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Fundamental Math Review 1
New SAT Online Resources
Fundamental Math Review
In this section, we will review the basic arithmetic that you will need to know for the Math Test. When you are confident with these basic math concepts, the SAT Math material becomes a lot easier. We encourage you to use these basic concepts as reference with the New SAT Guide 2.0. Much of the material covered in that book will rely on your understanding of the following principles. The concepts we will review in this section are:
- Properties of Integers
- Factors and Multiples
- Operations
- Fractions
- Ratios, Percentages, Proportions, and Rates
- Exponents and Radicals
- Scientific Notation
- Basic Geometry o Lines o Polygons, Triangles, and Quadrilaterals o Circles
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Integers
Part 1
Integers are positive and negative whole numbers, such as –2, –1, 0, 1, 2, etc. Fractions and decimals are not integers. Zero is an integer, but it is neither positive nor negative.
Here is a chart that summarizes types of integers:
Integer Properties
Word Definition Examples
Positive Greater than zero 2, 7, 23, 400
Negative Less than zero –2, –7, –23, –
Even Divisible by two 4, 18, 2002, 0
Odd Not evenly divisible by two 3, 7, 15, 2001
Prime Only divisible by itself and 1 2, 3, 5, 7, 11, 19, 23
Composite Divisible by numbers other than itself and 1 4, 12, 15, 20, 21
Consecutive Follow each other in numerical order 2, 3, 4, 5, 6
Factors and Multiples
A factor of a number is a positive integer that divides that number evenly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. When you divide 12 by any other integer, the result is not a whole number. For example, 12 ÷ 5 = 2.4. Since 2.4 is not a whole number, 5 is not a factor of 12.
Multiples of a number are the product of that number and any positive integer. For example, some multiples of 3 are 6 and 21 because 3 × 2 = 6 and 3 × 7 = 21.
Factoring or factorization is the process of writing a number as a product of its prime factors —the factors that are prime numbers. To factor a number, first divide it by any prime factor. Keep dividing the remainder by prime numbers until the remainder is a prime. Keep track of each factor, even if you divide by the same factor twice.
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- The factorization of 100 is 2 × 2 × 5 × 5.
- The factorization of 80 is 2 × 2 × 2 × 2 × 5.
- The shared prime factors are 2, 2, and 5.
- The GCF is the product of shared factors: 2 × 2 × 5 = 20.
- The LCM is the product of all the factors in the diagram. Only count the shared factors once: 5 × 2 × 2 × 5 × 2 × 2 = 400.
Fundamental Math Review 5
Operations
Part 2
The four main operations in arithmetic are addition, subtraction, multiplication, and division.
Operation Name of Result Words Numbers
Addition Sum The sum of 3 and 4 is 7. 3 + 4 = 7
Subtraction Difference The difference between 5 and 2 is 3. 5 – 2 = 3
Multiplication Product The product of 6 and 4 is 24. 6 × 4 = 24
Division Quotient The quotient of 40 divided by 5 is 8. 40 ÷ 5 = 8
If you perform operations with odd or even numbers, you can predict whether the result will be
odd or even:
- even + even = even
- odd + odd = even
- even + odd = odd
- even × even = even
- odd × odd = odd
- even × odd = even
Here are some properties you should know for addition and multiplication:
- Associative Property: ( a + b ) + c = a + ( b + c ) and a ( b c) = ( ab ) c
- Commutative Property: a + b = b + a and ab = ba
- Distributive Property: a ( b + c ) = ab + ac
And here are some properties you should know for subtraction and division:
- Subtracting a number is the same as adding its opposite: a – b = a + (– b )
- Adding a number to its opposite will give you zero: a + (– a ) = 0
- Dividing by a number is the same as multiplying by its reciprocal: a ÷ b = a ×^1 b
- Multiplying a number by its reciprocal will give you 1: a ×^1 a = 1
- You can’t divide any number by zero: a ÷ 0 is undefined
Fundamental Math Review 7
Fractions
Part 3
A fraction can be thought of in two ways. First, it is another way to represent division. When we write 3 4
, we mean three divided by four. But more importantly, fractions are used to express parts of a whole.
A fraction often takes the form
a b. The number on top,^ a , is the^ numerator,^ and the number on the bottom, b , is the denominator. Here are some properties of numerators and denominators:
- If a < b , then ab < 1. This is called a proper fraction.
- If a > b , then ab > 1. This is called an improper fraction.
- If a = b , then ab = 1.
You can compare two fractions by comparing their numerators and denominators:
- If two fractions have the same denominator , the fraction with the larger numerator is the larger
fraction:
7 >^
- If two fractions have the same numerator , the fraction with the smaller denominator is the
larger fraction:
3 >^
Mixed numbers are improper fractions written as an integer and a fraction. A mixed number is a sum of the integer and the fraction, not the product:
2
4 =^ 2 +
4 ≠ 2^ ×
A rational number is a number that can be expressed as a fraction of integers. Some examples include
-^23 , 5^14 , and 7. All of these can be expressed as a fraction of integers: 5^14 can be written as
4 , and 7
can be written as
Irrational numbers cannot be represented as a fraction of integers and will be expressed as a decimal or a symbol. Some examples include π, 81.52602934…, and any other non-repeating, infinite decimal.
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Equivalent Fractions
A fraction can be written different ways and still represent the same number. These different fractions
with the same value are called equivalent fractions. For example,
2 has the same value as
4 and
To rewrite a fraction without changing its value, multiply or divide the numerator and the denominator by the same number:
1 × 2
2 × 2 =
1 × 10
2 × 10 =
Multiplying the numerator and denominator of a fraction by the same number does not change the value of the fraction because you are really multiplying the fraction by 1:
1 × 2
2 × 2 =
2 ×
2 × 1 =
A fraction can be simplified by dividing its numerator and denominator by their greatest common factor.
For example, let’s simplify
- First, we’d factor 60 and 105:
= 2 × 2 × 3 × 5
= 3 × 5 × 7
60 and 105 have the common factors 3 and 5, so their greatest common factor is 3 × 5 = 15. Therefore, we can simplify the fraction by dividing the numerator and denominator by 15: 60 ÷ 15 105 ÷ 15 =^
Operations with Fractions
You can only add or subtract two fractions when they have the same denominator—in other words, when they have a common denominator.
To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator:
9 =^
9 =^
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Complex Fractions
Complex fractions are fractions that have one or more fractions in their numerator and/or denominator.
Example
2 –^12 3 +^56
To simplify a complex fraction, you can simplify the numerator and denominator and then divide. For example, to simplify the complex fraction above, you can first simplify the numerator:
2 –
Then, you can simplify the denominator:
3 +^56 =^186 +^56 =^236
And finally, you can divide the two fractions: 3 2 ÷
2 ×^
46 =^
Fractions and Decimals
If you divide the numerator by the denominator, you can convert a fraction into an integer or a decimal , which is a way of representing a fraction out of 10.
4 = 3 ÷ 4 = 0.
Decimals can be easier to compare than fractions if the fractions have different denominators or
different numerators. On the calculator section, you can use a calculator to convert fractions to decimals.
For the no-calculator section, you should know the common fraction-decimal conversions:
1 2 =.^
1 3 =.
� 2 3 =.
� 1 4 =.^
3 4 =.
The horizontal bar over the decimal means that it repeats infinitely: .3�^ = .33333…
Fundamental Math Review 11
Ratios, Percentages, Proportions, and
Rates
Part 4
Ratios
A ratio shows a relationship between two quantities. Ratios can be expressed as a fraction �
4 �, with a
colon (1:4), or with the word “to” (1 to 4). Ratios can be converted like fractions by multiplying or dividing each quantity by the same number. For example, if a jar has 12 red marbles and 15 blue marbles,
the ratio of red marbles to blue marbles would be
15 or 12:15. This can be reduced to 4:5.
Notice that ratios are not a “part-to-whole” relationship like a fraction unless one quantity is the total. In our jar of marbles, there is a total of 27 marbles. Therefore, if we wanted to write the fraction of
marbles that are red, we’d write
27 ,^ not^
Ratios can compare more than 2 quantities. For example, if a jar has 12 red marbles, 15 blue marbles, and 6 green marbles, the ratio of red to blue to green marbles is 12:15:6 or 4:5:2.
Percentages
A percentage is a ratio that compares a quantity to 100. For example, 20 is 80% of 25 because 20 25 =^
To convert a percentage to a fraction, re-write it as a fraction of 100 and simplify.
To convert a percentage to a decimal, divide the percentage by 100:
Fundamental Math Review 13
Then, we can cross-multiply and solve for x :
10 x
x
The correct answer is 87.5 minutes.
When solving problems involving rates, pay attention to the units in the question. The answer may be in different units than are given in the problem. To convert units, set up and solve a proportion between the different units.
Example If Juan is driving at 50 miles per hour, how many miles does he travel in 12 minutes?
First, set up a proportion to convert the rate to miles per minute:
50 miles 1 hour ×^
1 hour 60 minutes =^
5 miles 6 minutes
Then, find the distance travelled in 12 minutes:
5 miles 6 minutes =^
x miles 12 minutes
x =
6 = 10 miles
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Exponents and Radicals
Part 5
An exponent indicates that a number is being multiplied by itself a certain number of times. The number being multiplied is called the base. The raised digit is the exponent , and it tells you how many times a number is being multiplied by itself.
Example
In the expression 3 5 , 3 is the base and 5 is the exponent. This tells you that 3 is being multiplied by itself 5 times: 3 5 = 3 × 3 × 3 × 3 × 3
An exponent is also called a power. In the example above, we can say that 3 5 is the 5 th^ power of 3.
In the equation 3 2 = 9, 3 is the square root of 9, which we can also write as √9. This means that 3×3 = 9. The symbol we use for square roots (√ ) is called a radical. The number under the square root is called the radicand.
√9 is also the equivalent of 9
(^12)
. The x th^ root of any number a is written as a
(^1) x or √ x^ a. For example, √^4 81 = 81
1 (^4) = 3 because 3 4 = 81.
Exponent Rules
Here is a chart that shows some important rules for exponents and radicals. In this chart, a and b represent any number, and m and n represent any positive integers.
Exponent Rules
Rule Example
a^1 = a (^) 8 1 = 8
a^0 = (^1 8 0) = 1
a −m^ =
a m^
8 −^2 = 1
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Scientific Notation
Part 6
Place Values
Each digit in a decimal has a place value , which refers to where the digit is located in the number. In the number 123, 1 is in the hundreds place, 2 is in the tens place, and 3 is in the ones or units place. Digits to the right of the decimal point also have place values. In the number 0.45, 4 is in the tenths place and 5 is in the hundredths place.
Each place value represents a power of 10. Places to the left of the decimal point are products of 10 to a positive integer, while places to the right of the decimal point are products of 10 to a negative integer.
123.45 = 1 × 10^2 + 2 × 10^1 + 3 × 10^0 + 4 × 10−^1 + 5 × 10−^2
You can also write 123.45 as the product of a single decimal and a power of 10:
123.45 = 1.2345 × 10^2
Scientific Notation
Place values and powers of 10 are useful for writing very large or very small numbers in a shorter form called scientific notation. Scientific notation displays a number as a product of a decimal and a power of 10.
To convert a number to scientific notation, re-write the number as the product of the non-zero digits and the power of ten of the digit occupying the largest position. For example, the largest non-zero digit in 7,100,000 is 7, which is in the millionths place (10^6 ). 7,100,000 ÷ 10 6 = 7.1 , so 7,100,000 = 7.1 × 106.
You can write very small numbers as a product of ten to a negative exponent. For example, the
largest non-zero digit in 0.000003409 is the 3 in the millionths place (10−^6 ). 0.000003409 ÷
10 −^6 = 3.409, so 0.000003409 = 3.409 × 10−^6.
Notice that a number in scientific notation always consists of a decimal whose largest digit is in the ones place multiplied by a power of ten. Large numbers greater than 1 or negative numbers less than -1 are positive powers of 10. Numbers between -1 and 1 are negative powers of 10.
Fundamental Math Review 17
Operations with Scientific Notation
If two or more numbers have the same power of ten, you can add or subtract them by adding or subtracting the decimals:
5.4 × 10^8 – 2.93 × 10^8 = (5.4 – 2.93) × 10 8 = 2.47 × 10 8
If the result is larger than or equal to 10 or smaller than 1, you must adjust the power of ten so that the decimal has one digit to the left of the decimal point.
Example
What is the sum of 7.2 × 10−^3 and 5.5 × 10−^3?
First, you add the decimals:
7.2 × 10−^3 + 5.5 × 10−^3 = 12.7 × 10−^3
Then, you convert this result to correct scientific notation. Divide the right side by 10, moving the
decimal one to the left, and multiplying the 10−^3 by 10:
12.7 × 10−^3 = 1.27 × 10−^2
If the numbers are multiplied by different powers of 10, you must convert them to standard notation to add or subtract them. Then perform the arithmetic and convert them back to scientific notation.
You can multiply or divide numbers in scientific notation that have different numbers of ten. To multiply numbers in scientific notation, multiply the decimals and add the exponents on the powers of
- To divide numbers in scientific notation, divide the decimals and subtract the exponents on the powers of 10. Make sure the result is in correct scientific notation.
Example
What is (2.5 × 10^13 ) × (6.8 × 10−^5 )?
First, multiply the decimals:
2.5 × 6.8 = 17
Then, add the exponents:
1013 × 10-5^ = 1013 +(–5)^ = 10^8
Fundamental Math Review 19
Fundamental Geometry
Part 7
Lines
A line is a straight, one-dimensional object: it has infinite length but no width. Using any two points, you can draw exactly one line that stretches in both directions forever. For instance, between the points
A and B below, you can draw the line AB ⃖���⃗^. You name a line by drawing a horizontal bar with two arrows over the letters for two points on the line.
A line segment is a portion of a line with a finite length. The two ends of a line segment are called endpoints. To name a line segment, identify two points on the line, and draw a horizontal bar with above the letters for those two points. For instance, in the figure below, the points M and N are the endpoints of the line segment MN �����.
The point that divides a line segment into two equal pieces is called its midpoint. In the figure below, the point 𝑄𝑄 is the midpoint of the line segment PR ����.
Because Q is the midpoint, it divides the segment into two equal pieces. Therefore, you know that PQ = QR. PQ ����^ = QR ����.
A
B
M N
P Q R
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Polygons
A polygon is a two-dimensional shape with straight sides. Polygons are named for the number of their sides:
Types of Polygons
Name Number of Sides
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
A vertex of a polygon is a point where two sides meet. An interior angle of a polygon is an angle on the inside of the polygon formed by the intersection of two sides. A regular polygon has sides that are all the same length and interior angles that are all the same measure. Make sure to take a look at the New SAT Guide 2.0, Chapter 5, Part 1 for more details on angles and their properties.
To calculate the interior angles of any polygon with n sides, use the following formula:
Sum of interior angles = 180°( n – 2)
Using this formula, the sum of the interior angles in a hexagon is 180°( 6 – 2 )^ = 720°.
Two polygons are congruent if they have the same size and shape. Congruent polygons have an equal number of sides, equal lengths of corresponding sides, and equal measures of corresponding interior angles. Congruent polygons have an equal number of sides, equal lengths of corresponding sides, and equal measures of corresponding interior angles. For example, the quadrilaterals below are congruent because they are identical in shape and in size. One just happens to be rotated.
Congruent Polygons
