FOUNDATIONS OF MATH I In-Class Drills
FOUNDATIONS OF MATH I In-Class Drills Contents Page Warm Up: Order of Operations 2 Solutions to Warm Up 3 Equations Principles 4 Rules and Tips 5 Drills 6-7 Games (Divisibility and Exponents) Principles 8 Rules and Tips 9 Drills 10 Official Guide Problem Sets Problem Solving 11 Data Sufficiency 12
Copyright © 2007 MG Prep, Inc.
WARM UP: ORDER OF OPERATIONS The order of operations helps us determine the order in which we should perform computations that simplify a term. The six operations in the correct order are Parentheses, Exponents, Multiplication/Division and Addition/Subtraction. Parentheses Exponents Multiplication or Division
Addition or Subtraction
Simplify anything inside parentheses first. Apply exponents. For instance, you might square a number: 32 Multiply or divide. Remember that multiplication can be written in several ways: 3×2 = 3(2) = (3)2 = 6. 6 Also, division can be represented by fractions: 6 ÷ 3 = = 2. 3
There are “hidden parentheses” on the top and bottom of fractions: 6 + 9 = ( 6 + 9 ) ÷ 3 = 15 ÷ 3 3 Add or subtract numbers.
PEMDAS is a useful acronym you can use to remember the order in which operations should be performed. If you have two operations at the same level of importance, you should just do them in left-to-right order: 3 – 2 + 3 = 1 + 3 = 4. To override this order, you need to have parentheses: 3 – (2 + 3) = -2. Simplify the following expressions to a single number: 1.
7(3 + 2) – 9 =
P E M/D A/S
3. P E M/D A/S
2.
25 – 5(3 – 1)2 =
P E M/D A/S 3+1 9 − 3 + = 2 3
4.
3(2.3 + 1.7) − 4(2.3 − 1.3) =
P E M/D A/S
2
SOLUTIONS TO WARM UP: ORDER OF OPERATIONS 1.
7(3 + 2) – 9 =
2.
25 – 5(3 – 1)2 =
P E M/D A/S
7(5) – 9
P E M/D A/S
25 – 5(2)2 25 – 5(4) 25 - 20 5
35 – 9 26
7(3 + 2) – 9 = 26
3. P E M/D A/S
3+1 9 − 3 + = 2 3
4 6 + 2 3 2+2 4
3+1 9 − 3 + =4 2 3
25 – 5(3 – 1)2 = 5
4.
3(2.3 + 1.7) − 4(2.3 − 1.3) =
P E M/D A/S
3(4.0) – 4(1) 12 – 4 8
3(2.3 + 1.7) – 4(2.3 – 1.3) = 8
3
EQUATIONS Three basic skills serve as the foundation for successfully solving a variety of GMAT quantitative problems: 1) Solving for a variable 2) Combining equations together 3) Turning words into numerical relationships In order to comprehend, compute, and solve challenging problems in two minutes or less, these three skills should be automatic. Let’s start from the ground up… Solving for a variable: In order to solve for a variable, simply isolate the variable on one side of the equation. Get rid of numbers attached to the variable by reversing the original operations (for example, in order to isolate x in x + 5 = 7, you should subtract 5). In many ways, isolating a variable is similar to unraveling a problem. If you become unsure of the order in which you should work, remember to go in reverse of the order of operations, PEMDAS. First, get rid of numbers that are being Subtracted from or Added to the variable. Then get rid of numbers that the variable is being Divided or Multiplied by. Then get rid of Exponents. Finally, get rid of Parentheses. Additionally, you should take three other steps whenever the opportunity presents itself: simplify “blocks” of work within the equation, always combine like-terms, and always try to get variables out of denominators. Combining equations: Students new to the GMAT often make the mistake of trying to combine all the information in a long problem into one equation. Many problems are not built to be solved this way. It is often much easier to create simple equations first, because combining them later is generally not difficult. There are two common methods for combining equations: substitution, which you will use far more frequently, and elimination. The goal of both is the same: to end up with one equation with one variable. Turning words into numeric relationships: GMAT problems are often worded in a way that makes it difficult to translate them into numeric relationships. An important initial step is to correctly identify and label the unknowns. The rest of the problem is there to tell you something about the relationship between these unknowns. Be on the lookout for two common relationships: two parts that equal one another (tip: look for all forms of the word “is”), and two parts that add to a total. Checking your work: Successful test takers have a variety of efficient and effective methods for checking their work. These test takers know how to (1) estimate, (2) recognize limiting number properties, (3) eliminate unreasonable answer choices, and (4) plug in numbers. In addition, you should be comfortable walking back through a problem with your solution to make sure everything makes sense. Performed properly, this step will help you catch most simple computation errors. 4
RULES AND TIPS
EQUATIONS
Solving For A Variable
Combining Equations
Isolate a variable by reversing the order of operations (PEMDAS).
5(x − 1)3 − 30 = 10 A/S M/D E P
3
5(x – 1) = 40 (x – 1)3 = 8 (x – 1) = 2 x=3
Tip: Always combine in the easiest way possible, regardless of what variable you are asked to solve for.
If 3x + y = 10, and y = x – 2, y = ? Substitute from one equation into the other: 3x + (x – 2) = 10
While simultaneously performing three other actions whenever you can: (1) Simplify already-combined terms (2) Combine common variables (3) Cross-multiply to eliminate denominators
4x + 3 3 + 1 + =5 2x − 1 2 (1)
4x + 3 +2=5 2x − 1
A/S
4x + 3 =3 2x − 1
(3)
4x + 3 = 3(2x − 1)
(1)
4x + 3 = 6x − 3
(2)
3 = 2x − 3
A/S
6 = 2x
M/D
3= x
Solve: 4x – 2 = 10 4x = 12 x = 3. If x = 3, y = (3) – 2 = 1. Sometimes it is necessary to first isolate a variable before you substitute:
If 3x + y = 10, and y – 2x = - 5, x = ? Isolate: y = 2x- 5 Substitute: 3x + (2x – 5) = 10 Solve: 5x – 5 = 10 5x = 15 x=3 Sometimes you can eliminate a variable by adding or subtracting entire equations from one another. 3x + y = 10 + 2x – y = 5 5x = 15 x=3
Turning Words Into Numeric Relationships Common Terms: is, was, were, will be, same difference, less of, times, product average of x and y
= × (x + y)/2
total, sum, add… y less than x quotient, proportion ratio of x to y
+ x-y x/y x/y
5
EQUATIONS DRILL #1 1. h + k = 40, k = h + 18
2.
30t + 50 = 25, q = t + 5 q
3. 7x + 4y = 43, y + 8 = 2x
4. 3k − 2z = 16, 2z = 2k − 12
6
EQUATIONS DRILL #2 1. * There are four more women than men on Centerville’s board of education. If there are ten members on the board, how many are women? Variables: Equations: Combine and Solve: Check your work: 2. If Sam were twice as old as he is, he would be 40 years older than Jim. If Jim is 10 years younger than Sam, how old is Sam? Variables: Equations: Combine and Solve: Check your work: 3.* The average of 10, 30, and 50 is 5 more than the average of x, 20, and 40. What number is x? Variables: Equations: Combine and Solve: Check your work: *This problem is reprinted or slightly adapted from the Official Guide for GMAT Review (11th Edition or Math Supplement), published by the Graduate Management Admission Council.
7
GAMES Divisibility Questions require you to understand numbers in terms of their basic multiplicative building blocks. Problems that include the words “divisible by,” “factor,” and “multiple” are generally divisibility questions. The most basic multiplicative building blocks of a number are known as prime factors. The factors of that number are all the terms that divide into the number cleanly. Factors can be created by multiplying any combination of prime factors together. If the question asks for a total number of factors, you must always also count “1” as a factor. GMAT divisibility questions commonly work in reverse; instead of giving you a number, they give you the factors, and ask what you can determine about the number from these factors. In more difficult problems, factors often give overlapping information about the building blocks of your number (the prime factors), because the same building blocks can be used several times to create various factors. Exponents are simply shorthand for multiplying or dividing the same number by itself multiple times. Exponential terms consist of a base (the number being multiplied) and the exponent (the number of times the number is being multiplied). Exponential terms can only be combined if they have a common base or a common exponent. Common base problems are, for the lack of a better word, more common on the GMAT. It is often necessary to change the base of a term in order to have the common bases necessary to combine terms. The rules for combining exponents, like the rules of other types of shorthand, are difficult to comprehend and memorize unless you understand the logic behind them. When in doubt, think about exponent rules by writing out the multiplication or division that is involved. The fundamental rules of exponents involve basic multiplication and division. When multiplying exponential terms with common bases, you should add the exponents. When dividing exponential terms with common bases, you should subtract the exponents. Most other exponent rules are derived from these two.
8
RULES AND TIPS
GAMES Divisibility Example: x = 45
Prime Factors (the basic multiplicative building blocks): 3 × 3 × 5 Factors (all the different numbers you can form by combining the prime factors): 1, 3, 5, 9, 15, 45 1
3
5
3×3=9
3 × 5 = 15
3 × 3 × 5 = 45
Don’t forget that 1 is always a factor! Note: A number that doesn’t have any factors other than 1 and itself is a prime number.
Exponents a5 × a3 = (aaaaa) × (aaa) = a8
a 5 aaaaa = = a2 3 a aaa
(a5)3 = aaaaa × aaaaa × aaaaa = a15
a0 = 1
a −2 =
1 1 = 2 aa a
6a × 3a = 18a
1 2
a = a
6a = 2a 3a
9
GAMES DRILL 1. List the prime factors and factors for the number 36. Prime factors: Factors: 2. If x is divisible by 6 and 9, then x must be a multiple of I. 3
II. 18
III. 27
(A) I only (B) I and II only (C) II and III only (D) I and III only (E) I, II and III 3. Combine the following into one exponential term. A. 5 4 5 2 5 −1 =
g12 B. 4 3 = (g )
C. 3295 =
D. 58 × 28 =
4. If 103x = 1000y, x = A. y B. y/2 C. y/3 D. 3y E. y - 3
10
Official Guide Problem Solving Set Equations Combining Equations and Solving for a Variable 2015/13th Edition: 14, 42, 47, 54, 55, 72, 102, 187, 220 Quantitative Review: 2, 40, 41, 99, 107, 111 Word Problems 2015/13th Edition: 1, 4, 29, 60, 64, 76, 83, 88, 89, 93, 131, 137, 140, 153, 154, 167, 184, 203, 205 Quantitative Review: 3, 13, 19, 25, 51, 52, 54, 62, 75, 94, 115, 124, 126, 127, 171 Games Divisibility 2015/13th Edition: 2, 5, 26, 40, 74, 77, 87, 95, 110, 116, 118, 127, 155, 174, 204, 219 Quantitative Review: 68, 78, 98, 109, 112, 122, 125, 149, 164, 169, 172 Exponents 2015/13th Edition: 106, 150, 164, 180, 196, 217, 230 Quantitative Review: 47, 74, 86, 96, 106, 108, 147, 163, 166, 170
11
Official Guide Data Sufficiency Set Equations Combining Equations and Solving for a Variable 2015/13th Edition: 1, 17, 36, 60, 86, 98, 136, 156, 171 Quantitative Review: 15, 23, 35, 57, 61, 80, 94, 106, 124 Word Problems 2015/13th Edition: 9, 28, 44, 54, 57, 59, 65, 71, 78, 124, 126, 132, 141, 142, 147, 153, 158, 174 Quantitative Review: 6, 7, 9, 12, 13, 17, 26, 27, 29, 33, 84, 97, 108 Games Divisibility 2015/13th Edition: 58, 83, 101, 135 Quantitative Review: 3, 16, 39, 45, 64, 70, 82, 87, 90, 92, 115 Exponents 2015/13th Edition: 41, 53, 160, 162, 169, 172 Quantitative Review: 18, 25, 28, 54, 76, 79, 81, 100, 121
12
Copyright © 2007 MG Prep, Inc.
WARM UP: ORDER OF OPERATIONS The order of operations helps us determine the order in which we should perform computations that simplify a term. The six operations in the correct order are Parentheses, Exponents, Multiplication/Division and Addition/Subtraction. Parentheses Exponents Multiplication or Division
Addition or Subtraction
Simplify anything inside parentheses first. Apply exponents. For instance, you might square a number: 32 Multiply or divide. Remember that multiplication can be written in several ways: 3×2 = 3(2) = (3)2 = 6. 6 Also, division can be represented by fractions: 6 ÷ 3 = = 2. 3
There are “hidden parentheses” on the top and bottom of fractions: 6 + 9 = ( 6 + 9 ) ÷ 3 = 15 ÷ 3 3 Add or subtract numbers.
PEMDAS is a useful acronym you can use to remember the order in which operations should be performed. If you have two operations at the same level of importance, you should just do them in left-to-right order: 3 – 2 + 3 = 1 + 3 = 4. To override this order, you need to have parentheses: 3 – (2 + 3) = -2. Simplify the following expressions to a single number: 1.
7(3 + 2) – 9 =
P E M/D A/S
3. P E M/D A/S
2.
25 – 5(3 – 1)2 =
P E M/D A/S 3+1 9 − 3 + = 2 3
4.
3(2.3 + 1.7) − 4(2.3 − 1.3) =
P E M/D A/S
2
SOLUTIONS TO WARM UP: ORDER OF OPERATIONS 1.
7(3 + 2) – 9 =
2.
25 – 5(3 – 1)2 =
P E M/D A/S
7(5) – 9
P E M/D A/S
25 – 5(2)2 25 – 5(4) 25 - 20 5
35 – 9 26
7(3 + 2) – 9 = 26
3. P E M/D A/S
3+1 9 − 3 + = 2 3
4 6 + 2 3 2+2 4
3+1 9 − 3 + =4 2 3
25 – 5(3 – 1)2 = 5
4.
3(2.3 + 1.7) − 4(2.3 − 1.3) =
P E M/D A/S
3(4.0) – 4(1) 12 – 4 8
3(2.3 + 1.7) – 4(2.3 – 1.3) = 8
3
EQUATIONS Three basic skills serve as the foundation for successfully solving a variety of GMAT quantitative problems: 1) Solving for a variable 2) Combining equations together 3) Turning words into numerical relationships In order to comprehend, compute, and solve challenging problems in two minutes or less, these three skills should be automatic. Let’s start from the ground up… Solving for a variable: In order to solve for a variable, simply isolate the variable on one side of the equation. Get rid of numbers attached to the variable by reversing the original operations (for example, in order to isolate x in x + 5 = 7, you should subtract 5). In many ways, isolating a variable is similar to unraveling a problem. If you become unsure of the order in which you should work, remember to go in reverse of the order of operations, PEMDAS. First, get rid of numbers that are being Subtracted from or Added to the variable. Then get rid of numbers that the variable is being Divided or Multiplied by. Then get rid of Exponents. Finally, get rid of Parentheses. Additionally, you should take three other steps whenever the opportunity presents itself: simplify “blocks” of work within the equation, always combine like-terms, and always try to get variables out of denominators. Combining equations: Students new to the GMAT often make the mistake of trying to combine all the information in a long problem into one equation. Many problems are not built to be solved this way. It is often much easier to create simple equations first, because combining them later is generally not difficult. There are two common methods for combining equations: substitution, which you will use far more frequently, and elimination. The goal of both is the same: to end up with one equation with one variable. Turning words into numeric relationships: GMAT problems are often worded in a way that makes it difficult to translate them into numeric relationships. An important initial step is to correctly identify and label the unknowns. The rest of the problem is there to tell you something about the relationship between these unknowns. Be on the lookout for two common relationships: two parts that equal one another (tip: look for all forms of the word “is”), and two parts that add to a total. Checking your work: Successful test takers have a variety of efficient and effective methods for checking their work. These test takers know how to (1) estimate, (2) recognize limiting number properties, (3) eliminate unreasonable answer choices, and (4) plug in numbers. In addition, you should be comfortable walking back through a problem with your solution to make sure everything makes sense. Performed properly, this step will help you catch most simple computation errors. 4
RULES AND TIPS
EQUATIONS
Solving For A Variable
Combining Equations
Isolate a variable by reversing the order of operations (PEMDAS).
5(x − 1)3 − 30 = 10 A/S M/D E P
3
5(x – 1) = 40 (x – 1)3 = 8 (x – 1) = 2 x=3
Tip: Always combine in the easiest way possible, regardless of what variable you are asked to solve for.
If 3x + y = 10, and y = x – 2, y = ? Substitute from one equation into the other: 3x + (x – 2) = 10
While simultaneously performing three other actions whenever you can: (1) Simplify already-combined terms (2) Combine common variables (3) Cross-multiply to eliminate denominators
4x + 3 3 + 1 + =5 2x − 1 2 (1)
4x + 3 +2=5 2x − 1
A/S
4x + 3 =3 2x − 1
(3)
4x + 3 = 3(2x − 1)
(1)
4x + 3 = 6x − 3
(2)
3 = 2x − 3
A/S
6 = 2x
M/D
3= x
Solve: 4x – 2 = 10 4x = 12 x = 3. If x = 3, y = (3) – 2 = 1. Sometimes it is necessary to first isolate a variable before you substitute:
If 3x + y = 10, and y – 2x = - 5, x = ? Isolate: y = 2x- 5 Substitute: 3x + (2x – 5) = 10 Solve: 5x – 5 = 10 5x = 15 x=3 Sometimes you can eliminate a variable by adding or subtracting entire equations from one another. 3x + y = 10 + 2x – y = 5 5x = 15 x=3
Turning Words Into Numeric Relationships Common Terms: is, was, were, will be, same difference, less of, times, product average of x and y
= × (x + y)/2
total, sum, add… y less than x quotient, proportion ratio of x to y
+ x-y x/y x/y
5
EQUATIONS DRILL #1 1. h + k = 40, k = h + 18
2.
30t + 50 = 25, q = t + 5 q
3. 7x + 4y = 43, y + 8 = 2x
4. 3k − 2z = 16, 2z = 2k − 12
6
EQUATIONS DRILL #2 1. * There are four more women than men on Centerville’s board of education. If there are ten members on the board, how many are women? Variables: Equations: Combine and Solve: Check your work: 2. If Sam were twice as old as he is, he would be 40 years older than Jim. If Jim is 10 years younger than Sam, how old is Sam? Variables: Equations: Combine and Solve: Check your work: 3.* The average of 10, 30, and 50 is 5 more than the average of x, 20, and 40. What number is x? Variables: Equations: Combine and Solve: Check your work: *This problem is reprinted or slightly adapted from the Official Guide for GMAT Review (11th Edition or Math Supplement), published by the Graduate Management Admission Council.
7
GAMES Divisibility Questions require you to understand numbers in terms of their basic multiplicative building blocks. Problems that include the words “divisible by,” “factor,” and “multiple” are generally divisibility questions. The most basic multiplicative building blocks of a number are known as prime factors. The factors of that number are all the terms that divide into the number cleanly. Factors can be created by multiplying any combination of prime factors together. If the question asks for a total number of factors, you must always also count “1” as a factor. GMAT divisibility questions commonly work in reverse; instead of giving you a number, they give you the factors, and ask what you can determine about the number from these factors. In more difficult problems, factors often give overlapping information about the building blocks of your number (the prime factors), because the same building blocks can be used several times to create various factors. Exponents are simply shorthand for multiplying or dividing the same number by itself multiple times. Exponential terms consist of a base (the number being multiplied) and the exponent (the number of times the number is being multiplied). Exponential terms can only be combined if they have a common base or a common exponent. Common base problems are, for the lack of a better word, more common on the GMAT. It is often necessary to change the base of a term in order to have the common bases necessary to combine terms. The rules for combining exponents, like the rules of other types of shorthand, are difficult to comprehend and memorize unless you understand the logic behind them. When in doubt, think about exponent rules by writing out the multiplication or division that is involved. The fundamental rules of exponents involve basic multiplication and division. When multiplying exponential terms with common bases, you should add the exponents. When dividing exponential terms with common bases, you should subtract the exponents. Most other exponent rules are derived from these two.
8
RULES AND TIPS
GAMES Divisibility Example: x = 45
Prime Factors (the basic multiplicative building blocks): 3 × 3 × 5 Factors (all the different numbers you can form by combining the prime factors): 1, 3, 5, 9, 15, 45 1
3
5
3×3=9
3 × 5 = 15
3 × 3 × 5 = 45
Don’t forget that 1 is always a factor! Note: A number that doesn’t have any factors other than 1 and itself is a prime number.
Exponents a5 × a3 = (aaaaa) × (aaa) = a8
a 5 aaaaa = = a2 3 a aaa
(a5)3 = aaaaa × aaaaa × aaaaa = a15
a0 = 1
a −2 =
1 1 = 2 aa a
6a × 3a = 18a
1 2
a = a
6a = 2a 3a
9
GAMES DRILL 1. List the prime factors and factors for the number 36. Prime factors: Factors: 2. If x is divisible by 6 and 9, then x must be a multiple of I. 3
II. 18
III. 27
(A) I only (B) I and II only (C) II and III only (D) I and III only (E) I, II and III 3. Combine the following into one exponential term. A. 5 4 5 2 5 −1 =
g12 B. 4 3 = (g )
C. 3295 =
D. 58 × 28 =
4. If 103x = 1000y, x = A. y B. y/2 C. y/3 D. 3y E. y - 3
10
Official Guide Problem Solving Set Equations Combining Equations and Solving for a Variable 2015/13th Edition: 14, 42, 47, 54, 55, 72, 102, 187, 220 Quantitative Review: 2, 40, 41, 99, 107, 111 Word Problems 2015/13th Edition: 1, 4, 29, 60, 64, 76, 83, 88, 89, 93, 131, 137, 140, 153, 154, 167, 184, 203, 205 Quantitative Review: 3, 13, 19, 25, 51, 52, 54, 62, 75, 94, 115, 124, 126, 127, 171 Games Divisibility 2015/13th Edition: 2, 5, 26, 40, 74, 77, 87, 95, 110, 116, 118, 127, 155, 174, 204, 219 Quantitative Review: 68, 78, 98, 109, 112, 122, 125, 149, 164, 169, 172 Exponents 2015/13th Edition: 106, 150, 164, 180, 196, 217, 230 Quantitative Review: 47, 74, 86, 96, 106, 108, 147, 163, 166, 170
11
Official Guide Data Sufficiency Set Equations Combining Equations and Solving for a Variable 2015/13th Edition: 1, 17, 36, 60, 86, 98, 136, 156, 171 Quantitative Review: 15, 23, 35, 57, 61, 80, 94, 106, 124 Word Problems 2015/13th Edition: 9, 28, 44, 54, 57, 59, 65, 71, 78, 124, 126, 132, 141, 142, 147, 153, 158, 174 Quantitative Review: 6, 7, 9, 12, 13, 17, 26, 27, 29, 33, 84, 97, 108 Games Divisibility 2015/13th Edition: 58, 83, 101, 135 Quantitative Review: 3, 16, 39, 45, 64, 70, 82, 87, 90, 92, 115 Exponents 2015/13th Edition: 41, 53, 160, 162, 169, 172 Quantitative Review: 18, 25, 28, 54, 76, 79, 81, 100, 121
12
