GMATFoundations of MathLevel II

Foundations of Math II

In-Class Drills

CONTENTS PAGE(S)

Warm Up: Equations 2

Fractions, Decimals, & Percents

Principles 3

Rules and Tips 4

Drills 5

Geometry

Principles 6

Rules and Tips 7

Drills 8

Official Guide Problem Sets

Problem Solving 9-10

Data Sufficiency 11

2

WARM UP: EQUATIONS Remember the following principles of solving equations: 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). Combining equations: 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: 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: Be comfortable walking back through a problem with your solution to make sure everything makes sense. Also be ready to (1) estimate, (2) recognize limiting number properties, (3) eliminate unreasonable answer choices, and (4) plug in numbers Solve the following problems: 1. 3x3 + 10 = 34 x = ? 2. 2z – 5y = 3 y = z – 3 z = ? y = ? 3. Justin has five more pencils than highlighters. If he counts both pencils and highlighters, he has a total of thirteen writing instruments. How many pencils does he have?

3

FRACTIONS, DECIMALS & PERCENTS

Fraction, decimal and percent problems (FDP’s) that only require computation generally test just a few skills: your ability to add, subtract, multiply and divide fractions and decimals, and your ability to compare the value of different fractions and decimals. Word problems involving fractions and decimals generally work with this formula:

Part = Fraction (or decimal) Whole.

Advanced GMAT word problems tend to deliver this information in a disorganized fashion, and often you are required to perform multiple computations. However, the good news is that very few of these GMAT word problems go beyond this simple formula. Fractions: Most fraction problems on the GMAT require that you change the denominator of at least one fraction involved in the question before you add, subtract, or compare values. You can change the denominator of a fraction without changing the fraction’s overall value by multiplying or dividing the top and bottom by a common term. In general, you will need to divide the numerator and denominator by a common term in order to simplify a fraction. Also, you will need to multiply the numerator and denominator by a common term in order to get the common denominators necessary for addition, subtraction, or comparison. Many problems require you to simplify your answer to a lowest possible denominator, or to convert your answer into a decimal or percent. You can convert a fraction to a decimal by dividing the top by the bottom (using long division). Some fraction problems are about general principles, rather than computations. If the top (numerator) of a fraction increases while the bottom (denominator) stays constant, the number will get larger. If the top decreases while the bottom stays constant, the number will get smaller. In contrast, if the top stays constant, but the bottom increases, the number will get smaller. Likewise, if the top stays constant, but the bottom decreases, the number will get larger. Decimals and Percents: All percents should be converted into decimals before you perform any computations. Many questions require that you then reconvert from a decimal to a percent for the final answer. You can change a percent to a decimal by moving the decimal point two spots to the left; you can convert a decimal into a percent by moving the decimal point two spots to the right. Many FDP word problems can be made simpler by plugging in numbers. For fraction problems, pick numbers that are multiples of the denominators involved. For decimal problems, try to pick 100 as your starting point.

4

RULES AND TIPS FRACTIONS, DECIMALS AND PERCENTS

Fractions

Alteration: Change the denominator of a fraction without changing the fraction’s value by multiplying or dividing both the numerator and denominator by a common term.

24 24 12 2

36 36 12 3

1 1 1(2) 1 2 1 3

2 4 2(2) 4 4 4 4

Addition / Subtraction:

Only add or subtract terms that have a common denominator. Add or subtract the numerators while leaving the denominators constant.

2 1 3

4 4 4

2 1 1

4 4 4

Multiplication: Multiply the numerators together and the denominators together.

2 1 2

4 4 16

Division: Flip the bottom fraction and multiply.

33 2 6 14

9 4 9 36 6

2

Decimals and Percents

Alteration: Convert percents into decimals for all calculations. Many questions require you to

reconvert your decimal into a percent for the answer. Convert a percent into a decimal by shifting the decimal point two spots to the left; convert a decimal into a percent by shifting the decimal point two spots to the right.

80% → 0.8 0.06 → 6%

Addition / Subtraction:

Line up decimal points before performing calculations. 0.735 – 0.027 0.708

Multiplication: Ignore the decimals until you are done with the calculation. At that point, move the decimal point to the left one spot for every digit that was initially behind the decimal points.

0.0003 5.3 (ignoring the decimals) becomes →

3 53 = 159 but there were initially 5 digits behind the decimal points, so we move

the decimal five spots to the left → =0.00159

Division: To simplify a decimal division problem, move the decimal point in the same direction for both the dividend and the divisor.

0.03 divided into 5.1 → 3 divided into 510 = 170

5

FDP DRILL #1 1. 3 5 6

7 7 7

2. 3 1 1

4 2 3

3. 1

62

23

4. 2 4

(9) (15)3 5

5. 0.6 120 6. 15 0.05

7. 0.001 0.0003

8. 1/10 of 25% of 200 =

FDP DRILL #2 1. 25% of the students at a high school attended the pep rally. If there are 600 students total, how many students did not attend the pep rally? 2. After taxes, Sally keeps 4/5 of her gross pay. If her monthly taxes are $1400, what is her monthly gross pay? 3. Two jars that are both half full of liquid are poured into an empty bucket. If the bucket has three times the capacity of each jar, what portion of the bucket is now full? 4. As a salesperson, Phyllis can choose one of two methods of annual payment: either an annual salary of $35,000 with no commission or an annual salary of $10,000 plus a 20 percent commission on her total annual sales. What must her total annual sales be to give her the same annual pay with either method?*

*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.

6

GEOMETRY: CIRCLES, TRIANGLES AND RECTANGLES Circles: There are four main ways to measure a circle:

1) The radius represents the distance from the center of a circle to any point on its edge.

2) The diameter is the distance across the widest part of the circle. 3) The circumference is the perimeter of the circle (the distance around). 4) The area represents the space within the circle.

All four of these measurements are related to one another and can be derived from one another. Most GMAT circle problems require that you derive information about one measurement from another given measurement. A sector is one “pizza” slice of the circle. In order to determine the area of a sector or its arc length (the length of the sector’s “crust” or curved outer edge), first calculate circumference or area for the entire circle, and then multiply that by whatever proportion the sector represents. Often, the proportion of the sector relative to the entire circle can be determined by looking at the angle of the “pizza slice” – that is, the angle the sector makes with the center of the circle (also known as arc measure). If you compare this angle to 360, you will find the proportion of the sector. Triangles: Triangles are the most common polynomials on the exam. Additionally, problems that initially seem to be about other polynomials can often be solved using triangles. In a triangle, the length of any two sides must always be greater than the third. All angles must add up to 180, and the longest sides will always be opposite the bigger angles. In isosceles and equilateral triangles, equal sides are opposite equal angles. The area of a triangle is half the product of the triangle’s base and height. The most useful and commonly tested triangle is the right triangle. In a right triangle, the three side lengths must fit into a formula known as the Pythagorean Theorem: a2 + b2 = c2, where a and b represent the two smaller lengths and c represents the length of the longest side, the hypotenuse. Using this formula, you can always derive one length of the triangle from the other two. Trigonometry is the study of the exact relationship between the sides and angles of a triangle. The only trigonometry on this exam involves triangles with either angles of 30, 60 and 90, or 45, 45 and 90. The ratio of side lengths in these triangles is constant; therefore, if you know the length of one side, you can calculate the other two. Rectangles: Rectangles are the most common quadrilaterals on the exam. In a rectangle all

angles are 90, and opposite sides are equal in length. The area of a rectangle is l w, and its perimeter is 2(l + w). The area formulas for other quadrilaterals on this exam (square,

trapezoid, parallelogram) can be derived from the formula for the rectangle.

7

RULES AND TIPS GEOMETRY: CIRCLES, TRIANGLES AND RECTANGLES

Diameter = 2r Circumference = 2πr Area = πr2

The sum of two sides of a triangle is always greater than the length of the third side. The longest sides are opposite the longest angles. (Equal sides opposite equal angles). All angles add up to 180. The area is.

a2 + b2 = c2 Common triples: 3-4-5 5-12-13 8-15-17 Trigonometric Ratios: 30-60-90 1::2 45-45-90 1: 1:

Area = lw Perimeter = 2(l+w)

1 2

2

b bA h

r (radius)

If = 60, the sector would represent 60/360 = 1/6 of the entire circle. Then, Area of sector = (1/6) (πr2) Arc length = (1/6) (2πr)

b

h b

a

c

l

w

s

s

A = s2

b2

b1

h h A = bh

8

GEOMETRY DRILL 1. Fill in the missing values for the following circles: Diameter Radius Area Circumference Circle 1 12 Circle 2 5 Circle 3 16π Circle 4 20 π 2. Circle o has been split into eight equal pieces. If the area of the circle is 16π, what is the length of arc AB?

3. An artist wishes to paint a circular region on a square poster that is 2 feet on a side. If the area of the circular region is to be ½ the area of the poster, what must be the radius of the circular region in feet?* 4. A certain rectangular backyard has an area of 48 sq. feet and a perimeter of 28 feet. What is the distance from one corner of the yard to the opposite corner? *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.

A

B

9

OFFICIAL GUIDE SET: PROBLEM SOLVING

Note: GMAT Quant and Verbal Review supplemental guides are not included with enrollment in Manhattan Prep complete courses or on demand products. However, we have included problem numbers from the supplemental guides in this document for students who have purchased the books separately and would like to do additional review.

Fractions, Decimals and Percents Computations

2015/13th Edition: 17, 20, 21, 27, 31, 41, 46, 84, 85, 97, 122, 146, 156 2015/13th Quantitative Review: 6, 8, 9, 38, 39, 42, 46, 48, 50, 53, 60, 65, 88, 93, 176

2016 Edition: 22, 26, 28, 34, 36, 46, 49, 104, 105, 119, 146, 171, 178 2016 Quantitative Review: 16, 18, 23, 39, 40, 43, 45, 47, 50, 53, 60, 65, 90, 100, 176 2017 Edition: 3, 28, 34, 41, 52, 55, 69, 117, 118, 135, 164, 185, 191 2017 Quantitative Review: 20, 22, 39, 40, 42, 43, 46, 54, 58, 93, 101, 176

Word Problems

2015/13th Edition: 8, 11, 15, 19, 57, 58, 59, 65, 71, 94, 108, 114, 115, 123, 135, 141, 142, 144, 151, 194, 212 2015/13th Quantitative Review: 26, 34, 35, 49, 59, 61, 69, 73, 89, 95, 100, 101, 114, 118, 154, 155

2016 Edition: 6, 9, 19, 24, 62, 64, 65, 78, 90, 116, 130, 135, 138, 147, 160, 166, 167, 169, 175, 203, 216 2016 Quantitative Review: 30, 36, 37, 49, 59, 61, 64, 74, 91, 102, 106, 107, 118, 122, 156, 162 2017 Edition: 9, 12, 25, 30, 73, 75, 87, 102, 133, 147, 153, 156, 165, 176, 181, 182, 189, 218 2017 Quantitative Review: 45, 51, 52, 55, 69, 103, 106, 107, 118, 122, 156, 163

Geometry Circles

2015/13th Edition: 36, 69, 213 2015/13th Quantitative Review: 33, 145, 153, 162

2016 Edition: 41, 81, 217 2016 Quantitative Review: 35, 141, 154, 167 2017 Edition: 90, 127, 219 2017 Quantitative Review: 36, 140, 154, 169

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Triangles and Rectangles 2015/13th Edition: 75, 92, 206 2015/13th Quantitative Review: 44, 71, 76, 150, 157

2016 Edition: 94, 113, 210 2016 Quantitative Review: 44, 73, 76, 149, 164 2017 Edition: 106, 126, 212 2017 Quantitative Review: 65, 72, 150, 157

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OFFICIAL GUIDE SET: DATA SUFFICIENCY

Fractions, Decimals and Percents Computations

2015/13th Edition: 31, 46, 63, 92 2015/13th Quantitative Review: 2, 48

2016 Edition: 35, 58, 76, 107 2016 Quantitative Review: 13, 62 2017 Edition: 269, 298, 319, 352 2017 Quantitative Review: 193, 245

Word Problems 2015/13th Edition: 10, 25, 29, 40, 51, 55, 62, 76, 91, 94, 116, 157 2015/13th Quantitative Review: 5, 44, 49, 50, 53, 75, 93, 96, 102

2016 Edition: 5, 28, 33, 52, 63, 69, 75, 95, 106, 109, 128, 163 2016 Quantitative Review: 17, 65, 68, 69, 72, 88, 102, 104, 108 2017 Edition: 234, 253, 263, 290, 303, 311, 318, 339, 351, 372 2017 Quantitative Review: 197, 248, 250, 251, 254, 268, 278, 279, 282

Geometry Circles

2015/13th Edition: 35, 100, 102, 117, 122 2015/13th Quantitative Review: 58, 59, 95, 99

2016 Edition: 44, 116, 118, 129, 133 2016 Quantitative Review: 76, 77, 103, 105 2017 Edition: 360, 362, 373, 376 2017 Quantitative Review: 258, 259, 280

Triangles and Rectangles

2015/13th Edition: 73, 79, 113, 119, 128, 149, 152 2015/13th Quantitative Review: 19, 43, 65, 91 2016 Edition: 90, 98, 126, 131, 143, 159, 161 2016 Quantitative Review: 44, 64, 81, 101 2017 Edition: 231, 237, 334, 342, 370, 383, 396 2017 Quantitative Review: 229, 247, 263