The **ASVAB Arithmetic Reasoning** test measures a candidate’s ability to solve problems representing word problems and delivers mathematical questions and equations. These may not only be simple questions involving addition, subtraction, multiplication, or division but may also require reasoning skills to determine what is really being asked for and finding the best answer.

These word problems may have some technical terms, besides basic terms, such as area, perimeter, integer, or ratio, which are expected to be common mathematical knowledge. When solving Arithmetic Reasoning questions, you must pay attention not only to the numbers mentioned in the problem but also to the wording, the format of the paragraph, buzzwords, and more.

**Arithmetic Reasoning Tips**

**Finding “buzzwords”**

These words or phrases of emphasis indicate the action you will need to solve the problem. For example, if a problem mentions “difference” or “fewer” or “take away”, it may require you to use subtraction, while some words like “times” or “product” or “double” may stand for multiplication. Before solving the problems, make sure you carefully read and identify what process it requires. It will show you the direction you should take to solve the overall equation.

**Identify numbers**

Word problems can be simple with an addition or subtraction of 2 numbers, or they can include more complex numbers and operations. Pay your attention to all the given numbers and figures within the body of the paragraph. Read carefully these numbers, and then determine which of the numbers are relevant to solve the problem and which of them are misleading you.

Make sure you perform them in the right order. 6 – 8 and 8 – 6 bring two very different results and may affect your pass or fail. Be as careful as possible with the number to avoid unnecessary mistakes.

**Paragraph Format**

When dealing with the Arithmetic Reasoning section, you should notice that many word problems may contain irrelevant information that is used as a filler to distract you from the real question being asked. You must learn to scan all over the problem, disregard this misleading verbiage and focus on the portions that will help you answer the problem. Just because something is included in a paragraph doesn’t mean that it is important and must be used.

By identifying the format and context of the paragraph combining with the buzzwords and numbers, you can build a completed, simplified equation. Be sure that you select all necessary information, make a proper equation, and solve it.

If you run into a problem that stumps you, skip it to move ahead to another one and then come back to it if you have time. Do not waste too much time on a problem, try to quickly solve the other questions that you are certain about it.

**Steps to solving a word problem**

Here is the suggested route to answer the questions in the ASVAB Arithmetic Reasoning test.

**Carefully read the problem**

Because of the limited time, you may push yourself to solve a problem quickly. This easily leads to a disaster of failing the test. Word problems can be tricky, so you have to thoroughly read each to identify exactly what is being asked for.

**Determine the method used to answer**

After thoroughly understanding the problem, you’ll need to gather all the relevant data from the problem and decide what is the best way to solve the question it is asking.

**Setup the equations**

Once you have determined the method used to answer, you need to set all the relevant data into an equation that will lead you to the correct answer.

**Solve equations and review results**

When you have the equations for the question, solve it to find the final result. Then quickly review to make sure there is no regretful mistake in the solving progress.

**Basic Arithmetic Review**

Before starting practicing the Arithmetic problems, let’s review all the basic definitions, properties, and Arithmetic Reasoning formulas you may need in the ASVAB Arithmetic.

**Types of Numbers**

*NATURAL NUMBERS*

Natural numbers (i.e. counting numbers) are numbers that are used for counting and ordering. They can be expressed mathematically as {1, 2, 3, 4, 5, …}

*Even Number*

Even numbers are natural numbers that are divisible by 2.

2ℕ = { 2, 4, 6, 8, 10, 12, 14, … }

*Odd Number*

Odd numbers are natural numbers that are not divisible by 2.

2ℕ + 1 = { 1, 3, 5, 7, 9, 11, 13, 15…}

*Prime Number*

A prime number is a number greater than 1 that is only divisible by 1 and by itself.

Examples:

2, 3, 7, and 11 are prime numbers

P = { 2, 3, 5, 7, 11, 13, 17, 19,…}

*Composite Number*

Composite numbers are the product of some prime numbers. For example:

8 = 2 ⋅ 2⋅ 2

10 = 2 ⋅ 5

*WHOLE NUMBER *

In mathematics, the whole numbers are the basic counting numbers 0, 1, 2, 3, 4, 5, 6, … and so on.

*INTEGERS*

An integer number includes all positive whole numbers (a positive integer), and negative whole numbers (a negative integer), or zero.

We can put that all together like this:

Integers = { …, −4, −3, −2, −1, 0, 1, 2, 3, 4, … }

*FRACTION/RATIONAL NUMBER*

Fraction/rational number is a ratio of two integer numbers in the form of *A/B*, where *A* and *B* are integers and *B#0*.

A is called Numerator

B is called Denominator

*IRRATIONAL NUMBERS*

Real numbers that cannot be written as the quotient of two

integers but can be represented on the number line.

Example:

-2√3 , √2, π

*REAL NUMBERS*

Include all numbers that can be represented on the number

line, that is, all rational and irrational numbers.

**The Basic Number Properties**

Four basic properties of numbers include commutative, associative, distributive, and identity. You should familiarize yourself with each of these before taking the Arithmetic Reasoning subtest.

**Properties of addition**

*Identity Property of Zero *:

*a + 0 = a*

*Inverse Property*:

*a + (-a) = 0*

*Commutative Property *:

When adding two numbers together, the outcome (sum) is the same regardless of the order the numbers are placed in.

*a + b = b + a*

For example, the two following equations end up with the same result:

4 + 6 = 10 or 6 + 4 = 10

*Associative Property *:

When adding multiple numbers together, the outcome (sum) is the same regardless of the order the numbers are placed in.

*(a + b) + c = a + (b + c)*

**Properties of subtractions**

*Non-Commutativity*

Unlike addition, the order of two numbers in subtraction changes all the results. In other words, the subtrahend and minuend are distinct factors when subtracting and they cannot be switched order-wise (except subtrahend and minuend are equal).

*a – b # b – a*

For example :

8 – 6 = 2 is not the same as 6 – 8 = -2

*Non-Associativity*

When subtracting multiple numbers, the order of the numbers does matter. Subtracting numbers in different orders will result in different outcomes.

**Properties of multiplication**

*Property of Zero*:

*a × 0 = a*

*Identity Property of One*:

*a × 1 = a*, where *a # 0*

*Inverse Property*:

*a × 1/a = 1*, where *a # 0*

*Commutative Property*:

When multiplying two numbers together, the product is the same regardless of the order the numbers are placed in.

*a × b = b × a*

For example, the two following equations end up with the same result:

2 × 3 = 6 or 3 × 2 = 6

*Associative Property *:

When multiplying multiple numbers together, the product is the same regardless of the order the numbers are placed in.

*(a × b) × c = a × (b × c)*

For example :

(2 × 3) × 4 = 2 × (3 × 4) = 24

**Properties of division**

*Property of Zero *:

*0/a = 0*, when *a # 0*.

*Property of One *:

*a/a = 1* when *a # 0*.

*Identity Property of One *:

*a/1 = a × 1*.

**Absolute Value**

The absolute value of a number is always greater than 0.

If a > 0, |a| = a.

If a < 0, |a| = a.

For example, |8| = 8 and |-8| = 8. In each case, the answer is positive.

**Order of Operations **

*Step 1 *: Parentheses – Simplify any expressions inside parentheses.

*Step 2 *: Exponents (Powers, Roots) – Work out any exponents.

*Step 3 *: Multiply or Divide before you Add or Subtract

*Step 4 *: Addition and Subtraction These are done last, working from left to right.

For example:

10 – 8 × 4 + (6 ÷ 3) + 5 × 23

= 10 – 8 × 4 + 2 + 5 × 8

= 10 – 32 + 2 + 40

= 20

>> More: **General Science ASVAB Study Guide**

**Integers**

*Adding and subtracting with negatives*

– a – b = (-a) + (-b)

– a + b = b – a

a – (-b) = a + b

Example:

– 2 – 3 = (-2) + (-3) = -5

– 2 + 5 = 5 – 2 = 3

2 – (-3) = 2 + 3 = 5

*Multiplying and dividing with negatives*

-a × b = -ab

-a × -b = ab

(-a)/(-b) = a/b, b # 0

(-a)/b = -a/b, b # 0

Example:

-2 × 3 = -6

-2 × -3 = 6

(-2)/(-3) = ⅔

(-2)/3 = -⅔

**Fraction**

Fractions are another way to express division. The top number of a fraction is called the numerator, and the bottom number is called the denominator.

*Least common multiple*

The LCM of a set of numbers is the smallest number that is a multiple of all the given numbers. For example, the LCM of 5 and 6 is 30, since 5 and 6 have no factors in common.

*Greatest common factor*

The GCF of a set of numbers is the largest number that can be evenly divided into each of the given numbers. For example, the GCF of 24 and 27 is 3, since both 24 and 27 are divisible by 3, but they are not both divisible by any numbers larger than 3.

*Adding and subtracting fractions*

Fractions must have the same denominator before they can be added or subtracted.

If the fractions have different denominators, rewrite them as equivalent fractions with a common denominator. Then add or subtract the numerators, keeping the denominators the same. For example :

*Multiplying and dividing fractions*

When multiplying and dividing fractions, a common denominator is not needed. To multiply, take the product of the numerators and the product of the denominators :

Example :

⅔ × ⅛ = (2 × 1 )/(3 × 8) = 2/24 = 1/12

To divide fractions, invert the second fraction and then multiply the numerators and denominators :

⅔ ÷ ⅛ = (2 × 8)/(3 × 1) = 16/3.

Hope that all these study guides about arithmetic reasoning ASVAB help you prepare well for your coming exam! Take our free **ASVAB Arithmetic Reasoning practice test** now to practice all you have learned!

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