GRE Math Review Workshop - St. Lawrence · PDF fileGoals •Review the foundations and...

GRE Math Review Workshop Fall 2012 Quantitative Resource Center

Goals

• Review the foundations and concepts that are tested on the Quantitative Reasoning section of the GRE

• Discuss the types of Quantitative Reasoning questions on the GRE

Content Review

• Percentages • Simultaneous Equations • Symbolism • Special Triangles • Multiple and Oddball Figures • Mean, Median, Mode, and Range • Probability

Percentages

• Comparison of a number to 100. ▫ Example: 24% means 24 of 100

• Can also be expressed as a fraction ▫ Example: 24% =

• Can also be expressed as a decimal ▫ Example: 24% =

Percentages

• Most problems involve three quantities (whole, part, and percent). ▫ Percent =

• Example: What percent of 60 is 45?

Percentages

• Example: What is 20% of $80?

Percentages • Example: Last year, Julie’s startup company

showed a profit of $20,000. This year, the same company showed a profit of $25,000. If her company shows the same percent increase in profit in the coming year, what will that profit be? A. $27,000 B. $30,000 C. $31,250 D. $32,500 E. $35,000

Percentages • A hardware store is selling a lawnmower for $300. If

the store makes a 25% profit on the sale, what is the store’s cost for the lawnmower?

A. $210 B. $225 C. $240 D. $250 E. $275

Simultaneous Equations

• Means a set of equations with two variables, need to be solved at the same time (simultaneously)

• You can solve these equations using multiple methods.

• The easiest method is called elimination or combination, when you eliminate a variable by adding or subtracting equations.

Simultaneous Equations

• Follow the four steps to solve for one variable: 1. Write one equation under the other 2. If you cannot eliminate a variable by adding or

subtracting, multiply one equation by a number so you can do so.

3. Add or subtract the equations to eliminate a variable.

4. Solve for the remaining variable

Simultaneous Equations

• Example: If p + 2q = 14 and 3p + q = 12, then p = 1. Write one equation under the other 2. If you cannot eliminate a variable by adding or

subtracting, multiply one equation by a number so you can do so.

3. Add or subtract the equations to eliminate a variable.

4. Solve for the remaining variable

Simultaneous Equations

• What if we need to solve for q too? ▫ If p + 2q = 14 and 3p + q = 12, then p = 2

Simultaneous Equations

• In some cases, there is no single value for each variable in a system of simultaneous equations. ▫ Example: 4a + 2b = 10 12a + 6b = 30

• Try to eliminate one variable

Simultaneous Equations

• When you try to solve this system, you find out that the equations are equivalent. If you try to eliminate one variable, you end up eliminating both variables.

• This means that there is no single value for each variable; there are an infinite number of possible sets of values for a and b.

Simultaneous Equations

• Example: If x + y = 8 and y – x = -2, then y =

Simultaneous Equations

• Example: If m – n = 5 and 2m + 3n = 15, then m + n =

Simultaneous Equations

• Which of the ordered pairs of numbers (c, d) satisfies the simultaneous equations shown?

c + 2d = 6 -3c – 6d = -18 A. (-2, 4) B. (-1, -3) C. (0, 2) D. (2, 1) E. (3, -9)

Symbolism

• You are used to arithmetic symbols such as +, =, X, ÷, and %, but you are also likely to

encounter some very unusual symbols. You may be asked, for example, to find the value of

10 2, 5 7, or 65 2. • Such questions turn out to be not as difficult as

they first appear; you simply need to substitute normal math symbols for the unusual ones and then perform the operations.

Symbolism

Example: If a b = for all nonnegative numbers a

and b, what is the value of 10 6? A. 0 B. 2 C. 4 D. 8 E. 16

Symbolism Example: If a means to multiply a by 3 and a means to

divide a by -2, what is the value of ((8 ) ) ? A. -6 B. 0 C. 2 D. 3 E. 6 Don’t be fooled! If you take it one step at a time,

it’s much easier to manage.

Symbolism

Example: If x 0, let x be defined by x = x – . Then (-3) =

Special Triangles

• Knowing the properties of the most commonly tested figures prepares you to answer just about any geometry question you’ll encounter. For example, the sum of the angle measures in a triangle is always 180 degrees.

• Knowing the length of one side of a 30-60-90 triangle or 45-45-90 enables

you to find the lengths of the other sides.

Special Triangles

Equilateral Triangles: All interior angles are 60, and all sides are the

same length.

60 60

60

Special Triangles

Isosceles Triangles Two sides are the same length, and the angles

facing those sides are equal.

Special Triangles Right Triangles: Right triangles contain a 90 angle. The sides that form

the 90 are the legs; the side facing the 90 angle is the hypotenuse. The sides are related by the Pythagorean theorem a2 + b2 = c2, where a and b are the lengths of the legs, and c is the length of the hypotenuse.

c

b

a

Special Triangles

• Some triangles on the GRE are “special” right triangles whose side lengths always come in predictable ratios. If you can spot them, you won’t have to use the Pythagorean theorem to find a missing side length.

Special Triangles

• 3:4:5 Right Triangles Be on the lookout for multiples!

6:8:10, 9:12:15, … all Special Right Triangles • 5:12:13 (Occur frequently in Quantitative

Reasoning questions)

Special Triangles

• 45-45-90 Right Triangles (Isosceles Right Triangles)

• 30-60-90 Right Triangles

Special Triangles Example: In the triangle above, what is the length of side

BC? A. 4 B. 5 C. 4 D. 6 E. 5

A

B

C 7

4

45

Special Triangles

Example: (Cont.)

A

B

C 7

4

45

Special Triangles

Example: In triangle ABC, if AB = 4, then AC = A. 6 B. 7 C. 8 D. 9 E. 10

A

B

C

60 150

Special Triangles Example: If the perimeter of triangle ABC above is 16, what is its

area? A. 8 B. 9 C. 10 D. 12 E. 15

A

B

C 6

5

Special Triangles Example: A ladder 20 feet long is placed against a wall. If

the distance on the ground from the wall to the ladder is 12 feet, how many feet up the wall does the ladder reach?

A. 12 B. 15 C. 16 D. 21 E. 24

Multiple and Oddball Figures

You may see a combination of geometric shapes or an oddball figure on the GRE. While these problems may look difficult, you can often simplify them by first looking for familiar geometric shapes, such as special right triangles, squares, or circles.

Multiple and Oddball Figures How could you approach the following question? In the figure above, if the area of the circle with center 0 is 9,

what is the area of triangle POQ? A. 4.5 B. 6 C. 9 D. 3.5 E. 4.5

P

Q

O

Multiple and Oddball Figures Example: What is the perimeter of quadrilateral

WXYZ? A. 680 B. 760 C. 840 D. 920 E. 1,000

W Z

Y

X

200

180 300

Multiple and Oddball Figures Example: What is the value of x in the figure

below? A. 4 B. 3 C. 3 D. 5 E. 9

H

E

F

G 5

6

x

8

Multiple and Oddball Figures

Example: In the figure below, square PQRS is inscribed in a circle. If the area of square PQRS is 4, what is the radius of the circle?

A. 1 B. . C. 2 D. 2 E. 4

P

S

Q

R

Mean, Median, Mode, and Range

• Mean- Average of the numbers. (Also called mean, arithmetic mean, and average).

Average = It helps to remember that: Number of Terms Average = Sum of Terms

Mean, Median, Mode, and Range Mean Example: Nancy shopped at four department stores and spent an

average of $80 per store. If she wants to average no more than $70 per store over a total of six stores, what is the most she can average at the two remaining stores?

A. $40 B. $50 C. $55 D. $65 E. $70

Mean, Median, Mode, and Range • Median – the middle terms when all the terms

in a set are listed in sequential order. When there is an even number of terms in a set, the median is the average of the two middle terms.

• Mode – the term that occurs most frequently. It is possible to have more than one mode if the most frequent terms occur an equal number of times.

• Range – the different between the greatest term and the least term.

Mean, Median, Mode, and Range

Example: The only test scores for the students in a certain class are 44, 30, 42, 30, x, 44, and 30. If x equals one of the other scores and is a multiple of 5, what is the mode for the class?

A. 5 B. 6 C. 15 D. 30 E. 44

Mean, Median, Mode, and Range

Example: If half the range of the increasing sequence {11, A, 23, B, C, 68, 73} is equal to its median, what is the median of the sequence?

A. 23 B. 31 C. 33 D. 41 E. 62

Mean, Median, Mode, and Range

Example: If the average of x + 1, x + 2, x + 3 is 0, then x =

A. -2 B. -1 C. 0 D. 1 E. 2

Probability • Measures the likelihood of an event occurring and can be

represented as a fraction, decimal, or percent. For example, if rain today is just as likely as not, then the probability of rain today can be expressed as , 0.5, or 50%. You may also see a probability expressed in everyday language: “one chance in a hundred” means the probability is

• To find the probability of something happening, use this formula:

Probability =

Probability

Example: What is the probability of getting a 4 if you toss a six-faced die?

Probability • Keep in mind that the probability of an event not

happening is equal to 1 minus the probability of the event happening. This can be stated as: P(not event) = 1 – P(event)

• Every probability is expressed as a number between 0 and 1 inclusive, with a probability of 0 meaning “no chance” and a probability of 1 meaning “guaranteed to happen”.

• You can often eliminate answer choices on the GRE by having some idea of where the probability of an event falls between 0 and 1.

Probability

Example: If 14 women and 10 men are employed in a certain office, what is the probability that one employee picked at random will be a woman?

Probability

Example: A bag contains 6 red, 6 green, 8 yellow, and 5 white marbles. You pick one marble at random from the bag. What is the probability that the marble chosen is not green or yellow?

Probability

If Tom flips a fair coin twice, what is the probability that at least one head will occur?

Types of Questions

• Quantitative Comparison • Problem Solving • Data Interpretation

Reference • New GRE: Premier 2011-2012. (2012). New York, NY: Kaplan Publishing.

ISBN: 978-1-60714-849-4