GMAT Math Ultimate
Balancing Method for Mixtures/Dilutions: Example: How many liters of a solution that is 10% alcohol by volume must be added to 2 liters of a solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume?
(% diff b/w weaker solution and desired solution) x (amt weaker solution) = (% diff b/w stronger solution and desired solution) x (amt stronger solution) n(15-10) = (50-15)(2) 5n = 70 n = 14 L of 10% solution
Factoring Exponents
(5^k)−(5^k−1) (5^k)-(1/5)(5^k) (5^k)(1 - 1/5) (4/5)(5^k)
Balancing Method for Mixtures/Dilutions
(percent/price difference between weaker solution and desired solution) x (amt weaker solution) = (percent/price difference between stronger solution and desired solution) x (amt stronger solution)
Quadratics
(x + y)^2 = x^2 + 2xy + y^2 (x - y)^2 = x^2 - 2xy + y^2 (x+y)(x-y)=x^2 -y^2 When you see an equation in factored form in a question, immediately UNFACTOR it; vice versa.
Exponents
(x^r)(y^r)=(xy)^r (3^3)(4^3)=12^3 = 1728
Factorial of Zero
0! = 1
Solving Perpendicular Bisector Problems
1) Find slope of line segment = m 2) Find slope of perpendicular bisector = -1/m 3) Find midpt of line segment to identify a point (x,y) on the line of the perpendicular bisector 4) Use the slope + data point of the perpendicular bisector to plug into the slope-intercept equation to find the y-intercept 5) Use the y-intercept to produce an SI equation
Properties of Zero
Zero is an even integer. Zero is neither positive nor negative. Zero is a multiple of every number. Zero is a factor of no number.
Always Try to Factor!
ex: x^3 − 2x^2 + x = −5(x − 1)^2 x(x^2 − 2x + 1) = −5(x − 1)^2 x(x − 1)2 + 5(x − 1)^2 = 0 (x + 5)(x − 1)^2 = 0 x = −5, 1
Mixture Problem: How many liters of a solution that is 15% salt must be added to 5 liters of a solution that is 8% salt so that the resulting mixture is 10% salt?
n = total liters of solution 0.15n + 0.08(5) = 0.1(n + 5) 15n + 40 = 10n + 50 5n = 10 => n = 2 liters
Combinations (Order Does Not Matter)
nCr = n! / (r! (n - r)!) Where n is the total number of items in the set and r is the number of chosen items.
Permutations (Order Does Matter)
nPr = n! / (n - r)! Where n is the total number of items in the set and r is the number of chosen items.
Fractional Exponents
x^(r/s) = s root of (x^r) Ex: 4^(3/2) = sqrt(4^3)
Slope of a Line
y = mx + b m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
Data Sufficiency and Percent Change
All you need to compute a percent change is the RATIO of change: original; you don't need actual values. In fact, b/c original + change = new, you can compute the percent change using the ratio of ANY TWO of the following: original, change, new
Simplify the Base of Exponential Expression
Always try to simplify the base. • If 27^n = 9^4 • then (3^3)^n = (3^2)^4 => n = 8/3
Consecutive Integers
Even: 2n, 2n + 2, 2n + 4 Odd: 2n + 1, 2n + 3, 2n + 5
Exterior Angles in Triangles
Exterior angle d is equal to the sum of the two remote interior angles a and b. d = a + b
Intersection of 2 Lines
If 2 lines intersect on coordinate plane, at pt of intersections BOTH equations representing lines are true (eg pair of #s (x,y) that represents pt of intersection solves BOTH equations). To solve, use equation of 1st line to find value of y, and substitute that value for y in the 2nd equation. Then use the value found for x to plug in to find y.
Maximizing Area of Triangle or Parallelogram
If given 2 sies of a triangle/parallelogram, you can maximize the area by placing those 2 sides PERPENDICULAR to each other
Numbers Added or Deleted
Number added: (new sum) - (original sum) Number deleted: (original sum) - (new sum) Example: The average of 5 numbers is 2. After onenumber is deleted, the new average is -3. What number was deleted? CORRECT: Original sum: 5 x 2 = 10 New sum: 4 x (-3) = - 12 Number deleted = 10 - (- 12) = 22
Percent Table
Numbers Percentages/Fractions Part Whole 100 Part/whole = percent/100
Odds and Evens
Odd + Odd = Even Even + Even = Even Odd + Even = Odd Odd × Odd = Odd Even × Even = Even Odd × Even = Even
Factors of Odd Numbers
Odd numbers have only odd factors
Maximizing Area of Quadrilateral
Of all quadrilaterals w/given perimeter, the SQUARE has the LARGEST AREA Of all quadrilaterlas w/a given area, the SQUARE has the SMALLEST PERIMETER
DS: Sufficiency in Yes/No Questions
On "Yes/No" DS questions, if a statement answers the question conclusively in the affirmative or in the negative, then IT IS SUFFICIENT.
DS: Hard Questions
On harder DS questions, answer choices tend to be more sufficient than they might appear. • DON'T CHOOSE (E) if you have to guess. • Pick between (A) or (C), if you can eliminate (B). • Historically, (A) is slightly more common as the right answer.
Simple Interest
Simple interest = (principal)(interest rate)(time) I = Prt
Slope of a line
Slope = rise/run = difference between y-coordinates/difference between x-coordinates of 2 points on the line Slope of line containing (x1,y1) and (x2,y2) --> m = (y1 - y2)/(x1 - x2)
Multiplication Principle
The number of ways independent events can occur together can be determined by multiplying together the number of possible outcomes for each event.
Circular Permutations
The number of ways to arrange n distinct objects along a fixed circle is: (n - 1)!
4th Rule of Probability: Probability of A OR B
The probability of event A OR event B occurring is: the probability of event A occurring *plus* the probability of event B occurring *minus* the probability of both events occurring. P(A or B) = P(A) + P(B) - P(A and B)
Similar Triangle Areas
The ratio of the areas of two similar triangles is the *square* of the ratio of corresponding lengths. Triangle ABC has sides AB = 2 and AC = 4. Each side of triangle DEF is 2 times the length of corresponding triangle ABC (DE = 4, DF = 8) Triangle DEF must have 2x2, or 4, times the area of triangle ABC.
FOIL Method with Quadratics with Roots
Use FOIL Method with Quadratics with Roots n − 4√n + 4 => (√n − 2) (√n − 2) => x2 − 4x + 4
Number Added or Deleted
Use the mean to find number that was added or deleted. • Total = mean x (number of terms) • Number deleted = (original total) - (new total) • Number added = (new total) - (original total)
Volume of a Sphere
V = (4/3)(pi)(r^3)
Volume of a Cylinder
V = π(r^2)h
Chemical Mixtures: Table Method
Volume Original Change New Substance 1 Substance 2 ... Total Solution *Only insert actual amounts - compute percents off to the side
Squaring Fractions
When positive fractions between 0 and 1 are squared, they get smaller. Ex: (1/2)^2 = 1/4
DS: Common Trap
Do NOT use the information in one statement as an assumption in the second statement. • Statements are not necessarily related. • View separately!
Intersecting Sets
Draw Venn Diagram for sets A and B with overlap representing A intersect B |A union B| = |A| + |B| - |A intersect B|
45-45-90 Triangle
45-45-90 x (shorter legs), x(sqrt 2) (hypotenuse)
Area of Trapezoid
A = (sum of bases)(height)/2 A = {[(b1 + b2)/2](height)}/2
Simple Interest
A = P(1 + r)n A = amount accumulated P = principal r = annual rate of interest n = number of years
Area of a Rhombus
A = bh OR A = [(d1)(d2)]/2
A Common Digits Problem
A Common Digits Problem BA => 47 or 83 +AB +74 +38 CDC 121 121 A and B = 4 and 7 OR 3 and 8
Check for Prime
1. Pick a number n. 2. Start with the least prime number, 2. See if 2 is a factor of your number. If it is, your number is not prime. 3. If 2 is not a factor, check to see if the next prime, 3, is a factor. If it is, your number is not prime. 4. Keep trying the next prime number until you reach one that is a factor (in which case n is not prime), or you reach a prime number that is *equal to or greater than the square root of n.* 5. If you have not found a number less than or equal to the square root of n, you can be sure that your number is prime. Ex: the number n=19 has a square root of ~4.35. Test 2, 3, 4 --> you know 19 is prime because none of them are factors, and any other factor would be greater than sqrt(19).
Prime Factorization: Greatest Common Factor (GCF)
1. Start by writing each number as product of its prime factors. 2. Write so that each new prime factor begins in same place. 3. Greatest Common Factor (GCF) is found by multiplying all factors appearing on BOTH lists. 60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3 GCF = 2 x 2 x 3 = 12
Prime Factorization: Lowest Common Multiple (LCM)
1. Start by writing each number as product of its prime factors. 2. Write so that each new prime factor begins in same place. 3. Lowest common multiple found by multiplying all factors in EITHER list. 60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3 LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360
DS: Strategy
1.Focus on the question stem—thinking about the information needed to answer the question. 2. Look at each stem separately. 3.If neither statements was sufficient alone, look at both statements in combination. 4.Half of the Data Sufficiency (DS) answers on the GMAT come down to step 3.
Useful Percentages to Know
1/8 = 12.5% 1/6 = 16.6% 2/3 = 66.6% 5/6 = 83.3%
Common Right Triangles
3-4-5 or 6-8-10 or 9-12-15 5-12-13
Percent Example: 15 is 3/5 percent of what number?
3/5 percent = 3/500 15 = (3/500) x whole whole = 2500
30-60-90 Triangle
30-60-90 x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
DS: Rephrase
A good data sufficiency strategy is to rephrase the information in a question: z + z < z? => z < 0? (...or 0 < z < 1)
Prime Numbers
A prime number is a positive integer that has exactly two different positive divisors: 1 and itself. • 1 is NOT prime • 2 is both the smallest prime and the only even prime
DS: Strategy
AD or BCE: If you can determine that choice (A) is correct in your DS question, then you know that the ultimate answer must be either (A) or (D). If you can determine that choice (A) is not correct in your DS question, then you know that the ultimate answer must be (B), (C), or (E). Think of this as the before/after Christ distinction!!
DS: Yes/No Question Frequency
About 1/3 of DS questions are "Yes/No" questions.
Inscribed Angle, Minor Arc
An inscribed angle = two chords that have a vertex on the circle Inscribed angle with one chord as diameter = 35 degrees Minor arc = 2 x inscribed angle = 70 degrees
Set Problem: Each of 25 people is enrolled in history, math, or both. If 20 are enrolled in history and 18 are enrolled in math, how many are enrolled in both?
Answer: create a Venn diagram with one circle for history, one for math, and an overlapping space. Overlap = n History only = 20 - n Math only = 18 - n n + (20 - n) + (18 - n) = 25 38 - n = 25 n = 13 people in both history and math
Average Rate
Average A per B = (Total A)/(Total B) Average Speed = (Total Distance)/(Total Time)
1st Rule of Probability: Likelihood of A
Basic rule: The probability of event A occurring is the number of outcomes that result in A divided by the total number of possible outcomes.
Common Factors
Break down both numbers to their prime factors to see what factors they have in common. Multiply all combinations of shared prime factors to find all common factors.
DS: First Data Sufficiency Question
Calculate out the first DS questions to make sure they are correct. It is important to start out the section strong.
2nd Rule of Probability: Complementary events
Complementary Events: The probability of an event occurring plus the probability of the event not occurring = 1. P(E) = 1 - P(not E)
Evenly Divisible Problem: Determine the number of integers less than 5000 that are evenly divisible by 15
Divide 4999 by 15 => 333 integers OR => 5000/15 =333.something, so round DOWN to integer 333
3rd Rule of Probability: Conditional Probability
Conditional Probability: The probability of event A AND event B occurring is the probability of event A times the probability of event B, given that A has already occurred. P(A and B) = P(A) × P(B|A)
Work Problems
Consider work done in one hour (jobs/hour) Inverse of the time it takes everyone working together = Sum of the inverses of the times it would take each person working individually. For example, if worker A and B are doing a job, their combined rate of work is (1/A) + (1/B) = (1/T)
Determining # Integers within a Range of 1 - X that are Evenly Divisible by a Number N
Divide X by N and round down to the nearest integer. Ex: How many numbers less than 13 are divisible by 3? 13/3 = 4.33 --> 4 Proof: 3, 6, 9, 12
Compound Interest Example: If $10,000 is invested at 8% annual interest, compounded semiannually, what is the balance after 1 year?
Final balance = Principal x (1 + r/n)^(yn) Final = 10,000 x (1 + .08/2)^(1)(2) = 10,000 x (1.04)^2 = $10,816
Compound Interest
For Compound Interest: Divide interest by # of times compounded in 1 year to find interest for the compound period.
Rate x Time = Distance (rt = d)
For a fixed distance, the average speed is inversely related to the amount of time required to make the trip. Ex: Since Mieko's average speed was 3/4 of Chan's, her time was 4/3 as long. (3/4)r(4/3)t = d
DS: Equations
For a system with n variables: • If you have as many distinct linear equations as you have variables, you can answer ANY question about the system. • If you are only asked to solve for part of the system, you don't necessarily need all n equations. • If you are asked to solve for a relationship instead of the value of variables, you don't necessarily need all n equations.
Combined Events
For events E and F: • not E = P(not E) = 1 - P(E) • E or F = P(E or F) = P(E) + P(F) - P(E and F) • E and F = P(E and F) = P(E)P(F)
Gross vs. Net
Gross is the total amount before any deductions are made. Net is the amount after deductions are made.
Gross Profit
Gross profit = Selling Price - Cost
Group Problems Involving Both/Neither
Group1 + Group2 + Neither - Both = Total
DS: How Often will Problems be Both Insufficient?
Half the time statements (A) and (B) are both insufficient.
Factor Out and Simplify
Immediately try factoring/simplifying when possible. Example: Is 2x/6 + 24/6 an integer? => (2x + 24)/6 => x/3 + 4
DS: What is Being Asked?
In Data Sufficiency questions, you are usually being asked 1 of 3 things: 1. A specific value 2. A range of numbers 3. Yes/No Immediately write out the DS problem type (value, range, yes/no) on your scratch paper before you begin a DS problem.
Simple Interest Example: If $12,000 is invested at 6% simple annual interest, how much interest is earned after 9 months?
Interest = (12,000)(.06)(9/12) = $540
DS: Both Together
Only about half the time do you have to look at both statements in combination.
Percent Increase/Decrease Formula
Original x (1 - (percent decrease/100)) = New Original x (1 + (percent increase/100)) = New
Interest Problem: If $10,000 is invested at 10% annual interest, compounded semi-annually, what is the balance after 1 year?
P = 10,000 r = .10 y = 1 n = 2 FV = P (1 + r/n)^ny FV = 10,000 (1 + .1/2)^(2)(1) FV = 10,000 (1.1025)^2 = 10,000 (1.1025) = $11,025
Parallel & Perpendicular Bisectors
Parallel lines have equal slopes m1 = m2 Perpendicular lines have negative reciprocal slopes m1 = -1/m2 or m1m2 = -1 The midpoint between pt A (x1,y1) and B (x2,y2) is ([x1+x2]/2,[y1+y2]/2)
Discriminant
Quadratic equation: ax^2+ bx + c = 0 Dicriminant = b^2 - 4ac If discriminiant > 0, there are two roots (and two x-intercepts) If discriminant = 0, there is one root (and one x-intercept) If discriminant < 0, there are no (real) roots
Probability of Multiple Events
Rules: • A *and* B < A *or* B • A *or* B > Individual probabilities of A, B • P(A and B) = P(A) x P(B) ← "fewer options" • P(A or B) = P(A) + P(B) ← "more options"
Standard Deviation of n Numbers
STD measures the "spread" of data points vs the mean. Higher SD = Higher variation 1. Find arithmetic mean. 2. Find differences b/w mean and each of the n numbers. 3. Square each of the differences. 4. Find average of squared differences. 5. Take non-negative square root of this average. *Probably won't need to calculate this!
Problems Involving Either/Or
Some GMAT word problems involve groups with distinct "either/or" categories (male/female, blue collar/white collar, etc.). The key is to organize the information into a grid with the totals.
Approximations of Common Square Roots
Square root of 2 = 1.4 Square root of 3 = 1.7 Square root of 5 = 2.25
Sum of Consecutive Numbers
Sum = (average)x(number of terms)
Sum of Angles in a Regular Polygon
Sum of interior angles in a polygon with n sides =180(n - 2)
Average of Consecutive Numbers
The average of a set of evenly spaced consecutive numbers is the average of the smallest and largest numbers in the set. Average Set = (Smallest + Largest)/2
Compound Interest Formula - Compounding Annually
To compound annually: P = principal r = rate of interest (in decimal form) y = number of years New value = P (1 + r)^y
Compound Interest Formula - Compounding More Than Annually
To compound multiple times per year: P = principal r = rate of interest (in decimal form) y = number of years n = number of times per year (i.e., compounded every 3 months would be n = 4) FV = P (1 + r/n)^ny
Quadratic Formula
To find roots of quadratic equation: ax^2+ bx + c = 0 x = [−b ± √(b^2 − 4ac)]/2a
Indistinguishable Events (i.e., anagrams with repeating letters)
To find the number of distinct permutations of a set of items with indistinguishable ("repeat") items, divide the factorial of the items in the set by the product of the factorials of the number of indistinguishable elements. Example: How many ways can the letters in TRUST be arranged? (5!)/(2!) = 60 5! is the factorial of items in the set, 2! is the factorial of the number of repeat items ("T"s)
Powers and Roots
To multiply one radical by another, multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs. Example: 12√15/(2√5) = (12)/2 √15/√5 = 6√3 Example: (6√3 )2√5 = (6 × 2)(√3√5) = 12√15
DS: Solving a System of Equations Rule
To solve a system of n variables, you need n distinct linear equations. Example: What is the value of y? Given: x + y = 1 => insufficient without another distinct equation
Successive Percents
Two successive percent increases DO NOT result in a combined increase equal to the sum of both increases *To solve, choose real numbers - usually 100.
