value gets closer to 1.* Yesterday, at a certain school, the ratio of boys to girls was 1 to 3. Today, an equal number of boys and girls joined the school. The number that joined was greater than zero and no students left. Column A Number of Boys to Girls Now Column B 1/3

It's easy to see based off the rule that A will always be bigger

*Ratios can be cross multiplied to give us equations.* If 8 tigers were added to the zoo, the new ratio of lions to tigers would be 4 to 3. How many bobcats are at the zoo? A) 4 B) 8 C) 12 D) 24 E) 48 Lions 32% Leopards 16% Ocelots 20% Tigers 8% Bobcats 24%

L/T = 32/8; which simplified to 4:1 If we cross-multiply this we get that L=4T meaning that however many tigers there are (T) 4x that is the number of lions -Now we know that if T if increased by 8. T+8 then... L/(T+8)=4/3, which we can cross-multiply to get 3L = 4T+32. We know that L=4T so substituting that we get 4T+32=12T and we get T=4 -We know that Bobcats is 3x Tigers so 4*3=12

Check out saved image 63 Column A Number of different triangles possible using the given points as vertices. Column B 42

Notice that this is a counting problem. We have 7 points and any 3 points define a triangle but triangle dgb is the same gdb. Essentially this is a combination problem then. 7C3 will give us 35 which is less than 42

*Careful about "at least 2" vs "exactly 2". If it were exactly two then our answer would be 9/64. But now we have to consider the scenario in which all the rolls could be 6s.* 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?

P(at least 2 6s) = P(two 6s) + P(three 6s) P(2): 3C2((1/4)^2)(3/4) P(3): (1/4)^4 (9/64)+(1/64)=10/64= 5/32

Check out saved image 42 *Look at the big picture!*

Should be really easy once you do that and use the information given

*Don't assume; for this write down what is available then solve.* While driving from A-ville to B-town, Harriet drove at a constant speed of 115 kilometers per hour. Upon arriving in B-town, Harriet immediately turned and drove back to A-ville at a constant speed of 135 kilometers per hour. If the entire trip took 5 hours, how many minutes did it take Harriet to drive from A-ville to B-town?

T1+T2=5hrs D=115T1 D=135T2

*Realize that all the digits are set except for the 3s. Basically what we want is to select 2 spots of the 7 available and not repeat.* Sid intended to type a seven-digit number, but the two 3's he meant to type did not appear. What appeared instead was the five-digit number 52115. How many different seven-digit numbers could Sid have meant to type? 10 16 21 24 27

The combination for that is 7C2 = 7*6/(2)(1)=21 -Another is to count each scenario and remember that we can go down to the lowest which is 1 so... Our first choice with the 3's next to each other gives us 6 choices our next is 5 if we skip a blank since is 4 if we skip 2...down to 1 6+5+4+3+2+1=21

*Draw lines to see what happens first. Also do not make up numbers if you can just deal with normal given triangle relationships. 1-2-srt(3) The triangle in a diagram is equilateral. The smaller circle is tangent to all three sides of the triangle. The larger circle passes through all three vertices of the triangle. What is the ratio of the area of the smaller circle to the area of the larger circle?

This is very easy to solve once you make your own diagram and draw for a radius

*I tired this with formulas but quit since I thought it wouldn't work and then just relied on my made up version of D. For the formula I was on the right track.* Andy drove from Townville to Villageton at an average speed of 40 miles per hour. He then drove from Villageton to Townville at an average speed of 60 miles per hour. Column A Column B 50 The average speed of Andy's entire trip in miles per hour.

Total time = (d/40)=(d/60) Total distance = 2d 2d/(d/40+d/60)=48 since the d's cancel

*Start by questioning what the upper bound maximum area could be. That is the reason the limits are given.* 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

We have 3 conditions: 1) p<2 2) q<4 3) point is in the I quadrant This means that the lowest point in which our point could maximize the area is (2,0). If we get the area of this triangle and then compare we should see that all three options work.

*Here I at least was on the right track in thinking that this has to be a multiple/divisible problem.* The Sargon Corporation, which employs both men and women, offers an optional stock-option buy-in program to its employees. If 85% of the men and 77% of the women choose to participate in this plan, then which of the following could be the total number of employees? Indicate all possible values for the number of employees. 100 200 350 460 525 640 750 880

When the GRE says 77% of the women choose X, they DON'T mean approximately 77% (for example, 10 of 13 would be 76.9%, approximately 77%). The GRE does not mean approximately but exactly 77%. We could only have exactly 77% of something if the whole were 100 or divisible by 100. In the problem, we know that the number of women must be divisible by 100. For the men, So the number of men must be 20 or a multiple of 20. The total number of employees must be a number that we can write in the form (multiple of 100) + (multiple of 20) (A) 100 = this could be 100 women and no men, or 100 men and no women, but the first sentence tells us that Sargon "employs both men and women," so neither can be zero. This doesn't work. (B) 200 = this could be 100 women + 100 men (100 is a multiple of 20). This works. (C) 350 = the 50 in 350 can be represented neither as a multiple of 100 nor a multiple of 20. This doesn't work. (D) 460 = this could be 400 + 60 or 300 + 160 or etc., many combinations of (multiple of 100 + multiple of 20). This works. (E) 525 = the 25 in 525 can be represented neither as a multiple of 100 nor a multiple of 20. This doesn't work. (F) 640 = this could be 600 + 40 or 500 + 140 or etc., many combinations of (multiple of 100 + multiple of 20). This works. (G) 750 = the 50 in 750 can be represented neither as a multiple of 100 nor a multiple of 20. This doesn't work. (H) 880 = this could be 800 + 40 or 700 + 140 or etc., many combinations of (multiple of 100 + multiple of 20). This works.

*Write out comparisons neatly.* In a certain sequence of all positive terms, {a1, a2, a3, ...} each term equals the previous term times a constant factor. If (a1)(a5) = 900, what is the value of a3?

a1 a2=a1r a3=a2r=a1rr a4=a3r=a1rrr a5=a4r=a1rrrr (a5)(a1)=(a1rrrr)(a1)=900=(a1^2)(r^4)=(a1r^2)^2=(a3)^2=30

*Be very careful about how you take care of the left side.* If 2^2n + 2^2n + 2^2n + 2^2n = 4^24, then n =

a^2+a^2=2(a^2); similarly... the left side of the equation is: 4(2^2n)=4^24 now you can solve. The left side simplified to 2^2+2n *Beware! The left side DOES NOT EQUAL 8^2n. We cannot add the bases.*

*Think about what you know about mean. Mean = median if the set is consecutive or evenly spread out.* *Also if nothing stands out you should aim to plug in numbers.* If the average (arithmetic mean) of seven consecutive integers is k + 2, then the product of the greatest and least integer is k2 - 9 k2 - 2k + 1 k2 + 4k - 12 k2 + 6k + 9 k2 + 4k - 5

k+2 is then the median which means k+1 and k and k-1 are to the left of k+2

*Here you have to realize that 30minutes delay but arriving at the same time means that the new driving time is shorter by 1/2hour.* *Set D=rt in what you need first.* Every day at noon, a bus leaves for Townville and travels at a speed of x kilometers per hour. Today, the bus left 30 minutes late. If the driver drives 7/6 times as fast as usual, she will arrive in Townville at the regular time. If the distance to Townville is 280 kilometers, what is the value of x?

regular driving time = new driving time + 1/2hour 280/x = 280/(7/6x) + 1/2 t=D/x -The 1/2 is very important it is the part that is left on the table for being late and since we arrive at the same time it means that whatever the new time is, that plus 1/2hr must equal the old time

*Most of the info in this question is useless and plays no role. Best way is still to deal with variable but realize that the answer is simple.* *Do not try to write an equation with k as the variable that will lead to wrong answers. We don't know if k is in the terms we need. Write out how you would normally write out using your own variables and see if you can draw a connection.* If the average (arithmetic mean) of five consecutive negative integers is 2k - 1, what is the difference between the greatest and least of the five integers? 4 4k 4k + 4 4 - 4k 4k2 - 4k

x+x+1+x+2+x+3+x+4 The difference between the largest and smallest is x+4-x=4

*Remember that 2pir=pid. Use the variables to solve methodically.* *Again use the variables given and set equal to what you want to solve for. In this case we want to solve for the diameter or D.* If ABCD is a rectangle, BC = x and AB = 2x, then the circumference of the circle, in terms of x, is Check out saved image

x^2+(2x)^2=D^2 D=sqrt(5x^2)=sqrt(5)x So the circumference is sqrt(5)xpi