GRE Math Prep

How to work with factorials Ex: 8!/(6!x2!)

! symbol 0! = 1 4! = 4x3x2x1=24 Ex: 8x7x6!/6!x2x1 = 28

The average (arithmetic mean) of the numbers v, w, x, y, and z is j, and the average of the numbers x, y, and z is k. What is the average of v and w in terms of j and k ?

(5j-3k)/2

When given two coordinates for an isosceles triangle on an xy plane, how to find third point

(x1-x2)/y(should be same y)

The diagonal of a square creates what kind of triangle

a right triangle

Special Triangles

3:4:5 30 60 90 1, sqrt(3), 2 respectively 5:12:13 45 45 90 1, 1, sqrt(2) Respectively

How to use algebra to find the sum What is the sum of the 24 numbers

Sum= (average) x (number of terms) Sum = 17.5 x 24 = 420

Combined work problems ex: if it takes joe 4 hrs to pain a room and pete twice as long. How long would it take for both of them working together to paint a room?

Takes joe 4 hours to paint one room so 1/4 room is painted per hour for joe and double (1/8 for pete) 1/4+1/8=1/t 2/8+1/8=1/t 3/8=1/t t=8/3

Differences of squares

a^2 - b^2= (a+b) (a-b)

When solving for x with any fractions

always convert to like terms

How to find the original whole before the percent increase? Ex: after decreasing by 5%, the population is now 57,000. What was the original population

.95 x (original pop) = 57,000, solve and it equals 60,000

What are the factors greater then 1 do 135 and 225 have in common?

1) find the prime factors of each number 2) 135= 3x3x3x5 ; 225 =3x3x5x5 3) number shared are 3x3x5 4) thus, aside from 3 and 5, the remaining commom factors can be found by multiplying 3, 3, and 5 in every possible combination: 3x3 = 9, 3x5 = 15, and 3x3x5 = 45. 5) Common factors are 3, 5, 9, 15, and 45.

Standard Deviation

1. Find Average 2. subtract average from each individual figure 3. Square the result of each subtraction 4. Find the average of new figures

Example of probability problem where probabilities must be multiplied: If a fair coin is tossed 4 times, what's the probability that at least 3 of the 4 tosses will be heads? 2 ways to solve this problem.

1. List the different possible sequences where at least 3 of the 4 tosses are heads then do # desired outcomes/# total possible outcomes. 2. Use combinations formula. Probability of a head is 1/2 and probability of a tail is 1/2. Probability of any particular sequence of heads and tails resulting from 4 tosses is 1/2x1/2x1/2x1/2 = 1/16. Supposed that the result of each of the 4 tosses is recorded in 4 spaces: _____ _____ _____ _____ Number of ways of having exactly 3 heads among the 4 tosses is the number of ways of choosing 3 of the 4 spaces above to record an H: 4C3 = 4!/[3!(4-3)!] = 4!/[3!(1!)] = (4x3x2x1)/(3x2x1x1) = 4. The number of ways of having exactly 4 heads among the 4 tosses is 1. If we use the combinations formula, using the definition that 0!=1, then: 4C4 = 4!/[4!(4-4)!] = 4!/[4!(0)!] = (4x3x2x1)/(4x3x2x1x1) = 1 Thus, 4C3 = 4 and 4C4 = 1. So the number of different sequences containing at least 3 heads is 4+1=5. The probability of having at least 3 heads is 5/16

If x and y are the tens and the units digit, respectively, of the product 725,278×67,066, what is the value of x+y

12

What is the smallest positive integer that is a multiple of 12, 16, 18, 21, and 25?

12 = (2^2)(3) 16 = 2^4 18 = (2)(3^2) 21 = (3)(7) 25 = 5^2 (2^4)(3^2)(5^2)(7) = 25,200

Quantity A The sum of the distinct prime factors of 180 Quantity B The sum of the distinct prime factors of 150

180 ÷ 2 = 90; 90 ÷ 2 = 45; 45 ÷ 5 = 9; 9 ÷ 3 = 3. So 180 = 2 × 2 × 5 × 3 × 3. The distinct, that is different, primes are 2, 3, and 5, and the sum of these is 2 + 3 + 5 = 10. Find the prime factorization of 150 by repeatedly dividing 150 by small primes: 150 ÷ 2 = 75; 75 ÷ 5 = 15; 15 ÷ 5 = 3. So 150 = 2 × 5 × 5 × 3. The distinct, that is different, primes are 2, 3, and 5, and the sum of these is 2 + 3 + 5 = 10. The quantities are equal.

If 3^(n + 1) = 243, then (25^(n + 3))(43^(n − 4)) =

243 = 3 × 81 = 3 × 9 × 9 = 3 × 3 × 3 × 3 × 3 = 3^5 Thus, 3^(n + 1) = 3^5, so n + 1 = 5 and n = 4 Answer 4(10^7)

At store X, the price of every radio is the same, and the price of every table is the same. The total price of 24 radios and 16 tables is $375 more than the total price of 17 radios and 12 tables. What is the total price of 35 radios and 20 tables?

24r+16t=17r+12t+375 Move all of the variables to the left side of the equation and combine like terms: 24r−17r+16t−12t=3757r+4t=375 Here's the strategic part. What is the relationship between 7 radios and 4 tables and 35 radio and 20 tables? The second group is five times the first group. Now apply that mathematically to both sides of the equation: 5(7r+4t)=375(5)35r+20t=1,875

What is the least common multiple of 28 and 42?

28= 2 x 2 x 7 42= 2 x 3 x 7 Multiply each factor the greatest number of times it appears in a prime factorization, thus: LCM = 2 x 2 x 3 x 7=84

How to recognize MULTIPLES OF 2, 3, 4, 5, 6, 9, 10, and 12

2: last digit is even 3: sum of digits is a multiple of 3 4: last two digits are a multiple of 4 5: last digit is 5 or 0 6: sum of digits is a multiple of 3 and last digit is even 9: sum of the digits is a multiple of 9 10: last digit is 0 12: sum of digits is a multiple of 3 and last two digits are a multiple of 4

At a certain store, the cost of every table is $40 and the cost of every cabinet is $200. Kavya purchased p tables and q cabinets. Divit purchased q tables and p cabinets. The number of dollars that Kavya spent was exactly 3 times number of dollars that Divit spent. Quant a: p Quant b: q/7

40p+200q=3(40q+200p) 40p+200q = 120q+600p 80q = 560p 80q/560 = 560p/560 q/7 = p

In a certain game played by two teams, at any time the game is in progress, there are 5 players from each team in the game. If there are 7 players on each of team A and team B, how many different possible sets of 10 players may begin a game played by team A and team B?

441

Unit Digit

53^35

Similar Triangles

Angles are the same and sides are proportional

AREA of a TRAPEZOID Sides= 8 and 12 Height= 5

Area = (average of parallel sides) x (height) (8 + 12/2) x 5 = 50

AREA of a PARALLELOGRAM

Area = (base)(height)

find an avg rate ex: if the first 500 pages have an average of 150 words per page, and the remaining 100 pages have an average of 450 words per page, what is the average number of words per page for the entire 600 pages?

Avg a per b= total a/total b Total pages = 500 + 100 =600 Total words = (500 x 150) + (100 x 450)=75,000 +45,000=120,000

How to count integers The number of integers from a to b inclusive is: Example: How many integers are there from 73 through 419

B - A + 1 419 - 73 + 1 =347

How to solve a combination problem Ex:how many different ways are there to choose 3 delegates from 8 possible candidates

C=n!/k!(n-k)! where n=# in larger group and k=number you're choosing equals 56

The number of integers between 36 100 and 500 that are multiples of 11 Quantity A The number of integers between 100 and 500 that are multiples of 11 Quantity B 36

Easiest way to do this is to find the smallest multiple of 11 over hundred and the largest below 500. So in the calculator do {100}/{11}= 9.09. So the 10th multiple of 11 must be greater than 100. We know that the 10th multiple of 11 is 110. Similarly we do {500}/{11}= 45.45. Hence the 45th multiple of 11 is smaller than 500. We know that 45th multiple of 11 is 495. Hence number of multiples of 11 between 100 and 500 is 45 -10 + 1 = 36. Remember to add 1 as both 10th multiple and 45th multiple is counted. Hence both quantities are equal. Thus C is the correct answer.

Solving a dilution or mixture problem Ex: if 5 pounds of raisin that cost $1 are mixed with 2 pounds of almonds that cost $2.40 per pound, what is the cost per pound of the mixture?

Ex: 1(5lbs)+2lbs(2.40)=$9.80 $9.80/7pounds = $1.40

How to solve compound interest If 10,000 is invested at 8% annually what is the value in one year?

Final Balance= Principal x ((1+(int rate/c))^(time)(c) c = number of times the interest is compounded annually Answer: 10,816

How many positive integers less than 60 are equal to the product of a positive multiple of 5 and an even number?

Five

How to handle fractional powers

Fractional exponents relate to roots. For instance, x^(1/2) is sqrt(x) and x^(2/3) is third root(x^2)

How to determine a combined ratio Ex: The ratio of a to b is 7:3, ratio b to c is 2:5, what is the ratio of a to c?

Get terms to match that they have in common and then line up Common multiple of b is 6 so: 7:3 (2)= 14:6 2:5 (3)= 6:15 thus a to b to c = 14:6:15 Eliminate the common factor and this leaves a to c= 14:15

How to find mean of several integers when you have multiple variables (x,y)

Goal is to find (x+y/2)

How to solve an overlapping sets problem involving both/neither Ex: Of the 120 students at a certain language school, 65 are studying French, 51 are studying Spanish, and 53 are studying neither. How many are studying both?

Group 1 + Group 2 + Neither - Both = Total 65 + 51 + 53 - Both = 120 169- both = 120 both = 49

Isosceles Triangles

Have two identical angles and sides

How to use the PERCENT INCREASE / DECREASE FORMULAS The price goes up from $80 to $100. WHat is the percent increase

Identify the original whole and the amount of increase/decrease. Percent increase = amount of increase/original whole x 100% Percent decrease = amount of decrease/original whole x 100% Percent increase = 20/80 x 100% = 0.25 x 100% = 25%

Anisha and Dequon are 2 of the 10 people in a group. In how many different ways can this group of 10 people be divided into a group of 7 people and a group of 3 people if Anisha and Dequon are to be in the same group?

If Anisha and Dequon are in the group of 7 people, then the other 5 people in that group can be selected from the remaining 8 people. So, the number of ways in which that can be done is 8!5!/(8−5)! ways. This simplifies to 8(7)(6)(5!)/5!(3!)=8×7×6/3×2×1= 8×7=56.

Simple Interest

Interest= Principle x rt

When the length of a rect side is 4times greater have length equal to

L/4

How to simplify a radical. Example: Simplify sqrt(48)

Look for factors of the number under the radical sign that are perfect squares; then find the square root of those perfect squares. Keep simplifying until the term with the square root sign is as simplified as possible, that is, when there are no other perfect square factors inside the sqrt. Write the perfect squares as separate factors and unsquare them. sqrt(48) = sqrt(16) x sqrt(3) = 4sqrt(3)

Area of Rectangle

LxW

How many three-digit numbers can be formed with the digits 1, 3, and 5 each used only once?

Multiply the possibilities together: 3x2x1 = 6

In the y-coordinate plane, triangle RST is equilateral. Points of R and T are (0,2) (1,0) In the y-coordinate plane, triangle RST is equilateral. Points R and T have coordinates (0, 2) and (1, 0), respectively. Quantity A Quantity B The perimeter of triangle RST 3sqrt{5}

Now R and T are clearly two points on a triangle. So the distance between points R and points T is length of a side of the equilateral triangle Distance between R (0, 2) and T (1, 0) = sqrt{(0-1)^2 + (2 - 0)^2} = sqrt{5}. Now since RST is an equilateral triangle its Perimeter = 3 \times Side of triangle = 3 sqrt{5} Hence quantity A and B are equal. Thus Option C is correct.

How to solve an overlapping sets problem involving either/or categories Ex: At a conference with 130 attendees, 94 are doctors and the rest are dentists. 48 of the attendees are women and 1/4 of the dentists are women. How many attendees are male doctors?

Organize the information in a grid Use the information the question gives you and then use arithmetic to fill in the remaining boxes until you get what you are looking for Ex: 55

How to solve probability problems where probabilities must be multiplied ex:If 2 students are chosen at random to run an errand from a class with 5 girls and 5 boys, what is the probability that both students chosen will be girls?

Probability that 1st student chosen will be a girl: 5/10=1/2. Since there would be 4 girls and 5 boys left out of 9 students, the probability that the second student chosen will be a girl (given that the first is a girl) is 4/9. Thus, the probability that both students chosen will be girls is 1/2 x 4/9 = 2/9.

How to calculate simple probability

Probability= Number of Desired outcomes/number of total possible outcomes

In the equation ax^2 + bx + c = 0, a, b, and c are constants, and abc ≠ 0. If one root of the equation is -2, and b = 8a then which of the following is c equal to?

Since x = −2 is a solution to the equation ax^2 + bx + c = 0, we have a(−2 )2 + b(-2) + c = 0, and 4a − 2b + c = 0. The answer choices are all in terms of a, and we are given that b = 8a. Substituting 8a for b in the equation 4a − 2b + c = 0, we have 4a − 2(8a) + c = 0. Let's solve this equation for c in terms of a. We have 4a − 16a + c = 0, −12a + c = 0, and c = 12a.

Solving inequalities Ex: Rewrite 7 - 3x > 2

Solve like a reg equation 1. minus the seven 2. divide the 3 (sign changes direction) 3. equals x<5/3

Finding slope of the line

Rise/Run

Rectangular Solid Surface Area

S= 2lw +2lh +2wh or S= 2(lw+lh+wh)

The grass in a city park can be mowed by 5 gardeners in 6 hours. Working at the same rate, how many hours would it take 8 gardeners to mow that same grass?

Since 5 gardeners complete the task in 6 hours, it takes a total of 30 gardener-hours to cut the grass, which means it would take 1 gardener 30 hours to mow the lawn. Divide 30 by 8 to determine how long it will take 8 gardeners to complete the task: 30/8=3 3/4

Median

The middle number of a sequence. If there is an even number of figures then take the average of the two middle

How many positive integer factors does (2^5)(3^4)(5^7) have?

The possible positive integer factors of (2^5)(3^4)(5^7) are integers of the form (2^x)(3^y)(5^z), where x is one of the integers 0, 1, 2, 3, 4, or 5, y is one of the integers 0, 1, 2, 3, or 4, and z is one of the integers 0, 1, 2, 3, 4,. 5, 6, or 7. Since x can have 6 possible values, y can have 5 possible values, and z can have 8 possible values, the number of possible integer factors of (2^5)(3^4)(5^7) is 6 × 5 × 8 = 30 × 8 = 240.

Find the angle amount of the exterior part of a triangle

The sum of an interior angle and an exterior angle at each vertex is 180, so the sum of all of the interior and exterior angles is 3(180) = 540

How many different lists containing the numbers 1, 4, 5, 8, 17, 21, and nothing else are there in which each odd integer appears before any even integer?

There are 4 odd integers in the list (1, 5, 17, and 21) and 2 even integers (4 and 8). An acceptable list must have the 4 odd integers appearing before the 2 even integers. The odd integers will be arranged in the first four positions, in any order. The number of ways to arrange four items is 4! = 4 × 3 × 2 × 1 = 24. For each of those arrangements, the even numbers will be arranged in either of 2 ways (4, 8 or 8, 4). So, the total number of arrangements is 24 × 2 = 48. That makes (C) correct

Of a group of 100 diners in a restaurant, a diners had dessert, b diners had coffee, and 5 diners had both dessert and coffee. Quantity A The number of diners who did not have either dessert or coffee Quantity B 105 - a - b

To evaluate Quantity A, use the formula for overlapping sets: Total = Group A + Group B - Both + Neither. Fill in the known values to get 100 = a + b - 5 + Neither. Isolate "neither" on the right side: 105 - a - b = neither. The quantities are identical, so (C) is correct.

Average Speed

Total Distance/Total Time

Cube Volume Surface Area

V s =s^3 Surface area= 6S^2

Cylinder Volume Surface Area

V=(pie)r^2h S=S(pie)r^2 + 2(pie)rh

Sphere Volume Surface Area

V=4/3(pie)r^3 S=4(pie)r^2

How to add, subtract, multiply, and divide roots. Examples: sqrt(2) + 3sqrt(2) and sqrt(2) - 3sqrt(2)

You can add/subtract roots only when the parts inside the root are identical. sqrt(2) + 3sqrt(2) = 4sqrt(2) sqrt(2) - 3sqrt(2) = -2sqrt(2)

Determine combined percentage price increase/decrease ex: A price rises by 10% one year and 20% the next. What is the combined increase.

Yr one: 100 + (10% of 100) = 100 + 10 =110 Yr two: 110 + (20% of 110) = 110 + 22 = 132 from 100 to 132 is a 32% increase.

Weighted Average ex: Girls avg score is 30 Boys avg score is 24 twice as many boys than girls, what is the weighted avg

apply as if three parts (1/3 girls and 2/3 boys) ((1 x 30) + (2 x 24))/3= 78/3 = 26

A building is constructed in the form of a rectangular box. The lengths of its sides are 20 feet and 100 feet. Its height is 50 feet. The sides, and only the sides, must be faced with glass. If glass siding costs $2 per square foot, and no glass is broken or misused, what will be spent on glass siding?

area to be faced 2(100 ft×50 ft)+2(20 ft×50 ft) =2(5,000 ft2)+2(1,000 ft^2) =10,000 ft2+2,000 ft^2 =12,000 ft2 12,000 ft^2×$2ft/2=$24,000

1/a/b=

b/a

diagonal of a solid

d^2 = a^2 + b^2 + c^2

Quickly squaring integers 45^2 (xy^2)

first: find x (x+1) Second: square y Third combine (dont add, place the value of the first step first then place the value the second step afterwards.)

How to find the maximum and minimum lengths for a side of a triangle

if you know the lengths of two sides of a triangle, you know that the third is somewhere between the positive difference and the sum of the other two sides

a, b, c, and d are integers. 1 < a < b < c < d abcd = 210 Quantity A C Quantity B 5

look at the prime factors of 210. You have 210 = 2 × 105, then 105 = 5 × 21, and 21 = 3 × 7. Therefore, the prime factorization is 210 = 2 × 3 × 5 × 7. Given the ordering set up by 1 < a < b < c < d, it must be that a = 2, b = 3, c = 5, and d = 7. No other combinations of distinct integers multiply to 210.

Finding different points on coordinate plane using special triangle

make a right triangle and then use rules

How to solve a permutation problem

nPk = n!/(n-k)! n=number in the larger group k=number you're arranging Number of ways to arrange elements sequentially Order is important

How to use original average and new average to find out what was added or deleted ex: The average of the five numbers is 2. after one number is deleted, the new average is -3. what number was deleted?

number added = new sum - original sum number deleted = original sum - new sum original sum= 5 x 2 = 10 new sum = 4 x (-3) = -12 difference between them = 10 - (-12) = 22

Finding the new average when a number is added or deleted ex: Michaels avg score after four tests is 80, if he scores 100 on the fifth test, what is his new average

original Sum = 4 x 80= 320 Add fifth New Sum= 320 + 100= 420 New Avg= 420/5=84

For all real numbers a and b, the operation # is defined by a#b = |a| - 2b + |b|. Quantity A 7#(−10) Quantity B 10#(−7)

quant a 7#(-10) = |7| - 2(-10) + |-10| = 7 - (-20) + 10 = 7 + 20 + 10 = 37 quant b 10#(-7) = |10| - 2(-7) + |-7| = 10 - (-14) + 7 = 10 + 14 + 7 = 31 Quantity A is greater, so (A) is correct

How to use actual numbers to determine a ratio ex: the ratio of 20 oranges to 12 apples

ratio = of/to 20/12 or 5/3 or 5:3

something method

remember

How to find the sum of consecutive numbers Ex: What is the sum of the integers from 10 through 50, inclusive?

sum = (avg) x (# of terms) Average = (10+50) / 2 = 30 Number of terms = 50-10+1 = 41 Sum = 30x41 = 1230

Find avg speed ex: rosa drove 120 miles at 40 mph and returned the same 120 miles at 60 mph. what was the avg mph for the whole trip

total distance/total time 120 at 40 mph= 3hrs 120 at 60 mph =2hrs 240/5hrs = 48