Page 1 Arithmetic Reasoning Study Guide for the ASVAB

How to prepare for the ASVAB Arithmetic Reasoning Test

General Information

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.


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 Mentioned

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. 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.

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, if you are uncertain about a specific question or section, feel free to move ahead to a section you are more comfortable with, and then circle around to the more difficult questions. 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

There are three basic kinds of numbers:

  • Whole numbers: The set of whole numbers (with no fractional part) from 0 to infinity:

  • Counting numbers: The same as the whole numbers, except this set does not include 0, because 0 can’t be counted:

  • Integers: The set of all positive counting numbers, their opposites which are negative, and 0. Positive integers are greater than 0, and negative integers are less than 0.

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.

Important Special Numbers

  • Prime numbers: Any number which is divisible only by itself and one:

  • Even numbers: Any number that when divided by 2 leaves no remainder; it is exactly divisible by 2:

  • Odd numbers: Any number that when divided by 2 leaves a remainder of 1:

  • Factor: A factor is an integer which 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: These are sets of numbers that increment by exactly one member in their set. For example these are five consecutive integers: and these are five consecutive even numbers:

  • Fractions: Part of a whole with a numerator on the top representing the number of parts, the horizontal bar to indicate part of a whole is represented, and the whole number of parts in the denominator. For example: If Jimmy ate 3 out of 5 apples in a bag, he ate of the apples.

  • Mixed Number: A fraction that contains a whole number and fractional part. If Johnny ate one whole bag containing five apples and also ate 3 out of 5 apples in a second bag, the fraction this represents would be mixed:

  • Improper Fraction: With an improper fraction, the numerator is bigger than the denominator. To convert a mixed fraction to an improper fraction multiply the denominator by the whole integer part, then add this to the numerator. This is the new numerator of the converted improper fraction. The original denominator is retained. For example, with , multiply 5 by 1, and add this product to 3 to get:

To convert an improper fraction to a mixed fraction, divide the denominator into the numerator to find the whole number part.
The remainder from this division becomes the numerator:

Basic Properties of Numbers

There are three properties of numbers we must be aware of when dealing with the four mathematical operations: 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 what order you add two or more numbers in, the answer is always the same:
  • The Commutative Property of Multiplication: It doesn’t matter what order you multiply two or more numbers in, the answer is always the same:
  • 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:
  • 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:
  • The Distributive Property: To evaluate a number times other numbers added inside parentheses, multiply it by each of the addends inside the parenthesis, then add these products:

The short way to describe this property is: distribute the number outside the parenthesis 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:

With the distributive property we can factor out 6x to get: