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Get better scores with our complete help for ASVAB (Armed Services Vocational Aptitude Battery) including Arithmetic reasoning and Mathematics Knowledge.

Our comprehensive lessons on ASVAB test cover help on topics like Fractions, Factoring trinomials, Exponents, Solving linear equations, Rational expressions, Solving systems of equations, and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest ASVAB test problems. Then, strengthen your understanding with tons of ASVAB practice questions.

All our lessons are taught by experienced math teachers. Let’s finish your preparation in no time, and ACE that ASVAB test.

Watch, practice, and improve. Learn all the ASVAB Test Prep topics, including:

  • Number Theory
  • Fractions
  • Linear Inequalities
  • Systems of Linear Equations
  • Radicals
  • Exponents and Exponent Rules
  • Rational Expressions
  • Probability
  • Rates, Ratios, and Proportions
  • Exponential Functions

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Multiplying and Dividing Fractions

Fraction is a word from Latin meaning “broken” — a part of a whole. There are two types of fractions: the proper fractions and improper fractions. Proper fractions are those that have a numerator that is smaller than the denominator; whereas the improper fractions, on the one hand, have a bigger numerator than the denominator. Then, there are also the mixed numbers, as known as compound fractions, which are numbers that is a combination of both a whole number and a fraction like 512 5\frac{1}{2} , 3945 39\frac{4}{5} , etc.

Multiplying a whole number with a fraction We will start off by looking at how to multiply fractions. Multiplying fractions with whole numbers allow us to solve for problems when we want to know how much a portion of the whole is. For example, what is a quarter of an hour or what is 45 \frac{4}{5} of 20. When we do the multiplication, all we need to do is multiply the numerator with the whole number and then divide the product by the denominator. Multiplying proper fractions is another kind of multiplication of fractions we will encounter. The calculation process is very much like the same, minus the part where we need to convert the whole number into a fraction. So, we only need to multiply all the numerators and also multiply all the denominators. Here’s an example: 23×14=2×13×4=26 \frac{2}{3} \times \frac{1}{4} = \frac{2\times1}{3\times4} = \frac{2}{6} . Multiplying improper fractions and mixed numbers has a very similar process but requires one more step. We need to convert the mixed numbers into improper fractions before we can do the multiplication.

Then, we will be tackling how to divide fractions. When we are dealing with divisions that involve fractions, we simply need to flip the number(s) after the division sign around to get the reciprocal of the whole number or fraction, and then change the division sign to the multiplication sign. For example, 12÷3=12×13 \frac{1}{2} \div 3 = \frac{1}{2} \times \frac{1}{3} or 4÷78=4×87 4 \div \frac{7}{8} = 4 \times \frac{8}{7} . Then, we just follow the same steps as we would take when we multiply fractions. The same process is used for all fraction divisions, whether it involves whole numbers, proper fractions, or improper fractions. Same as the way how we treat mixed numbers in multiplications, we will need to convert the mixed numbers into improper fractions before we can do the divisions.

As always, we will also discuss how we can apply the skills of multiplying and dividing fractions in our daily life by looking at some word problems.

Here's what we cover in Multiplying and Dividing Factions:
  • How to multiply and divide fractions with whole numbers
  • How to multiply and divide improper fractions with mixed numbers
  • How to solve word problems by adding, subtracting, multiplying, and dividing fractions
  • Get Started Now

Solving Systems of Linear Equations

In this chapter, we are going to further our exploration of linear relations, linear equations, and linear functions, and look into systems of linear equations. As the name suggests, system of linear equations is a set of linear equation. By solving systems of equations, it means that we are to find out if there is an ordered pair (x and y), such that it satisfies both of the linear equations. Graphically speaking, we want to see if these equations will or will not intersect with each other when they are plotted on a coordinate plane.

The 3 cases to consider when graphing the solutions to a system of linear equations.
The 3 cases to consider when graphing the solutions to a system of linear equations

To begin, we will first learn how to determine the numbers of solutions in a system of equations. For a linear system with two equations, we basically want to see if the two straight intersect. There are three cases to consider: One solution; no solutions; and infinite solutions. If the two lines have different slopes, then they will intersect with each other once somewhere on a Cartesian plane. In this case, this system of linear equations is said to have one solution. However, if the lines are parallel to each other, they will have the same slope and just different y-intercepts. These linear equations do not intersect, and therefore, there are no solutions to this system. The last scenario is when the two lines have the same slope and same y-intercept. In this case, the two lines lie on top of each other, and this linear system has infinite solutions.

Solving systems of equations comes in three methods. We can use the graphing method, the elimination method, and the substitution method. In this chapter we will focus on these different methods and at the same time learn the different applications of solving linear systems.

In solving systems of equations by graphing, we simply graph the two equations out and look at the values of x and y, and examine where they intersect. It also is possible that the two lines do not intersect at all. In solving systems of equations by elimination, we can use addition or subtraction to eliminate one variable and then solve for the other. After solving for the value of one variable, we would substitute that value to one of the equations to solve for the other variable. This way, we are able to find if and where these lines would intersect. Solving systems of equations by substitution means that we pick one of the equations from the system and use that to solve for either x or y. After that, we substitute the value of the chosen variable to the other equation to solve for the other variable.

In the last part the chapter, we will learn all about the different applications of linear equations like solving problems related to money, distance problems, and rectangular shapes.

Here's what we cover in Solving Systems of Linear Equations:
  • How to find the number of solutions to a system of linear equations
  • How to solve systems of linear equations by graphing, elimination, and substitution
  • How to solve for word problems involving money, distance and time, unknown number, and rectangular shapes with linear systems
  • Get Started Now


Exponents are always used in mathematical expressions. Exponents tell us how many times a number is multiplied to itself. For example, if we were asked to solve 57 5^7 , we would multiply 5 to itself seven times. In the given expression, 5 is called the base and 7 is called the exponent or power. With the help of exponential calculators, you could easily compute the solution by just plugging in the given numbers. However, if we come across more complicated expressions or expressions that involve more exponents, it would be best to acquaint ourselves with the exponent rules. These rules help us understand the exponent properties and would guide us through the course of simplifying exponents and expressions that contains exponents.

What are exponent rules? In this chapter, we will look at all the rules of exponents. The first rule that we will look at is the product rule, which can be denoted as (ax)(ay)=a(x+y) (a^x)(a^y) = a^{(x+y)} . By using the product rule, when we want to multiply powers with a common base, we can simply do so by adding exponents together. Then, there is the quotient rule which states that, if we are to divide two powers with each other and their bases are the same, we can just subtract the exponents to complete the division. The formula of the quotient rule looks like this: axay=a(xy) \frac{a^x}{a^y} = a^{(x-y)} . There is also the power rule, (ax)y=a(xy) (a^x)^y = a^{(x \cdotp y)} . It comes in handy when an exponent is raised to another exponent. In cases like this, we only need to multiply the exponents without doing anything to the bases. Lastly, we would also look at the negative exponent rule: a1=1a a^{-1} = \frac{1}{a} or amn=1amn a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} . The negative exponent rule suggests that for any base raised to a negative exponent, we only need to flip the number over to get rid of the negative sign. Each of these laws of exponent will be discussed in depth as to how we can integrate and combine them all together to solve more advanced expressions.

Exponent Rules Table
For a ≠ 0, b ≠ 0
Product Rule Product rule for exponents
Quotient Rule Quotient rule for exponents
Power Rule Power rule for exponents
Power of a Product Rule Power of a  product rule for exponents
Power of a Fraction Rule Power of a  fraction rule for exponents
Zero Exponent Exponent rule for a zero exponent
Negative Exponent Exponent rule for a negetive exponent
Fractional Exponent Exponent rule for a fractional exponent

This chapter would also discuss scientific notation, a technique used to express very large or very small numbers through the use of the base 10, raised to an exponent. We would also be converting rational exponents into radical exponents and vice versa and solving problems that involve exponents.

Here's what we cover in Exponents:
  • How to simplify and solve for exponents using the product rule, quotient rule, power rule, and negative exponent rule
  • How to convert between radicals and rational exponents
  • How to write and operate with numbers in scientific notation
  • Get Started Now

Factoring Trinomials

Having learned about the basics of the polynomial expressions and factorization from previous lessons, we are now more equipped to proceed a more complex concept. In this chapter, we are going to learn about factoring polynomial expressions.

Factoring by taking out the GCF Polynomials can be factored by looking at the common factors. However, factoring out common factors one by one from a polynomial can be time consuming. Therefore, we will make use of our knowledge of prime factorization to factor polynomials by taking out the greatest common factor.

Another method we will learn is factoring by grouping. When we are factoring polynomials that have an even number of terms, it is very likely that we can factor these polynomials by grouping. The tricks are to look for the common factors in the polynomials, group them together and then take out the greatest common factors from each group.

Factoring difference of squares is another way to factor polynomials. As we can tell from its name, we would use this factoring method when we have a polynomial of difference of two squares. If we have a polynomial which looks like a2b2 a^2 - b^2 , we know that we can factor this binomial by using the formula:

The difference of two squares

This method would also test our understanding of the FOIL method which we use when we multiply one polynomial to another. This is because, at the end of our factoring process, we can counter check the answers by applying the FOIL method.

Lastly, we will go further and look at two formulas that help us factor polynomials that have a degree of three: difference of cubes, and sum of cubes. Similar to the formula of difference of squares, difference of cubes deals with polynomials that are in the form of a3b3 a^3 - b^3 , while the sum of cubes is used when the polynomials are in the form of a3+b3 a^3 + b^3 . The formulas are as follow:

Difference of cubes
The difference of two cubes
Sum of cubes
The sum of two cubes

There is a mnemonic “SOAP” that helps us keep track of the signs when we factor using these two formulas.

Use the SOAP mnemonic to help you track signs when factoring cubic polynomials
Here's what we cover in Factoring Trinomials:
  • How to factor by taking out the greatest common factor, and grouping
  • How to factor difference of squares
  • How to factor difference and sum of cubes
  • Get Started Now

Rational Expressions

We know that there are different systems of numbers, and one of the numerous number systems is called Rational numbers. Rational numbers are defined as any number that can be expressed as a fraction form, where the denominator can never be equal to zero. In this chapter, instead of discussing about rational numbers, we will go further and look into rational expressions. From the definition we have for rational numbers, we can deduce that a rational expression is simply an expression in a form of ratio.

Adding & subtracting fractions review When solving for mathematical problems and equations, we all know that the more simplified the expressions are, the easier for us to get the answers correctly. Therefore, before we learn how to do any calculations on rational expressions, it is crucial for us to understand how to simplify them. We will learn how to simplify rational expressions by applying our knowledge of prime factorization and factoring algebraic expressions. Like simplifying fractions, the aim of simplifying rational expressions is simply to reduce both the numerator and denominator as much as possible.

As such, we will look at how to simplify rational expressions as well as how to find non-permissible values of the variables when simplify them. If you would review the definition of rational numbers, one of the criteria that needs to be satisfied is that the denominator should not be equal to zero. The reason why the denominator of a rational expression cannot be zero is exactly the same as the reason why the denominator of a fraction cannot be zero — it is just undefined. This is why there are non-permissible values in rational expressions. Non-permissible values are values that would make the algebraic expression in the denominator equal to zero.

Like any other expressions, rational expressions can be simplified and also be combined through the four operations. In other words, adding and subtracting rational expressions, multiplying rational expressions, and dividing rational expressions are all possible and we will learn how to do that one by one. More importantly, we will look at how we can apply our knowledge on rational expressions to practical use in our daily life.

Here's what we cover in Rational Expressions:
  • How to simplify and find the restrictions of rational expressions
  • How to add, subtract, multiply, and divide rational expressions
  • How to solve and apply rational equations in word problems
  • How to simplify complex fractions
  • How to perform partial fraction decomposition
  • Get Started Now


Probability is perhaps one of the most interesting topics in math because you get to apply them in so many real life issues, like the probability of rain, the probability of winning the lottery, and so on.

Probability is the measure of the chance of events to occur in a particular sample space. It can be expressed in percent or fraction form. Probability is more like a simulation of events in a mathematical way in order to predict possible outcomes. Theoretical probability can be computed using the formula:

Formula for theoretical probability

Events can either be independent or dependent. Independent events are not affected by other events. For example, if we flip a coin, the fact that we get a heads in a trial will not increase or decrease our chances of getting a heads or a tails in another trial. That is, the probability of getting a heads or tails is always 1/2 {1}/{2} or 50%, regardless of what we get before. Dependent events, on the other hand, are affected by past events. For example, the probability of choosing a queen from a deck of cards depends on whether a queen has been chosen out of the same deck of cards before.

A tree diagram of probabilities In this chapter, we will be learning how to determine probability using tree diagrams, tables, and Venn diagrams. They are all very useful to help visualize all possible outcomes. By using the tree diagrams, it is much easier for us to look for the favorable outcomes, and then calculate the probability of an event. However, when it comes to complex probabilities, tree diagrams are just not quite enough to fully represent the relationships between different events visually. In cases that involve more variables and complex relationships, we will need to make use of the Venn diagrams. Venn diagrams are visual tools that help us see how each event is related to and overlapped with (if any) each other before we start evaluating any probabilities regarding to the data set.

A venn diagram of probability
venn diagram of probability
Here's what we cover in Probability:
  • How to use tree diagrams, Venn diagrams, and tables to calculate probabilities
  • How to find the probability of independent events
  • Get Started Now

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Common Questions

How to study for the ASVAB?

You can first test yourself with this sample ASVAB practice test to see which areas and topics you need to focus more on. Then, watch our video lessons and work on the practice questions for review and better understanding.

What to expect when taking the ASVAB test?

It is a multiple choice test and consisted of 9 subtests in total, including General Science (GS), Arithmetic Reasoning (AR), Word Knowledge (WK), Paragraph Comprehension (PC), Mathematics Knowledge (MK), Electronics Information (AS), Auto and Shop Information (AS), Mechanical Comprehension (MC), and Assembling Objects (AO). Our AVSAB test prep mainly focuses on reviewing materials from the two subtests that are on math: arithmetic reasoning and mathematics knowledge. You will be given 36 minutes to finish 30 questions on arithmetic reasoning and 24 minutes to answer 25 questions on mathematics knowledge. Calculators are not allowed anytime during the test.

What is the AFQT score?

AFQT stands for Armed Forces Qualification Test. AFQT scores are reported in terms of percentile and calculated based on your scores from 4 subsets - Arithmetic Reasoning (AR), Mathematics Knowledge (MK), Paragraph Comprehension (PC), and Word Knowledge (WK).

What is a good ASVAB score?

All services use AFQT to determine if an applicant is eligible for enlistment. However, the minimum AFQT score required by each service varies. Contact a military recruiter for more detail: Air Force, Army, Navy, Marines, and Coast Guard.

How often can I take the ASVAB test?

If you want to resit for the test, you will need to wait for one calendar month to retake the test after your initial ASVAB. For the second time, you need to wait for two calendar months. After the second resit, you will need to wait for six calendar months so that you can take the test again. The scores are valid for application for enlistment for up to two years.

How much does the ASVAB test cost?

It is free to take the test.

How long is the ASVAB test?

Each subset has a different time limit. Also, there are two ways of how the ASVAB is administered: Paper and Pencil; and Computer (CAT-ASVAB). For the paper and pencil version, you will be allowed 149 minutes to answer 225 questions whereas you will have 154 minutes to finish 154 questions for the CAT-ASVAB version. The CAT-ASVAB test is computer-adaptive. So, the difficulty level of questions given to you changes based on how well you answer the previous ones. Therefore, one important thing to keep in mind when taking the CAT-ASVAB version is that, unlike the paper and pencil ASVAB, you will not be able to review or change an answer once it is submitted.

Where can I take the ASVAB?

It is usually taken place in schools by test administrators from the federal government. If you are a current high school or post-secondary student, contact your teacher or counselor for more information. The ASVAB is offered in schools as part of the ASVAB Career Exploration Program, which is designed to allow students to know more about the civilian and military work. However, if you are not currently enrolled in school, you can take the test at Military Entrance Processing Stations (MEPS), which are a Department of Defense joint-service organization. Yet, if you do not live close to a MEPS, you can take the test at a Military Entrance Test (MET) site which are usually located in Federal government office buildings, National Guard armories, and Reserve centers. For more information, contact a military recruiter. Find a recruiter near you here.

What is ASVAB?

ASVAB stands for Armed Services Vocational Aptitude Battery. It is an aptitude test developed by the U.S. Department of Defense to determine a person’s qualification to be enlisted in the United States Armed Forces and job position assignment in the military. There are two versions of ASVAB test: ASVAB at MEPS, and Student ASVAB. The ASVAB at MEPS is the enlistment version that is used for recruiting purposes only whereas the Student ASVAB is part of the ASVAB Career Exploration Program (CEP) that is carried out in high schools and community centers.

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