Arithmetic Reasoning Study Guide for the ASVAB

General Information

If you take the paper-and-pencil version of the ASVAB, you will have 30 items on this section of the test to complete in 36 minutes. If taking the CAT version, you’ll need to complete 16 problems in 39 minutes.

The Arithmetic Reasoning section of the ASVAB test focuses on word problems and delivers mathematical questions and equations in a format that must be synthesized. These may not only be simple questions involving basic multiplication but may also be as complex as completing a physics equation listed in paragraph or word-block format. When deconstructing these word problems, pay close attention to all aspects of the question, including all numbers mentioned, buzzwords, and the format of the paragraph itself. The following pointers will give you a more in-depth analysis of each component of word problems.

Specific Strategies

Buzzwords

Mathematical reasoning requires you to focus on all aspects of the word problem, with particular attention to buzzwords indicating the action to take (multiplication, division, etc.). Seek out important phrases, such as difference, minus, and take away for subtraction, or product, times, and double for multiplication. Before tackling the question as a whole, look for such buzzwords and identify what process the word problem requires. From there, you can continue to solve the overall equation.

Numbers in Problems

Pay close attention to all numbers and figures mentioned within the body of the paragraph. Identify these numbers, set them aside, and identify which of the numbers are relevant and which of the numbers are tossed in for the purpose of throwing you off or misleading you. While paying attention to the numbers, also pay attention to the order. Identifying a 7 and a 9 as the two elements of a word problem is only effective if you are able to correctly identify the proper sequence of the numbers. The expressions \(9 - 7\) and \(7 - 9\) yield two very different results and may be the difference between passing and failing a test.

Paragraph Format

The format of a paragraph is also a key component of mathematical reasoning. A physics equation, for instance, will be formatted far differently than a simple algebraic equation; a physics word problem will likely involve a vehicle of some type (car, train, plane, etc.), while a simple addition equation may involve a myriad of different situations in many different contexts. To determine the nature of the equation, you must first identify the format and context of the paragraph. From there, you can combine the buzzwords and numbers to form a completed, simplified equation.

Basic Procedures

When preparing for the Arithmetic Reasoning section of the ASVAB test, always remember to scan the question first, identify all necessary parts, and discard the rest. Because word problems are lengthier than the actual equation or function, there will likely be a significant portion of text that is not relevant to solving the equation, but is used as filler to throw you off track.
When dealing with these problems, identify all necessary information, formulate a workable equation, and solve the equation. As always, whenever you are uncertain about a specific question or section, if you are taking a paper-and-pencil test, you can feel free to move ahead to a section you are more comfortable with, and then circle around to the more difficult questions. In any case, while taking the test, exercise patience and reasoning to achieve a higher score.

While preparing for this test, be sure you have a basic understanding of the following math ideas. If something listed here still stumps you, seek out further explanation and practice.

Basic Arithmetic Definitions

Number Types

You need to be aware of the different types of numbers.

Whole numbers: The set of numbers (with no fractional part) from 0 to infinity: \((0,1,2,3…)\).
Counting numbers: The same as the whole numbers, except this set does not include 0, because 0 can’t be counted: \((1,2,3...)\). These are also called natural numbers.
Integers: The set of all positive counting numbers, their opposites, which are negative, and 0: \((...-3,-2,-1,0,1,2,3…)\). Positive integers are greater than 0, and negative integers are less than 0.
Natural numbers: The natural numbers (also called counting numbers) are \((1, 2, 3…)\) extending forever. There are infinitely many of them. They are the positive whole numbers.
Rational numbers: Rational numbers are any number that can be written as one integer divided by another. All fractions are included, such as \(\frac{3}{5}, \frac{7}{-4}\), and \(\frac{119}{23}\). All whole numbers are included too, since they can be written over \(1\) (e.g., \(\frac{12}{1}\)). All non-repeating decimals are also included. For example, \(2.8\) can be written as \(\frac{28}{10}\), a ratio of two integers.
Irrational numbers: Irrational numbers are those that cannot be written as a ratio of two integers. Examples are \(\sqrt{5}\) and \(\pi\).
Real numbers: Real numbers include all of the rational and irrational numbers, which would include every number on a number line.
Imaginary numbers: An imaginary number is any number of the form \(ai\), where \(a\) is any real number, and \(i\) is \(\sqrt{-1}\). For example \(\sqrt{-4}\) can be written as \(\sqrt{4} \cdot \sqrt{-1}\) which can be written as \(2\sqrt{-1}\) or \(2i\).
Complex numbers: A complex number is any number of the form \(a+bi\), where \(a\) and \(b\) are real numbers. For example \(2+5i\). Sometimes these are called imaginary numbers.

Important Special Numbers

Prime numbers: any number which is divisible only by itself and one: \((2, 3, 5, 7, 11…)\); since the number 1 is only divisible by 1, it is not considered a prime number
Even numbers: any number that when divided by 2 leaves no remainder; it is exactly divisible by 2: \((0,2,4,6,8...)\)
Odd numbers: any number that when divided by 2 leaves a remainder of 1: \((1,3,5,7,9…)\)
Factor: an integer that evenly divides another integer, and when multiplied by itself or other factor(s), will equal this bigger number: 2 is a factor of 10, and when multiplied by 5 equals 10
Multiple: the product of two or more integers: 10 is a multiple of 2 because 5 times 2 equals 10
Consecutive Numbers: sets of numbers that increment by exactly one member in their set; for example, these are five consecutive integers: \((4,5,6,7,8)\) and these are five consecutive even numbers: \((4,6,8,10,12)\)

The Four Basic Operations

When using the four operations with numbers, we need to know how to talk about the result:

Addition: Adding two or more numbers results in a sum.
Subtraction: Subtracting two numbers results in a difference.
Multiplication: Multiplying two numbers results in a product.
Division: Dividing two numbers results in a quotient.

Basic Properties of Numbers

There are three properties of numbers we must be aware of when dealing with the four mathematical operations: the Associative Property, Commutative Property, and Distributive Property. These form the basis for many problem-solving methods and procedures.

The Commutative Property of Addition: It doesn’t matter in what order you add two or more numbers, the answer is always the same:

\[2 + 3 + 5 =10 \text{ is the same as }3 + 5 + 2 = 10\]

The Commutative Property of Multiplication: It doesn’t matter in what order you multiply two or more numbers, the answer is always the same:

\[2\cdot 5\cdot 3 = 30 \text{ is the same as } 5\cdot 3\cdot 2 = 30\]

The Associative Property of Addition: It doesn’t matter how you group three or more numbers to be added, the answer is always the same:

\[(2+5) +3 = 10 \text{ is the same as }2+(5+3) = 10\]

The Associative Property of Multiplication: It doesn’t matter how you group three or more numbers to be multiplied, the answer is always the same:

\[(2 \times 5) \times 3 = 30 \text{ is the same as }2 \times (5 \times 3) = 30\]

The Distributive Property: To evaluate a number times other numbers added inside parentheses, multiply it by each of the addends inside the parentheses, then add these products:

\[2(5+3) = 2\cdot 5 + 2\cdot 3 = 10 + 6 = 16\]

The short way to describe this property is: distribute the number outside the parentheses over all the numbers inside it. This distributive property also means that you can factor some things out of an expression leaving the larger part inside the parentheses, and the smaller outside of them. For example:

\[2x + 6x^2\]

With the distributive property we can factor out \(2x\) to get:

\[2x(1 + 3x)\]

Identity Properties: For the operation addition, there is one number that, if added to any given number will not change the given number. That one number, of course, is 0. Zero is called the Additive Identity. Likewise, for the operation multiplication, there is one number that, if multiplied times a given number, will not change the given number. That one number is 1. One is called the multiplicative identity.
Inverse Operations: For each mathematical operation, there is a reverse operation that undoes the original operation. To undo addition, use subtraction and vice versa. To undo multiplication, use division and vice versa. Squaring and taking the square root are another example of a pair of inverse operations.
Closed and Open Systems: For certain sets of numbers, there are operations that always give a result that is in the set. For example, in the set of even numbers, if you add any two, you always get another even number, never an odd number. We say that the set of even numbers is closed under addition (also true under multiplication, by the way). On the other hand, if you divide one even number by another, you will likely get a fraction, and fractions are not even numbers. We say that the set of even numbers is open under division.