GMAT Quant
5/8
0.625
5/6
0.833
7/8
0.875
finding GCF
1) List out all factors, find the GMF 2) Find the prime factorizations, take the smaller of exponents ex: GCF(24,36) 24 = 2³3 36 = 2²3² GCF(24,36) = 2²3 = 12
finding LCM
1) List out all multiples, find the LCM 2) Find the prime factorizations, take the larger of exponents ex: LCM(24,36) 24 = 2³3 36 = 2²3² LCM(24,36) = 2³3² = 72
√3
1.73
√5
2.24
discriminant
more than 0 → 2 solutions equal 0 → 1 solutions less than 0 → no solutions
sum of squares
n(n+1)(2n+1)/6 ex: 3²+2²+1² → (3)(4)(7)/6 = 14
the number of powers of number k, in the n!
n/k+n/k²+n/k³... till n>kˣ ex: p = product of 1-30 inclusive, greatest integer k for which 3ᵏ is a factor of k 30/3+30/3²+30/3³ ≈ 10+3+1 = 14 → 3¹⁴ is the greatest factor of 30
properties of one
not prime; factor of every number
perfect squares factors
positive integers that are perfect squares have an odd number of factors; not perfect squares have even number of factors ex: 16 → {1,16,2,8,4) → 5 factors
binomial probability (combination and probability)
start with the combination to get total options, then use the probability to multiply ex: If you flip a fair coin 4 times, what is the probability that you will get exactly 2 tails? 1) Find combination C(4,2) = 4!/2!2! = 6 = ways to get 2 tails from 4 flips 2) Find probability P(2T/4) = (.5)⁴ = probability of getting 2 tails out of 4 flips 3) Multiply to find binomial probability probability to get 2 tails out of 4 flips = (.5)⁴*6 = 3/8
cone surface area
the area of the base + sector area (of larger circle area with slant as radius)
dependent probability
the incidence of one event does affect the probability of the other event they can both occur at the same time; applies to group formula (of 2) P(A or B) = P(A)+P(B)-P(A and B) P(A and B) ≠ P(A)P(B) P(A and B) = P(A)P(B⎮A) ex: 6 blue and 5 small shirt with 10 items total P(B or S) = 6/10+5/10-1/10 = 1 P(B and S) = 6/10*1/6 = 1/10 ≠ 6/10*5/10 ≠ 30/100 ex: not replacing marbles
complete the square
the process of adding a term to a quadratic expression to make it a perfect square trinomial
LCM and GCF
the product of any two numbers is equal to the product of their LCM and GCF
solve equations using structure
when solving complex equations, try substitutions for factors and substitute back in ex: (x²+4)²−11(x²+4)+24 = 0 → a = x²+4 → a²-11a+24 = 0 → a = 8 and a = 3 → x²+4 = 8 and x²+4 = 3 → x = ±2
dual percentage
x(1±y/100)(1±z/100) x = staring value y = percent difference z = percent difference ex: $100 goes up 30% then down 10%, what is the final value? 100(1.3)(.9) = 117
(x+y)²
x²+2xy+y²
(x-y)²
x²-2xy+y²
(x+y)(x-y)
x²-y²
1/8
0.125
mixture problems
(Amount of Mix₁)*(% of Mix₁) + (Amount of Mix₂)*(% of Mix₂) = (Amount of Solution)*(% of solution)
divisibility rules
*11 - subtract and add digits left to right and answer is 0 or 11 *12 - apply rules 3 or 4
perfect squares up to 30
*32² = 1024 *64² = 4096
powers of 3, 4, 5, and 6
*3³ = 27 *5³ = 125
multiply to perfect square/cube
- For perfect square, all prime factors should have an even power - For perfect cube, all prime factors should be have a multiple of 3 power ex: What is the smallest positive integer that can be multiplied by 360 to make it a perfect square? 360 = 3²2³5 → 2*5 → 3²2⁴5² = √3600 = 60
objects moving
- If in the same direction, subtract to find the relative speed ex: A=80mi/hr, B=60mi/hr, and A is 100mi behind B, how long for A to be 20mi behind? 80mi = (20mi/hr)t → t = 4hrs - If in opposite direction, add to find the relative speed ex: A=30mi/hr, B=20mi/hr, and A and B are 100mi apart, how long for A and B to meet? 100mi = (50mi/hr)t → t = 2hrs
similar triangles with shared parts
- In two similar triangles, their perimeters and corresponding sides, medians and altitudes will all be in the same ratio. ex: finding x → 2/1 = 3/x → x = 3/2 - In two similar triangles, the ratio of their areas is the square of the ratio of their sides. ex: finding x (using areas) → (1/2)*2*1/(1/2)*3*x = 2/3x = 2²/3² → x = 3/2
1/6
0.166
3/8
0.375
exponent rules
1) Multiplication, add the powers. ex: 2²*2⁴ = 2⁶ 2) Division, subtract the powers. ex: 2⁵/2² = 2³ 3) When raising one power to another, multiply them. ex: (2⁵)² = 2¹⁰ *When raising to the same power for different bases, multiple/divide the numbers and raise to power ex: (2*3)² = 2²*3² = 6² ex: (3/2)² = 3²/2²
ratio (current to future)
1) Set up current and future ratios 2) Combine equations by substitution ex: The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers? current ratio → s/t = 30/1 → 30t = s future ratio → s+50/t+5 = 25/1 substitution → 30t+50 = 25t+125 → t = 15
subtracting inequalities
1) Switch one of the inequalities by flipping the sign and making the values opposite. 2) Match the inequalities to have the same sign. 3) Add the inequalities. ex: 5>4 and 3>1 → 5>4 and -3<-1 → 5>4 and -1>-3 → 4>1
standard deviation
1. Find the mean 2. Find distances between observations and the mean 3. Square each deviation 4. Add up the squares of each deviation and divide by the sample number (if random sample do n-1) 5. Find square root of result ex: {4, 6, 8} → 2²+0²+2² = 8/3 → σ = √(8/3)
√6
2.45
√7
2.65
√8
2.83
right triangle
3-4-5 5-12-13 8-15-17 7-24-25
permutation example
Create a 3 digit code (0-9 inclusive) in which the 1st digit cannot be 0 or 1, the second must be 0 or 1, and the second and third digit cannot be both 0 8(not 0 or 1)*1(must be 1)*10(any number) + 8(not 0 or 1)*1(must be 0)*9(not 0) = 152 *create two scenarios and add them (different arrangements)
combination example
From 7M and 5W, how many groups of 4M and 2W can be created? 1) Create combination for one group 4M group from 7M → 7!/4!3! = 35 2) Create combination for second group 2W group from 5M → 5!/2!3! = 10 3) Multiply combinations group of 4M and 2W → 35*10 = 350 *create 2 groups and multiple them (within same selection)
permutation example (path)
In a laboratory experiment, a rat must travel from start (top left corner) to finish (bottom right corner) to earn a reward. If the rat can travel only on the paths shown. How many distinct routes of minimum possible length can the rat travel? 1) Consider each step an option. The rat mush travel 3 steps down (DDD) and 3 steps right (RRR). 2) Calculate the permutation to arrange the desired options. The rat must take 6 steps in total (DDD and RRR). How many arrangements of DDD and RRR are possible? P(6,6) with 2 sets of 3 items repeated = 6!/0!3!3! = 20 = permutation for choosing 6 items out of 6 items with repetitions
paramaters
When scaling each item in a set: mean/median - add or subtract x → increase or decrease by x; multiple or divide x → multiple or divide by x standard deviation/IQR - add or subtract x → remains same; multiple or divide by x → multiple or divide by x
remainder formula
X = QN + R X/N = Q +R/N X = dividend (# being divided) Q = quotient (number yielded when X divided by N) N = divisor (number being divided by) R = remainder
permutation example (restriction)
You need to put your reindeer, Gloopin, Quentin, Ezekiel, and Lancer, in a single-file line to pull your sleigh. However, Quentin and Gloopin are best friends, so you have to put them next to each other, or they won't fly. 1) Find the permutation if they were next to each other; instead of 4 items count the ones next to as one → 3 total P(3,3) = 3!/0! = 6 = permutation for choosing 3 items out of 3 items 2) Find the combination of the two items being flipped → 2 ways 3) Find total arrangements with restriction = 2*6 = 12 4) Find total arrangements = P(4,4) = 24 5) Subtract total minus restriction permutations if 2 items must be next to each other = 24-12 = 12
terminating decimal
any fraction with only prime factors 2, 5, or both
rational number
any number that can be expressed as a ratio of two integers
right pyramid volume
applies to any Base shape, including circles (cone)
central angle and inscribed angle in a circle
central angle is twice any inscribed angle subtended by the same arc
combination vs permutation
combination: order does not matter; remove the same combination in different order; you use combination when creating a committee n = total items r = group size ex: ways to create a group of 2 with ABC → 3!/2!*(3-2)! = 3 → AB, AC, BC permutation: order matters; remove sets of repetitions; you use permutation when configuring codes n = total items r = group size ex: ways to arrange BBAA → 4!/2!2! = 6 → BAAB, ABBA, AABB, BBAA, ABAB, BABA
solving absolute value inequalities
create 2 inequalities with one positive and negative
diagonal formula in rectangular solid
d = √l²+w²+h²
exterior angle of a triangle
equals sum of opposite interior angles
sum of consecutive numbers
ex: The sum of 15 consecutive integers is equal to x. If the sum of the last 5 integers in the sequence is equal to 370, what is the value of x? sum of last 5 integers = 5(a+10+a+14)/2 = 5a+60 = 370 → a = 62 sum of 15 integers = 15(a+a+14)/2 = 15a+105 → 15(62)+105 = 1035
equivalent equations
ex: [−4x−6y=9 and 3x+y=−4] equivalent to [−4x−6y=9 and −x−5y=5]? -4x-6y=9 + 3x+y=-4 → -x−5y=5 → equivalent
explicit formula for the arithmetic sequence
explicit finds at any point ex: Find an explicit formula for the arithmetic sequence 12 , 5 , −2 , −9 , ... f(n) = 12+(n-1)(-7)
explicit formula for geometric sequence
explicit finds at any point ex: Find an explicit formula for the geometric sequence 3 , 15 , 75 , 375 , ... f(n) = 3(5)ⁿ⁻¹
compound interest
finding interest? A-P = [yield] = P(1+r/n)ⁿᵗ-P [yield] = P ((1+r/n)ⁿᵗ)-1) ex: Given the yield is $500, the annual rate is 8%, what is the initial investment? 500 = P(1.08) - P → 500 = .08P → P = 6250 ex: If $500 is invested at annual rate of %5 that compounds monthly, what is the value after 10 years? A = 500(1+.05/12)¹²*¹⁰ = 823.5
inverse function
f⁻¹(f(x)) = x for all domains of x taking the inverse of the function reverses each other
inscribed angles in a circle
inscribed angle subtended by the same arc are equal
properties of zero
integer; neither positive nor negative; even; not prime
number of factors
prime factor, write with exponents, add 1 to each exponent, then multiply ex: 96 = 2⁵3¹ → (5+1)(1+1) = 12 → 12 factors: 96, 48, 32, 24, 16, 12, 8, 6, 4, 3, 2, 1
recursive formula for the arithmetic sequence
recursive finds based on previous point ex: Find the recursive formula of the geometric sequence 12 , 5 , −2 , −9 , ... f(1) = 12 → f(n) = f(n-1)-7
recursive formula for the geometric sequence
recursive finds based on previous point ex: Find the recursive formula of the geometric sequence 3 , 15 , 75 , 375 , ... f(1) = 3 → f(n) = f(n-1)(5)
independent probability
the incidence of one event does not affect the probability of the other event they can both occur at the same time; applies to group formula (of 2) P(A or B) = P(A)+P(B)-P(A and B) P(A and B) = P(A)*P(B) ex: coin toss (H) and rolling dice (even) P(H or E) = .5 + .5 - (.5*.5) = .75 P(H and E) = .5*.5 = .25 P(not H and not E) = .5*.5 = .25 ex: replacing marbles in a bag
mutually exclusive probability
they cannot both occur at the same time P(A or B) = P(A)+P(B) P(A and B) = 0 ex: cannot get a H and T from coin toss P(H or T) = .5 + .5 = 1 P(H and T) = 0
group formula (of 3)
total = [group 1] + [group 2] + [group 3] - [only 2] - 2[all 3] + [neither] total = [only 1] + [only 2] + [only 3] + [neither]
group formula (of 2)
total = [group 1] + [group 2] - [both] + [neither] total = [only group 1] + [only group 2] + [both] + [neither] *[group 1,2] = [only 1,2] + [both]
transformations
translation - figure slid in any direction reflection - figure flipped over a line rotation - figure turned around a point dilation - figure enlarged or reduced
probability with combination (group)
use combinations to get the favorable outcomes by total outcomes ex: There are 7 students in a class: 5 boys and 2 girls. If the teacher picks a group of 4 at random, what is the probability that everyone in the group is a boy? 1) Find desired combination C(5,4) = 5!/4!1! = 5 = combination for choosing 4 boys out of 5 boys 2) Find total combination C(7,4) = 7!/4!3! = 35 = combination for choosing any 4 students out of 7 students 3) Divide desired outcome/total outcome probability that everyone in the group is a boy = 5/35
