GMAT Math Ultimate Review 2023

GMAT Math Ultimate Review 2023 Common Factors - ANS-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. Gross Profit - ANS-Gross profit = Selling Price - Cost Combined Events - ANS-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) Multiplication Principle - ANS-The number of ways independent events can occur together can be determined by multiplying together the number of possible outcomes for each event. 1st Rule of Probability: Likelihood of A - ANS-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. 2nd Rule of Probability: Complementary events - ANS-Complementary Events: The probability of an event occurring plus the probability of the event not occurring = 1. P(E) = 1 - P(not E) 3rd Rule of Probability: Conditional Probability - ANS-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) 4th Rule of Probability: Probability of A OR B - ANS-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) Probability of Multiple Events - ANS-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" Indistinguishable Events (i.e., anagrams with repeating letters) - ANS-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) Combinations (Order Does Not Matter) - ANS-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) - ANS-nPr = n! / (n - r)! Where n is the total number of items in the set and r is the number of chosen items. Circular Permutations - ANS-The number of ways to arrange n distinct objects along a fixed circle is: (n - 1)! Slope of a Line - ANS-y = mx + b m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1) 30-60-90 Triangle - ANS-30-60-90 x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse) 45-45-90 Triangle - ANS-45-45-90 x (shorter legs), x(sqrt 2) (hypotenuse) Common Right Triangles - ANS-3-4-5 or 6-8-10 or 9-12-15 5-12-13 Number Added or Deleted - ANS-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) Factors of Odd Numbers - ANS-Odd numbers have only odd factors Quadratic Formula - ANS-To find roots of quadratic equation: ax^2+ bx + c = 0 x = [−b ± √(b^2 − 4ac)]/2a Discriminant - ANS-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 Exponents - ANS-(x^r)(y^r)=(xy)^r (3^3)(4^3)=12^3 = 1728 Prime Factorization: Greatest Common Factor (GCF) - ANS-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) - ANS-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 Check for Prime - ANS-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). Rate x Time = Distance (rt = d) - ANS-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 Factoring Exponents - ANS-(5^k)−(5^k−1) (5^k)-(1/5)(5^k) (5^k)(1 - 1/5) (4/5)(5^k) Squaring Fractions - ANS-When positive fractions between 0 and 1 are squared, they get smaller. Ex: (1/2)^2 = 1/4 Approximations of Common Square Roots - ANS-Square root of 2 = 1.4 Square root of 3 = 1.7 Square root of 5 = 2.25 Inscribed Angle, Minor Arc - ANS-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 Area of Trapezoid - ANS-A = (sum of bases)(height)/2 A = {[(b1 + b2)/2](height)}/2 Area of a Rhombus - ANS-A = bh OR A = [(d1)(d2)]/2 Compound Interest Formula - Compounding Annually - ANS-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 - ANS-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 Interest Problem: If $10,000 is invested at 10% annual interest, compounded semiannually, what is the balance after 1 year? - ANS-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 Compound Interest - ANS-For Compound Interest: Divide interest by # of times compounded in 1 year to find interest for the compound period. 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? - ANS-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 Evenly Divisible Problem: Determine the number of integers less than 5000 that are evenly divisible by 15 - ANS-Divide 4999 by 15 => 333 integers OR => 5000/15 =hing, so round DOWN to integer 333 Determining # Integers within a Range of 1 - X that are Evenly Divisible by a Number N - ANS-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 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? - ANS-n = total liters of solution 0.15n + 0.08(5) = 0.1(n + 5) 15n + 40 = 10n + 50 5n = 10 => n = 2 liters Fractional Exponents - ANS-x^(r/s) = s root of (x^r) Ex: 4^(3/2) = sqrt(4^3) Prime Numbers - ANS-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 Always Try to Factor! - ANS-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 Intersecting Sets - ANS-Draw Venn Diagram for sets A and B with overlap representing A intersect B |A union B| = |A| + |B| - |A intersect B| Standard Deviation of n Numbers - ANS-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! Consecutive Integers - ANS-Even: 2n, 2n + 2, 2n + 4 Odd: 2n + 1, 2n + 3, 2n + 5 Properties of Zero - ANS-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. FOIL Method with Quadratics with Roots - ANS-Use FOIL Method with Quadratics with Roots n − 4√n + 4 => (√n − 2) (√n − 2) => x2 − 4x + 4 Similar Triangle Areas - ANS-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. Exterior Angles in Triangles - ANS-Exterior angle d is equal to the sum of the two remote interior angles a and b. d = a + b Gross vs. Net - ANS-Gross is the total amount before any deductions are made. Net is the amount after deductions are made. Useful Percentages to Know - ANS-1/8 = 12.5% 1/6 = 16.6% 2/3 = 66.6% 5/6 = 83.3% Odds and Evens - ANS-Odd + Odd = Even Even + Even = Even Odd + Even = Odd Odd × Odd = Odd Even × Even = Even Odd × Even = Even Simplify the Base of Exponential Expression - ANS-Always try to simplify the base. • If 27^n = 9^4 • then (3^3)^n = (3^2)^4 => n = 8/3 Powers and Roots - ANS-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 Problems Involving Either/Or - ANS-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. Factor Out and Simplify - ANS-Immediately try factoring/simplifying when possible. Example: Is 2x/6 + 24/6 an integer? => (2x + 24)/6 => x/3 + 4 Volume of a Sphere - ANS-V = (4/3)(pi)(r^3) Sum of Angles in a Regular Polygon - ANS-Sum of interior angles in a polygon with n sides =180(n - 2) Group Problems Involving Both/Neither - ANS-Group1 + Group2 + Neither - Both = Total Numbers Added or Deleted - ANS-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 Balancing Method for Mixtures/Dilutions - ANS-(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) 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? - ANS-(% 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 Average Rate - ANS-Average A per B = (Total A)/(Total B) Average Speed = (Total Distance)/(Total Time) A Common Digits Problem - ANS-A Common Digits Problem BA => 47 or 83 +AB +74 +38 CDC 121 121 A and B = 4 and 7 OR 3 and 8 Factorial of Zero - ANS-0! = 1 Compound Interest Example: If $10,000 is invested at 8% annual interest, compounded semiannually, what is the balance after 1 year? - ANS-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 Simple Interest - ANS-Simple interest = (principal)(interest rate)(time) I = Prt Simple Interest Example: If $12,000 is invested at 6% simple annual interest, how much interest is earned after 9 months? - ANS-Interest = (12,000)(.06)(9/12) = $540 Sum of Consecutive Numbers - ANS-Sum = (average)x(number of terms) Average of Consecutive Numbers - ANS-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 Percent Example: 15 is 3/5 percent of what number? - ANS-3/5 percent = 3/500 15 = (3/500) x whole whole = 2500 Work Problems - ANS-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) Simple Interest - ANS-A = P(1 + r)n A = amount accumulated P = principal r = annual rate of interest n = number of years Quadratics - ANS-(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. DS: How Often will Problems be Both Insufficient? - ANS-Half the time statements (A) and (B) are both insufficient. DS: Strategy - ANS-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. DS: Rephrase - ANS-A good data sufficiency strategy is to rephrase the information in a question: z + z < z? => z < 0? (...or 0 < z < 1) DS: What is Being Asked? - ANS-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. DS: First Data Sufficiency Question - ANS-Calculate out the first DS questions to make sure they are correct. It is important to start out the section strong. DS: Common Trap - ANS-Do NOT use the information in one statement as an assumption in the second statement. • Statements are not necessarily related. • View separately! DS: Hard Questions - ANS-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. DS: Equations - ANS-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. Continues...