General Concepts

Question Types

There are two timed sections of the SAT exam Math section: one during which you may use a calculator and the other during which you may not. The calculator-allowed section contains 38 questions, and you will be given 55 minutes to complete them. There are 20 questions in the no-calculator section, which is timed for 25 minutes. There is more information on calculator use in the SAT’s Calculator Policy.

Within both of these sections are questions pertaining to the following four areas of math. For study time allocation purposes, we show the number of questions posed on each topic:

• Heart of Algebra — 19

• Problem Solving and Data Analysis — 17

• Passport to Advanced Math — 16

• Additional Topics — 6

Most of the questions in both sections are standard multiple-choice, but you will also encounter a few questions that require you to “grid in” your answer. This means that there will not be answers to choose from. You must arrive at the answer and write or type it in the boxes provided. By section, the numbers of grid-in questions are:

• Calculator Section — 8

• No-Calculator Section — 5

The test is very consistent with the type of mathematics questions that it uses, year after year. The following are the types of mathematical questions that you are likely to encounter:

• Arithmetic

• Divisibility

• Multiplication

• Addition

• Subtraction

• Evens and Odds

• Prime Numbers

• Percents

• Square of a Number

• Exponents

• Roots

• Averages

Interpret, Simplify, and Solve

To interpret, or evaluate, an expression is to determine its numerical value. Sometimes these expressions will include variables, but in order to evaluate the value of the expression, the value of each of the variables must also be known.

To simplify an expression is to apply algebraic rules in order to find a more simple expression to work with. For example, we can write 4x as x + x + x + x, but it is often more convenient to simplify the second expression to the first. Consider the following example:

-3(x + y - 3) + 2x - 4y can be simplified by first applying the distributive property, and then combining like terms:

-3x - 3y + 9 + 2x - 4y = -x - 7y + 9

The expression on the right is easier to work with than the expression on the left.

Only equations, or statements including a comparison of two expressions, can be solved. The solution to an equation or set of equations is the collection of values that, when plugged into the original equation(s), yield a true statement. For example, 4 is the solution to the equation x - 0 = 4 because only 4 yields a true statement (4 = 4), when substituting for x.

Arithmetic

Arithmetic skills refer to the questions that can be solved by using addition, subtraction, multiplication and/or division. Since calculators are permitted in the test, the questions will obviously not be purely arithmetic - they’re not out to measure your ability with a calculator. So in this style of question, you’ll need to recall your order of operations. A good trick to recall your order of operations is “Please Excuse My Dear Aunt Sally”…before you say “huh?” recognize the first letters in this phrase:

• Work within Parenthesis

• Simplify Exponents

• Multiplication and Division

• Addition and Subtraction

The majority of arithmetic questions will require you to take multiple steps, and will likely test other skills as well, instead of being purely arithmetic. Often, the questions will be presented in the form of word problems, where you will need to decide when to add, subtract, multiply and divide.

For example:

How many egg cartons are needed to hold 300 eggs, if each carton can hold one dozen (1 dozen = 12)

A. 15

B. 18

C. 22

D. 25

E. 28

Note: the answer is 25 

Divisibility

The factors of integer X are the integers by which X can be divided without leaving a remainder. Thus, X is divisible by its factors.

For example:

The number 10 is divisible by both 5 and 2. 10 can be divided by both of these integers without leaving a remainder.

To review the rules of divisibility, have a look at the following:

1. Numbers divisible by 2 end in even numbers.

2. Numbers divisible by 3 can be determined by adding the sum of their digits and checking if that number is divisible by 3 (for example the number 123: 1 + 2 + 3 = 6, 6 is divisible by 3 with no remainder).

3. Numbers divisible by 4 can be identified if their last two digits will divide by 4 without a remainder (for example, the number 624: the last two digits are 24, which are divisible by 4 with no remainder).

4. Numbers divisible by 5 end only in 5 or 0.

5. Numbers divisible by 9 occur when the sum of its their digits are divisible by 9 (for example, the number 639: 6 + 3  +9 = 18, which is divisible by 9).

6. A number is only divisible by 10 if it ends in 0

The following is an example of a divisibility question:

Which of the following integers divides into both 200 and 150?

A. 3

B. 7

C. 30

D. 50

E. 300

Note: The correct answer is (D) 

Multiplicity

The following are a few simple rules to keep your multiplications on track:

Positive x Positive = Positive

Negative x Negative = Positive

Negative x Positive = Negative 

Addition

Here are some rules to be certain that there are no slips while doing addition:

Positive + Positive = Positive

Negative + Negative = Negative

Negative + Positive = either positive or negative (you must use the absolute value of both: subtract the smaller from the larger and keep the sign of whichever absolute value was larger) 

Subtraction

The definition of subtraction is: A - B = A + (-B)

A minus B is the same as A plus (the opposite of B)

X > 0, means that X is a positive number

X < 0, means that X is a negative number

-(A - B) = -A + B = B - A

(-X)2 = X2

If X - 0, X2 > 0

If, on the number line, one number occurs to the left of another number, the number on the left is the smallest number.

Therefore, when studying the line above, you will know that X < Y and Y < Z.

For example:

Use the number line to make conclusions with regards to whether each number is positive or negative.

In this situation, you will have an easier time if you implement specific numbers to fit the problem. For example, let X = -7, Y = -2, and Z = 3. Be certain to utilize some negative numbers while substituting.

The following is an example of a subtraction question:

Y - X

Solution: Positive Y is greater than X.

-2 - (-7) = -2 + 7 = 5 

Evens and Odds

An even number is any word that is divisible by 2: numbers that are within the set {…-6, -4, -2, 0, 2, 4, 6,…}. Remember, though, that an even number is divisible by 2 and not have any remainder. Keep in mind also that 0 is an even number. Consecutive even numbers are all located 2 units apart. For example, if x is an even number, then the next consecutive even number would be represented as X + 2.

Odd numbers, on the other hand, are numbers within the set {…-5, -3, -1, 1, 3, 5,…}.

The following charts demonstrate the properties of odd and even numbers. To check the property of a number, you can simply substitute the appropriate numbers.

Consider the following example:

If R is an odd integer, what are the next two consecutive odd integers?

A) T and V

B) R and R + 1

C) R + 1 and R + 2

D) R + 2 and R + 4

E) R + 1 and R + 3

Note: the correct answer is (D)

Here’s another example:

If x is an odd integer and y is an even integer, tell whether each expression is odd or even.

A. x2 

B. xy

C. y2

D. x + y

E. 2x + y

Note (A) is odd. (B) is even. (C) is even. (D) is odd, and (E) is even. 

Prime Numbers

A prime number is defined as an integer that is greater than 1, and has only two positive factors, 1 and itself.

For example, 7 is a prime number, as its only factors are 1 and 7. However, 6 is not a prime number, because its factors are 1, 2, 3, 6

The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Note, though that 1 is not a prime number, and both the smallest and the only even prime number is 2.

Prime factorization is the process by which you express a number as a result of only prime numbers.

For example:

To create the prime factorization of 24, you’d represent it as:

2 x 2 x 2 x 3 or 23 × 3

To create the prime factorization of 15, you’d represent it as:

5 x 3

An example of a factor question is:

If xy = 13 and both x and y are positive integers, then what is the sum of x + y?

A. 13

B. 14

C. 16

D. 20

E. 26

Note: the answer is B

Here is another example:

What is the sum of the first 5 prime numbers?

A. 18

B. 28

C. 30

D. 34

E. 38

Note: The first five prime numbers are 2, 3, 5, 7, 11 and their sum is 28. The answer is B. 

Percents

The word percent means “hundredths” or a number which is divided by 100. Converting a number into a percentage involves multiplying the number by 100.

A percent can be determined by performing the division of the part by the total and multiplying it by 100:

Percent = Part x 100

For example, if Wendy missed 12 out of 80 examination questions, what is the percent of questions she missed?

Percent = missed questions x 100 = 12/80 x 100 = 0.15 x 100 = 15%

The phrase “X is N percent of Y” can also be written mathematically as

X = N x Y

The word “is” means equal (=), while the word “of” means “multiply”

However, before multiplying, you must change a percent into a decimal or fractional format.

For example:

5 is 20% of 25, means 5 = 0.20 x 25

To change the fraction into the percent, you must first change the fraction into a decimal, and then multiply by 100 (or move the decimal point by 2 places to the right)

For example:

Change the fraction 1/5 into a percent.

First, change the fraction 1/5 into the decimal 0.2, and multiply by 100 (move the decimal 2 places to the right). Therefore:

1/5 x 100 = 20%

The following table provides the common percentages that you will use on a regular basis, and may wish to memorize.

Please note that numbers over 1 achieve percentages that are greater than 100%

Consider the following example:

What is 20% of 50?

A. 5

B. 8

C. 10

D. 12

E. 15

Note: the answer is C

To solve this question, you must rewrite it as an algebraic question.

Therefore, let x represent the unknown number.

X = 0.20 x 50

Keep in mind that to change the percent to a decimal, and that the

word “of” means that you should multiply.

X = 10

Here is another example:

5 is what percent of 2?

A. 2.5%

B. 25%

C. 100%

D. 250%

E. 500%

Rewrite this as an algebraic equation. 5 = n × 2

Solve for n and remember to change the answer to a percent.

n = 5/2 = 2.5 = 250%

Therefore, the answer is (D) 

Ratio

Ratios are used to compare one quantity to another quantity.

In the SAT exam math section, for example, there are 19 questions in the Heart of Algebra section, 17 questions in the Problem Solving and Data Analysis section, 16 questions in the area of Passport to Advanced Math, and 6 questions in the area of Additional Topics in Math, for a total of 58 math questions.

When we compare the quantity of one part to another part or one part to the total, the comparison or relationship is called the ratio.

If we want to show the ratio of the number Heart of Algebra to the Problem Solving and Data Analysis (PSDA) questions, we write it as 19:17 (read as: “19 is to 17”). This can also be written as a fraction, 19/17.

To find the ratio of the number of questions in the PSDA to the whole SAT exam math section, we write it as 17:58 or 17/58.

Proportion

Ratios and proportions are related math concepts. Ratios that are equal are said to be in proportion, or proportionate, to each other.

If the ratio of an employee’s incentive to her basic daily wage is proportionate to the ratio of overtime hours rendered over the normal 8-hour day, we present this as:

Incentive/daily wage = overtime hours/8 hours

The value for incentive is not equal to the value for overtime hours, but the ratio incentive/daily wage is equal (or proportionate) to the ratio overtime hours/8 hours.

Square of a Number

Squaring a number means to multiply that number by itself. The notation for squaring a number (x) is as follows:

x2

When squaring an integer, the result obtained is called a perfect square.

When preparing for the test, make sure that you are fully capable of understanding and reproducing the following table, as well as recognizing the numbers that are perfect squares and perfect cubes.

Squared numbers and special properties

• x2 > 0 always, except for x = 0

• x2 > x for x > 1

• x2 < x for 0 < x < 1

• x2 = x for x = 1 or 0

• The square root of x2 equals the absolute value of x.

• If x2  = y2, then either x = y, or y = -x, or x = -y.

The following is an example:

Of the following numbers, which is a both a perfect square and a perfect cube?

A. 4

B. 8

C. 9

D. 16

E. 64

Note: the answer is (E) 

Exponents

The mathematical notations for numbers which are the result of a number that is multiplied by itself a number of times is called exponents.

Examples:

x3 = x × x × x

x5 = x × x × x × x × x

The expression of xn is also called the nth power of x. The x is the base, while the n is the exponent. Math questions will usually only utilize integral exponents. x2 is read as x-squared, and x3 is read as xcubed. All others are read as a power of x. x4 is read as the 4th power of x.

When it comes to the power of 10, there is a simple, quick rule that simplifies the powers of 10, by writing it as 1, followed by the number of zeros as specified by the power.

Examples: 105 = 1 followed by 5 zeros. 100000 = 100,000.

An example you may find is:

Represent 32,456 to the power of 10.

The solution would be as follows:

32,456 = 3 × 104 + 2 × 103 + 4 × 102 + 5 × 101 + 6 × 100

Consider the following example:

Solve for x: (x - 3)2  = 49.

You could use algebra and take the square root of both sides or since 49 is a perfect square you could guess integers for x. Just remember x - 3 must be positive or negative.

If you try guessing, the integers 10 and -4 work. To get an algebra

solution, do the following:

(x - 3)2  = 49

x - 3 = 7 or x - 3 = -7

x = 10 or x = -4

It is your goal to get problems correct quickly. Sometimes guessing (Guessing in this case means substituting in numbers to see which satisfy the equation.) is faster than solving an equation, if you train yourself to use the technique. Of course, if you cannot "see" the answers fast enough, use other approaches to answer the problem. 

Roots

The test will require you to manipulate both square roots and cube roots. Some of the questions will measure whether or not you understand these expressions.

You should remember that none of the following should ever occur:

1. No perfect square can be left underneath a radical (square root) sign.

2. No radical can be within the denominator.

3. No fractions may occur within the radical sign. 

Averages

There are three basic components that comprise an average problem:

1. Total

2. Average (also known as a mean)

3. # of numbers

The average is the total of elements that are within the set.

To discover the average, simply divide the total by the # of numbers.

For example:

Jenna’s last four test scores were 35, 56, 75, and 28. What is the average of Jenna’s test scores?

A. 43

B. 48.5

C. 52.5

D. 54

E. 47

Note: the answer is (B).

35 + 56 + 75 + 28 = 194

194 / 4 = 48.5

Five things to remember when solving averages:

1. If a number that is the same as the average is added, the new average will not change.

2. If a number is added and it is less than the average, the average will decrease.

3. If a number is added and it is greater than the average, the average will increase.

4. If a pair of numbers are added, and they are “balanced” on both sides of the average, the arithmetic mean is the middle value.

5. To discover the average between two evenly spaced numbers, add the first and the last terms and divide them by 2. 

Properties of Operations

An important math property to remember is the PEMDAS acronym for the order of operations, which stands for parentheses, exponents, multiplication/division, and addition/subtraction. It means that operations enclosed by parentheses must be performed first, before other operations, expressions containing exponents follow next, and so on. Multiplication has the same priority as division, and the key is to always proceed from left to right. The same is true for the order of addition and subtraction.

The three basic properties of adding and multiplying numbers are associative, commutative, and distributive properties. A good foundation on these properties makes it easy for you to manipulate many areas of math.

The associative property simply says that the grouping of numbers in addition or multiplication will not affect the result of operations. For example:

(3 x 5)4 = 5(3 x 4)

m + (n - o) = (m + n) + -o

Note in the second example: Does the property also apply to subtraction, then? Well, think of it as addition of a negative number, which may actually be written as:

m + (n + -o) = (m + n) + -o

The commutative property states that the elements in the operation can be moved around without affecting the result.

3 x 5 x 4 = 4 x 5 x 3

m + n - o = -o + n + m

When we refer to the distributive property of numbers, we mean performing multiplication distributed over addition.

3(x + y) = 3x + 3y

A(2B - C) = 2AB - AC

All these properties apply only to addition and multiplication, including the addition of negative values, but never to division.

Analysis and Rewriting

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Polynomials can also be written as a product of their factors. The factors of polynomials are simply smaller polynomials. For example, let’s look at this:

x3 - x2 + 2x - 2

This can be factored:

x3 - x2 + 2x - 2 = (x3 + 2x) - (x2 + 2) = x(x2 + 2) - (x2 + 2) = (x2 + 2)(x - 1)

It’s not always easy to see what a polynomial’s factors are, especially since we often will have like terms that combine when the polynomial is expanded.

If you’re given one form of a polynomial and need to find the other, you can always use unknown numbers in the unknown form, compare it to the known form, and then deduce what the numbers must have been. For example, suppose we are given:

x3 + ax2 + 2x - 2 = (x2 + b)(x - 1) = x3 - x2 + bx - b

where a and b are constant numbers, and we are asked to find them. We can expand the right side:

x3 + ax2 + 2x - 2 = (x2 + b)(x - 1) = x3 - x2 + bx - b

Now we can compare coefficients, i.e. match the numbers multiplying like powers of x. One side has a multiplying x2 and the other has -1, so a = -1. Comparing either of the last two coefficients similarly shows that b = 2.

Area, Surface, and Volume

Any closed two-dimensional figure will possess an area. The area of a figure is simply the amount of space enclosed by the figure. Because areas are two-dimensional, each area measurement will have units raised to the second power.

It is worthwhile to commit the following area formulas to memory:

• area of a circle — pi * r2 , where r is the radius of the circle

• area of a triangle — 1/2(b * h), where b is the base, and h is the height

• area of a quadrilateral — l * w, where l is the length (or base), and w is the width (or height)

• area of trapezoid — 1/2(b1 + b2) * h, where b1 and b2 are the two bases, and h is the height

In cases where you are asked to find the area of a figure composed of many different shapes, break the figure apart into its constituent shapes, calculate the area of each of the shapes, then add each of the areas together to find the total area.

Three-dimensional figures possess both a surface area and a volume. The surface area is the sum of the areas of each of the figure’s surfaces. In order to find the surface area of a figure, first, calculate the area of each surface composing the figure and then add the areas together.

The volume of a three-dimensional figure is the amount of space enclosed by the figure. In the same way that an area is composed of two dimensions and represented by units squared, a volume is composed of three dimensions and represented by units cubed.

It is worthwhile to commit the following volume formulas to memory:

• volume of a sphere — 4/3 * pi * r3, where r is the radius of the sphere

• volume of a cylinder — pi * r2  * h, where r is the radius of the base, and h is the height

• volume of a rectangular prism — l * w * h, where l is the length, w is the width, and h is the height

Coordinate Geometry

The coordinate plane combines two perpendicular coordinate axes to visually represent geometric figures. The horizontal axis is commonly used to represent the independent variable (usually x), and the vertical axis is commonly used to represent the dependent variable (usually y, or f(x)).

The origin is the intersection of the x-axis and the y-axis and is located at point (0, 0). The intersection also creates four quadrants:

• quadrant I contains positive x and y values.

• quadrant II contains negative x and positive y values.

• quadrant III contains negative x and y values.

• quadrant IV contains positive x and negative y values.

Points on the xy coordinate plane are represented as follows:  (x, y), where the x value is the positive or negative distance from x = 0, and the y value is the positive or negative distance from y = 0.