ASVAB – Arithmetic Reasoning

ASVAB - Arithmetic Reasoning

Arithmetic reasoning is just a fancy word for math.

Arithmetic reasoning is just a fancy word for math. It can be broken up into basic math and applied math.

Beginning with the basics, we have a set of:

Whole numbers (0, 1, 2): no decimals/fractions, and all positive including zero
Integers (-1, 0, 1, 2): no decimals/fractions, and all positive and negative numbers
Rational numbers (-⅖, 1, ¾): any number written as a fraction with integer numerator and denominator
Irrational numbers (Pi, √2): not a rational number, cannot be written as a fraction
Real numbers (-9, 0, √7): set of all rational and irrational numbers

Knowing the difference between these types of numbers plays a key role in doing well on this exam.

Let’s take a closer look at irrational numbers, as they can be tricky to understand. When dealing with irrational numbers, try to mentally picture them in fraction form.


It can be written as a fraction where both numerator and denominator are integers



Cannot be written as a fraction or ratio of two integers

7 → can be expressed as 7/1, where the numerator is 7 and denominator is 1

0.3333 → all recurring decimals are rational

√16 → can be simplified as 4, where numerator is 4 and denominator is 1


See how both the 7 and 1 are whole numbers?
And how when you simplify the square root of 16 it becomes 4, which is also a whole, rational number?

√2 → cannot be simplified. The √2 is 1.41421356. There’s no way you can put that into fraction form

0.311311131 → decimals are neither recurring nor ending

Pi → the decimal is never-ending, never repeating, and has no pattern


These numbers cannot be simplified or turned into fractions


Learn more about irrational numbers with our learning decks.

But math isn’t just looking at numbers, it includes being able to effectively add, subtract, multiply, and divide every single type of number mentioned. To practice this, come play our learning decks.

Next we have fractions. There are 4 different types of fractions:

  1. Proper → numerator is smaller than denominator (⅔, ½)
  2. Improper → numerator is larger or equal to denominator (6/3, 4/4)
  3. Mixed → combination of whole and fractional numbers (1 ¼, 2 ¾)
  4. Equivalent → when two fractions mean the same quantity (½ = 2/4)  

Depending on what type of fraction you are working with will determine how you will add, subtract, multiply, and divide them. 

You will also need to know how to convert and simplify fractions.

For example, if you have a fraction that is 4/8, you need to be able to simplify it and keep dividing the numerator and denominator by the same number until you cannot go any further. The number 4 goes into both 4 and 8 evenly. So 4 divided by 4 is 1, and 8 divided by 4 is 2. So the fraction becomes ½.


Now let’s say we have a fraction that is 38/8. This is an improper fraction and we must convert it to a mixed number. But first, we must simplify the top and bottom, as we did in the previous example. The hardest part is figuring out what number goes into both 8 and 38 evenly, so let’s start with something simple, like 2. If both numbers are even, it’s always a safe bet to start with 2 and go from there. 38 divided by 2 is 19 and 8 divided by 2 is 4. So now our fraction is 19/4. But we are still not done.

To convert our fraction, always start with figuring out how many times our denominator (4) fits into the numerator (19) evenly and not above it. 4 goes evenly into 19 only 4 times, which is 16. If you try to fit it 5 times, 4 x 5 is 20, which is greater than 19. The remainder now is 3, because 19 - 16 = 3. So now, our mixed number will be 4 3/4.


Remember to always simplify and convert your fractions because the answers will always be written in the simplest form.

When dealing with applied math, the real focus is on real-world problems such as calculating simple interest and compound interest. These types are more likely to be found on the exam than basic math. To find more of these types of questions, check out our learning deck.


As with anything, the more practice and repetition you allow for yourself, the more likely you’ll ace this exam. So remember to practice, practice, practice!