GMAT Quant Questions (TTP)

1/2 = 0.5 1/3 = 0.3333 1/4 = 0.25 1/5 = 0.2 1/6 = 0.166... 1/7 = 0.14285 1/8 = 0.125 1/9 = 0.111... 1/10 = 0.1 1/11 = 0.090909090.... 1/12 = 0.0833... 1/16 = 0.0625

Common fraction denominators: 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/16

IMPORTANT True. √4 = 2, not ± 2. This is confusing because while the squares of both +2 and -2 is 4, the square root of 4 can only take the value +2. Therefore, when a " √ " is present, the result must be a positive number. However, if there is not (if we are looking at x^2 for example), we can consider 2 and -2. Instead, ±√4 = ± 2, not √4 = ± 2

IMPORTANT T/F: To denote the principal square root, we use the square root symbol (√ ). We consider only the non-negative square root of the number, which is called the principal square root. In other words, the √(positive number) will be positive.

See picture

Make sure how to do synthetic division!!

Proper = 1/3 Improper = 3/2

Proper vs Improper Fraction

True (anything over 0 is undefined)

T/F: 0 does not have a reciprocal

True. 1. Essential GMAT Quant Skills 1.21 Introduction to Factorials

T/F: 0! = 1

True. 1. Essential GMAT Quant Skills 1.16 PEMDAS: the order of mathematical operations

T/F: PEMDAS for MD and AS goes left to right, not multiplication first and then division or addition and then subtraction.

True. 3. Properties of Numbers 3.4.1 Addition and subtraction rules for even and odd numbers and 3.4.2 Multiplication rules for even and odd numbers and 3.4.3 Division rules for even and odd numbers

T/F: When adding or subtracting two numbers, if both numbers are even, or both numbers are odd, the result will be even. Otherwise, the result will be odd. Also Multiplication Rules: even * even = even even * odd = even and odd * even = even odd * odd = odd Also Division Rules: even/even may be even or odd (depends on the division. Ex. 6/3=2 vs 4/2 = 2) even/odd is even if it produces an integer odd/odd is odd if it produces an integer Therefore is not odd/even

True. 2. Linear and Quadratic Equations 2.9.3 Three common quadratic identities

T/F: When multiplying square roots, √x * √y = √ (x*y)

Initial Answer: C Correct Answer: D I got this wrong because the question is asking "could Mr Stewart evenly divide the class into 3 study groups." For both statements, the answer will always be "yes". 3. Properties of Numbers 3.13.7 Word Problems involving divisibility

Don't look at the work for this question: There is a certain number of students in Mr. Stewart's class. Could Mr. Stewart evenly divide the class into 3 study groups? 1) If Mr. Stewart reduced the number of students in his class by 16% he could evenly divide the class into groups of 9. 2) If Mr. Steward reduced the number of students in his class by 6% he could evenly divide the class into groups of 3.

Initial Answer: C Correct Answer: C 0. Intro to Quant Section 0.9 Triangulating Between Two Statements in Data Sufficiency Problems REMEMBER, THINK OF THE STATEMENTS AS... A n B and not A U B

If m is a positive number less than 60, what is the value of m? 1) m is a multiple of 5 2) m is a multiple of 11

True 1. Essential GMAT Quant Skills 1.2.2 Using the bow tie method to compare the size of fractions

In general, given two positive fractions a/b and c/d, a/b > c/d if ad > bc If given two negative fractions, a/b > c/d if ad > bc (or ad < bc in magnitude)

True. 3. Properties of Numbers 3.3.2 Properties of One

Other Properties of One Rule 1: One is a factor of all numbers, and all numbers are multiples of 1 Rule 5: One is the only number with exactly 1 factor Rule 6: One is not a prime number. (The first prime number is 2)

True. 3. Properties of Numbers 3.3.1 Properties of zero

Other Properties of Zero Rule 7: Zero is the only number that is equal to its opposite. (0 = -0) Rule 8: All numbers are factors of zero (equivalently, zero is a multiple of all numbers). - If you divide zero by any non-zero number, the result will always be zero. In other words, any non-zero number can be multiplied by zero to give zero as the product. This is because zero multiplied by any number yields zero. Therefore, all numbers can be considered factors of zero, and zero can be seen as a multiple of all numbers. (ex. The first nonnegative multiple of 2 is 0, since 2 * 0 = 0) Definition of prime number: A whole number greater than 1 that cannot be exactly divided by any whole number other than itself and 1 (e.g. 2, 3, 5, 7, 11, etc.) Rule 9: Zero is the only number that is equal to all its multiples Rule 10: Zero is not a factor of any number except itself --> a is a factor of b means that b = a * k (for some integer k). Thus, since 0 = 0 * k, we see that 0 is a factor of 0 Rule 11: Any number (except zero) raised to the zero power is equal to 1 (the result is not zero). Rule 12: Zero is an even number

1. 200 2. First, note that digit in the hundredths place of 0.397 is 9 (0.397). Next, we find the digit directly to the right of the hundredths place, which we see is 7 (.0397). A 7 indicates that we round up by adding 1 to 9. Since this would produce 10, we instead make the 9 a 0 and add 1 to the digit to he left of 0. Thus the 3 in the tenths place becomes a 4. Then we remove the digits to the right of the hundredths place. Thus, 0.397 rounded to the nearest hundredth rounds up to 0.40.

Rounding Up When the Original Digit Is 9 1. What is 195 rounded to the nearest ten? 2. What is 0.397 rounded to the nearest hundreth?

True. 3. Properties of Numbers 3.13.8 Divisibility rules

T/F: Number divisible by 9: A number is divisible by 9 if the sum of all the digits is divisible by 9. For example, 479,655 is divisible by 9 because the sum of the digits (4 + 7 + 9 + ... + 5 = 36) is divisible by 9. Number divisible by 11: A number is divisible by 11 if the sum of the odd-numbered place digits minus the sum of the even-numbered place digits is divisible by 11. Ex) 253 is divisible by 11 because (2+3) - 5 = 0, which is divisible by 11.

True. 4. Roots and Exponents 4.2 The Square Root

T/F: The Square Root When the radical symbol ( √ ) is used, we consider the non-negative square root of the number The √(positive number) will be positive. The square root of a variable squared is equal to the absolute value of that variable. In other words, √(x^2) = abs(x) REMEMBER, this is different from something like the quadratic equation or looking at x^2 = 4 and seeing how x can be 2 or -2. IF YOU SEE A √, then you can't think like that. And REMEMBER, when doing √(x^2), the output is abs(x), not just x.

True. 2. Linear and Quadratic Equations 2.2.6 Choosing which method to use

T/F: A system of two linear equations can be solved by substitution or the combination method.

THIS IS JUST A REMINDER (YOU'VE BEEN THINKING LIKE THIS ALREADY) True. Let's say that in a DS problem, in which we were asked to determine the value of x, statement one allowed us to conclude that x = 1 or x = 0, while statement two allowed us to conclude that x = 1 or x = 5. Of course, neither statement one alone nor statement two alone would answer question. However, would the correct answer be C or E? If we are not careful, we may assume that because statement one and statement two both produced multiple answers fo x, using the statements together would also produce multiple values for x. Therefore, we may incorrectly choose option E. However, that assumption is not correct. Any time neither statement one alone nor statement two alone is sufficient to answer the question, but there is a single (unique) possible value shared between both statements, both statements together are sufficient to answer the question.

T/F: Any time neither statement one alone nor statement two alone is sufficient to answer the question, but there is a single (unique) value shared between both statements, then both statements together are sufficient to answer the question (choosing C instead of E)

True. 1. Essential GMAT Quant Skills 1.10.2 Squaring and. taking the square root of a number between 0 and 1

T/F: If 0 < x < 1, it must be true that x^2 < x < √(x)

True. 3. Properties of Numbers 3.15.3 A formula for division

T/F: If an integer x is the dividend (numerator), an integer y is the divisor (denominator), Q is the integer quotient of the division, and r is the nonnegative remainder of the division, then: x/y = Q + (r/y)

True. 3. Properties of Numbers 3.9.4 The number of unique prime factors in a number does not change when that number is raised to a positive integer exponent.

T/F: If some number x has y unique prime factors, then x^n (where n is some positive integer) will have the same y unique prime factors.

True. 3. Properties of Numbers 3.12.1 If we know the LCM and the GCF of two positive integers, x and y, we know the product of x and y.

T/F: If the LCM of x and y is p, and the GCF of x and y is q, then xy = pq. That is, xy = LCM(x,y) * GCF(x,y)

True. 3. Properties of Numbers 3.8 Multiples

T/F: If y does not equal 0, x is a multiple of y if and only if x/y is an integer. Ex: y = 2 2*0 = 0 2*1 = 2 2*2 = 4 etc. An infinite number of multiples exist for every number.

True. Ex. (3+4)/(9 times 4) = 7/36, not 1/3 1. Essential GMAT Quant Skills 1.16.1 The order of operations in fractions

T/F: In a fraction with addition and/or subtraction in the numerator and/or denominator, that operation must be completed prior to dividing the numerator by the denominator

True. 2. Linear and Quadratic Equations 2.8.1 Be cautious when an expression is set equal to zero

T/F: In an equation such as x(x+100) = 0, don't make the invalid assumption that x is not zero. That is, don't divide both sides of the equation by x to remove the x outside the parentheses. Instead, use the zero product property to solve the equation. In this case, it must be true that x = 0, or x + 100 = 0

True

T/F: In data sufficiency question, we are not being asked to calculate a numerical answer. Instead, we are being asked only whether we could produce a unique answer. Take your analysis only to the point at which you are sure you could or could not answer the question.

True. 3. Properties of Numbers 3.16. Determining the number of trailing zeros in a number 3.17 Leading Zero Stuff

T/F: In whole numbers, trailing zeros are created by (5x2) pairs. Each pair creates one trailing zeros. Thus, the number of trailing zeros of a number is the number of (5x2) pairs in the prime factorization of that number. Ex) 5^18 * 2^20 --> 18 trailing zeros Also, any factorial >= 5! will always have zero as its units digit. Leading Zeros If X is an integer with k digits, and if is not a perfect power (this is not the same thing as a multiple) of 10, then 1/X will have k-1 leading zeros. If X is an integer with k digits, and if X is a perfect power (not multiple) of 10, then 1/X will have k-2 leading zeros.

True. 3. Properties of Numbers 3.9 Prime numbers

T/F: Prime numbers have only two factors: 1 and themselves. Therefore, they are only divisible by two numbers. The number 2 is the only even prime number, and it is also the smallest prime number.

True. 3. Properties of Numbers 3.15.8 Adding and Subtracting Remainders

T/F: Remainders can be added or subtracted (but we must correct for any excess remainders at the end of the multiplication) in order to find remainders of larger numbers. Ex) 12 + 13 + 17 divided by 5 12/5 gives remainder 2 13/5 gives remainder 3 17/5 gives remainder 2 2 + 3 + 2 = 7. 7 - 5 = 2.

True. 3. Properties of Numbers 3.15.7 Multiplying Remainders

T/F: Remainders can be multiplied (but we must correct for any excess remainders at the end of the multiplication) in order to find remainders of larger numbers. Ex) 500 * 600 * 700 by 8 500/8 gives remainder 4 600/8 gives remainder 0 700/8 gives remainder 4 4*0*4 = 0, therefore, 500 * 600 * 700 by 8 has 0 remainder Another ex) 12 * 13 * 17 by 5 12/5 gives remainder 2 13/5 gives remainder 3 17/5 gives remainder 2 2*3*2 = 12 - 5 - 5 = 2, therefore 12 * 13 * 17 by 5 gives remainder 2

True. 3. Properties of Numbers 3.12.2 The LCM provides us with all the unique prime factors of some set of positive integers

T/F: The LCM of a set of positive integers provides us with all of the unique prime factors of the set. Thus, it provides all the unique prime factors of the product of the numbers in the set.

True. 3. Properties of Numbers 3.21 Terminating Decimals

T/F: The decimal equivalent of a fraction will terminate if an only if the denominator of the reduced fraction has a prime factorization that contains only 2s or 5s, or both. If the prime factorization of the reduced fraction's denominator contains anything other than 2s or 5s, the decimal equivalent will not terminate. Ex) 3/40 will terminate 5/60

True. For example, the principal square root of 16 = 4.

T/F: The non-negative square root of a number (the square root that is either zero or positive) is called the principal square root.

True. 3. Properties of Numbers 3.14.3 Products of Consecutive Even Integers

T/F: The product of n consecutive even integers will always be divisible by 2^n * n!

True (yeah yeah)

T/F: Two fractions a/b and c/d are equivalent fractions if... a*d = b*c

True. 2. Linear and Quadratic Equations 2.10.2 Equation Trap: When One Equation Is Sufficient To Determine Unique Values For Two Variables

T/F: We'll often be presented with problems involving a great number of variables than unique equations. For example, a problem may involve two variables but only one equation. Don't assume that the same number of unique equations as variables is required to determine the values of those variables. Sometimes, we'll be able to determine the values of a greater number of variables using fewer unique equations containing those variables. We'll encounter situations in which we're presented with two different variables and only one equation, yet using only that equation, we'll be able to determine the values of those variables.

True. Ex) 18,117^2. Look at the units digit. 1. Essential GMAT Quant Skills 1.13 Look for Clever, Time-Saving Ways to Solve Problems

T/F: When presented with a large number raised to the second power, if the answer choices all have unique units digits, then we can square just the units digit of the original number to determine the units digit of the full result. By doing so, we will determine the answer.

True. ex) If x^2 = 100x, what is x? You cannot divide by a variable unless you know that it's not zero (does it state it in the problem, like "x is positive" or "x is not zero" etc.) 2. Linear and Quadratic Equations 2.10.6 Equation trap: assuming the value of a variable cannot be zero.

T/F: You can divide by a variable ONLY when you know that the variables is not equal to zero. Similarly, you can divide by a variable expression, like (x+1), ONLY when you know that the expression is not equal to zero. x * 30/x, I know x can't be 0 in a particular problem, so I can cancel

Basically, it's like testing values (x = 1, x = 2, x = 3, etc.) and seeing the same answer (either all yes, all no, or all this unique value) and automatically concluding you are right because you haven't seen something that contradicts you. Therefore, it is important to test different types of numbers (ex. if looking at x^2, test not only positive integers but also fractions, negative numbers, etc.) In general, there are five major types of numbers that we should consider testing: 1) Positive integers 2) Positive proper fractions (fractions between 0 and 1) 3) Zero 4) Negative proper fractions (fractions between -1 and 0) 5) Negative integers Remember, be strategic with your numbers. Ex: If the questions states that "x is an integers," they don't test fractional values for x. It is also important to note that not all DS questions are most easily solved by testing strategic numbers

What is the Black Swan Trap?

Finding the GCF Step 1: Find the prime factorization of each integer Step 2: Of any repeated prime factors among the integers in the set, take only those with the smallest exponent (so if there is like 3^3 and 3^2, coming from different integer prime factorization trees, we choose 3^2). If we're left with two of the same power (ex. 3^2 and 3^2), just take that number once. If there are no repeated prime factors, the GCF is 1, not 0. Multiply the numbers together and that is the GCF Ex) LCM of 60 and 72 ONLY DO THIS IF THE NUMBERS ARE HARD

What is the process of finding the GCF of any set of positive integers? (ONLY DO THIS IF THE NUMBERS ARE HARD)

Finding the LCM Step 1: Find the prime factorization of each integer Step 2: Of any repeated prime factors among the integers in the set, take only those with the largest exponent (so if there is like 3^3 and 3^2, coming from different integer prime factorization trees, we choose 3^3). If we're left with two of the same power (ex. 3^2 and 3^2), just take that number once. Note: "Repeated" for the purpose of finding LCM means shows up more than once, Step 3: Of what is left, take all non-repeated prime factors of the integers and multiply together this with step 2. Result is the LCM. Ex) LCM of 24 and 60 is 24 = 2^3 * 3, 60 = 2^2 * 3* 5. Therefore, the set is {2^3, 3^1, 2^2, 3^1, 5^1} We now choose that largest exponents from the prime factors that are repeated across all integers, a.k.a the set (so 2^3 and 3^1, taking 3^1 only once) Multiply this by the remaining non-repeated prime factor of 5 Therefore, 24 * 5 = 120! ONLY DO THIS IF THE NUMBERS ARE HARD

What is the process of finding the LCM of any set of positive integers? (ONLY DO THIS IF THE NUMBERS ARE HARD)

Finding the Number of Factors of a Particular Number Step 1: Find the prime factorization of a number Step 2: Add 1 to the value of each exponent, and then multiply these results. The product will be the total number of factors of the number. Ex) 2160 Step 1: 2^4 * 3^3 * 5 Step 2: (4+1)(3+1)(1+1) = 5 * 4 * 2 = 40

What is the process of finding the total number of factors that a number has? Side Note: # of prime factors vs # of unique prime factors ex) 32 = 2 * 2 * 2 * 2 * 2 32 has 5 prime factors and 1 unique prime factor

Initial Answer: E Correct Answer: C 0. Intro to Quant Section 0.9 Triangulating Between Two Statements in Data Sufficiency Problems Don't make the mistake of concluding that we cannot determine a unique value for x. Noice that only 5 is common to both statements. Since we have determined a single value for x, the statements together are sufficient REMEMBER, THINK OF THE STATEMENTS AS... a unique value from A n B and not A U B

What is the value of x? 1) x = 4 or x = 5 2) x = 5 or x = 6