ASVAB Study Guide - Arithmetic Knowledge (r)

ASVAB STUDY GUIDE
– ARITHMETIC
KNOWLEDGE

UNDERSTANDING ARITHMETIC KNOWLEDGE SECTION OF THE
ASVAB

PRESENTED BY: P DENSE SMITH

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

The Mathematics Knowledge section of the exam measures your knowledge of various math areas, such
as algebra and geometry. You may be asked to find the square root of a number or the volume of a brick
with given dimensions. Algebraic problems may require finding the value of “y” in a given equation. A
review of math symbols—such as ≠, ≤, and √—can help you solve the given problems much faster, and
using our ASVAB math study guide to practice answering the algebra and geometry questions on the
test can help increase your overall AFQT score. The CAT-ASVAB has 16 questions in 20 minutes; the
paper-and-pencil version has 25 questions in 24 minutes.

Fractions

Like percentages, a fraction is part of a whole but presented differently. It has a numerator and a
denominator. In the fraction 1/2, 1 is the numerator which represents the part, while 2 is the
denominator which represents the total number of parts. Fractions, percentages, and decimals are
actually different ways of presenting the same concept of parts of a whole. For instance:
The fraction 1/100 is the same as the percentage 1% and the decimal [0.01].

The fraction 1/2 is the same as 50% and [0.5].
Converting Fractions to Percentage or Decimal
ASVAB questions involve converting fractions to percentages or decimals, or the other way around. The
methods are quite straightforward, actually, and are as follows:
To convert a fraction to its percentage form: divide the numerator by the denominator, multiply by 100,
and then add the % sign.
Thus, converting 1/2 to a percentage involves the following steps:

Divide the numerator by denominator:
1/2=0.512=0.5
(Note: This is the decimal form of 1/2.)

Multiply by 100: 0.5 x 100 = 50
Add the percentage sign: 50%

To revert 50% to fraction form:
Write 50% as 50/100.

Reduce 50/100 to its lowest form, which is 1/2.
Simplifying Fractions
Many questions require simplifying fractions. It means reducing a fraction to its simplest form.

Take the following fractions:
3/6,5/10,7/14,4/8,,5/10,7/14,4/8,2/4

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

They all actually reduce to 1/2. Did you find it easy simplifying those fractions to 1/2 If not, here’s how:

Let’s take the fraction: 7/14

Factor the numerator and denominator:

Find the Greatest Common Factor then divide the numerator and denominator by it.
✓ Factors are the 2 numbers that you multiply to get the number in this case to get number 7 and
number 14
✓ Choose the biggest (otherwise known as Greatest) factor that they both have in common (the
same factor they both have)
✓ Divide by that factor
✓ In our example 7 is the Greatest Common Factor or GCF

By dividing the 7 in the numerator and denominator, it leaves the reduced fraction: ½.

Equivalent Fractions

Equivalent fractions look different at first glance, but are actually the same in value. In the previous
illustration, for instance, 1/2 is equivalent to:

7/14,4/8,2/4,3/6,5/10,

They are all equivalent fractions. Equivalent means equal, because when they are reduced they equal
the same fraction. In the case above they all equal 1/2

In a question, you may need to find a number that is equivalent to 1/2 but has a denominator of 10.
You will get the missing numerator by using this relationship:

1/2=numerator/10

You will find the numerator by cross-multiplying 10 and 1 = 10, and dividing 10 by 2 = 5. The numerator
is 5.

This gives you the equivalent fraction of 5/10.

LCD

The LCD, or least common denominator, is the least common multiple (LCM) of the denominators in two
or more given fractions. How’s that again?

To illustrate, let’s look at the fractions:

3/8,4/5,1/10

The denominators are 8, 5, and 10. The multiples of each of those denominators are:

8: 8, 16, 24, 32, 40, 48,…
5: 5, 10, 15, 20, 25, 30, 35, 40, 45,…
10: 10, 20, 30, 40, 50,…

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

The smallest multiple is 40; hence, it is the LCD of the given fractions. The LCD is a very important
concept when learning about fractions, as you will see later.
Mathematical Operations Involving Fractions
ASVAB questions may ask for addition, subtraction, multiplication, and division of fractions, too. Here
are the basic rules:
Addition with the same denominators:
Simply add the numerators and copy the common denominator.
Example:
1/9+4/9=5/9
Addition with different denominators:

• Determine the LCD.
• Find the equivalent fractions so that all fractions have the same denominator (which is the LCD).
• Proceed to addition.
Example:
2/5+3/10=?
The LCD is 10 (10 is the smallest common multiple or denominator – both denominators go into 10).
The equivalent fraction of 2/5 using the LCD of 10 is 4/10
✓ We multiply 5 x 2 to get 10 therefore we have to multiply 2 x 2 which is 4.
The question is now rewritten as:
4/10+3/10 (now we have the same denominators so we can add).
We get 7/10.
Subtraction of fractions with the same denominators as well as different denomintors:
The procedure is similar to
✓ Addition involving fractions with the same denominators.
✓ Subtraction with different denominators:
Multiplication:
Multiply the numerators (or top numbers).
Multiply the denominators (or bottom numbers).
Simplify or reduce, if the resulting fraction can still be simplified or reduced.

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

Example:
4/5⋅6/22= (4*6)/(5*22)=24/110=12/55 (simplify by GCF 2 therefore divide top and bottom by 2)
Division:
Turn the second fraction upside down (called the reciprocal form of the fraction), and then proceed to
multiplication (s0me remember this process by Keep Change Flip or KFC).
Example:
12/55÷4/5=12/55 * 5/4=60/220 = 6/22 ( we simplified by the GCF of 10, so we divided by 10)

Percentages
Percent means a number out of 100, or a part of a whole. So 20% simply means 20 out of 100,
or 20/100.
You should familiarize yourself with a number of concepts to be adequately prepared for all percentage
questions. These concepts include the proportion method of solving percentages, the different parts of
percentage equations, and the language used when describing percentages or percentage equations.
Proportion Method
Many questions in the ASVAB Mathematics Knowledge section can be solved using the proportion
method of solving percentages, which may be summed up as follows:
Percent/100=part/whole
Take this question, for instance:
What percentage of 20 is 5?
In this case, the percentage is unknown and may be represented by x.
The number 5 is the part and 20 is the whole.
Plugged into the relationship above, the equation would look like this:
x/100=5/20
By Cross multiplying It will then be an easy to solve for x,

✓ 20x = 500 (20 * x) = (100 * 5)
✓ X= 25, the percentage is 25%.
Percentage Decrease or Increase
The proportion method described above comes in handy when solving for more difficult percentage
questions. Try using it in the following question.
The laptop now sells at $499 but its original price was $699. By what percent was the laptop discounted?
Solve first for the decrease in price:

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

699–499=200

This is the part of the whole price that was discounted from the original price. The percent discount x,
then, is:

x/100=200/699

Cross multiply and Solve for x, the percent representing the discount is 28.61%.

Properties

Knowing and understanding the terms associated with properties is of vital importance. Become familiar
with terms such as prime number, composite, whole number, factorization, greatest common factor,
least common multiple, cubes, roots, and reciprocal.

Though this may seem hard, most terms are easy to understand and piggyback off of one another. A
prime number, for instance, is a number only divisible by itself and 1, while a composite number is a
number divisible by multiple numbers. Studying these terms and how to apply them to mathematical
equations can prepare you for the 25 questions found on the ASVAB.

Algebra

The ASVAB requires you to have a basic knowledge of algebra and algebraic equations. For instance,
students may be required to solve for x in an equation such as:

x+8=35

You always want to isolate the variable (x here) on one side of the equation so that your answer will be
on the other side. To do this, you need to perform the opposite operation (of the number that needs
removing) to both sides of the equation. This is how that would look for this problem:

x+8−8=35−8

x=27

Another example:

x/4=8

Here, the opposite operation of the division shown in x/4 is multiplication. So:

4*x/4=8 *4

x=32

You may also be asked to find the square root of a number or the reciprocal of a number. Study and
practice completing basic algebraic equations to prepare yourself for this part of the mathematics test.

Exponents

Exponents are also called powers. They can be positive or negative numbers. They can even be fractions,
which will be discussed under “Roots.”

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

A positive exponent on a number is the number of times a number is multiplied by itself.
85 is the same as: 8⋅8⋅8⋅8⋅8 (8 multiplied by itself 5 times, the number 5 is the exponent)
A negative exponent is the number of times 1 is divided by the number with the exponent.
8–5 is the same as: 1/8⋅8⋅8⋅8⋅8 (or you can just compute 85 and make the answer a fraction)
So if you see an expression like this:
(a4)(c–2)/b–3
it is actually the same as:
(a4)(b3)/c2 WHY???

✓ Your answer can never contain a negative exponent
✓ If a negative exponent is in the numerator remember it really is a fraction therefore you would

move it to the denominator
✓ If a negative exponent is in the denominator it should be moved to the numerator
Multiplying terms with the same variables but different exponents simply involves copying the
variables and adding the exponents.
(x+1)5⋅(x+1)6=(x+1)5+6=(x+1)11
In multiplying terms with constants and different variables, simply multiply the constants separately,
multiply the similar variables according to the rule, and copy dissimilar variables and their exponents.
3x2(2x3y2)=(3⋅2)(x2⋅x3)(y2)=6x5y2
When you need to raise a number or variable with an exponent to another power, such as (x5)3, simply
copy the number or variable and raise it to the product of the exponents.
(x5)3=x5 * 3=x15
Roots
You may be familiar with roots, also called radicals, when they are presented inside the symbol √ or
radix. Recognize them, too, when they are presented as fractional exponents, as in:
x1/2 which is the same as x−−√x (which is pronounced as the square root of x)
251/2= √25=5 (square root of 25 =5)
y3/2=(y3)1/2=√y3 (square root of y to the 3rd power)

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

Logarithms
While logarithm questions do not commonly appear in the ASVAB pencil-and-paper test, they often do
in CAT-ASVAB. Students often eye logarithms suspiciously. Well, you shouldn’t. It’s simply a variation of
exponents.
The expression:
log2(8) simply asks the question: “How many times will you have to multiply 2 (called “base”) by itself to
get 8?” The answer is 3.
Therefore, log2(8) =3 (Read as: “The log base 2 of 8 is 3.”)
Its exponential form is: 23=8. So its sort of like doing exponents backwards
Logarithmic expressions often appear in equations where you must solve the value of a variable.
Consider the following question:
Solve for x if:
log3(2x+1)=log3(9x)
The logarithmic expressions on the left and right sides of the equation have the same base (3), and the
only way for the expressions to be equal is for the “arguments,” or the terms inside the parentheses, to
be equal. Thus, you can proceed to solve with:
2x+1=9x which will give you the value of x=1/7

Polynomials
Polynomials are algebraic expressions consisting of several terms, such as constants, variables, and
exponents, combined together by addition, subtraction, multiplication, and division, excluding division
by variables.
Factoring polynomials
This is very useful in solving various problems involving polynomials.
Factor the polynomial:
x2–7x+12
Factor means find the the factors or numbers and variables when multiplied together equal the answer.
In this case the answer or product is x2–7x+12
The factors are set up like this:
(x+?)(x+?)

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

Start with the general format for factoring a polynomial
✓ The variable x has an exponent of 2.
✓ We know that x * x = x2 (because of the exponent rules we discussed earlier…When you
multiply variables you add the exponents)
✓ Therefore we set up the factoring as (x +? )(x + ?)
✓ Now we need to replace the ? with 2 different numbers
✓ Think of two numbers which when multiplied will result in +12, and when added together will
result in –7. That’s quite easy:

–3 *⋅–4=+12 and –3+(–4)=–7

Plug –3 and –4 into the general format:

(x–4)(x–3)

These are the factors of the polynomial.

Solving polynomials

When a problem tells you to solve the polynomial, it is the same as telling you to find the roots of the
polynomial, or solving for the values for x. Using the same illustration as above and continuing to find
the roots, set each of the factors equal to zero and solve for x:

We already know
✓ (x -4) (x-3)

We take each parenthesis and make an equation equal to zero

✓ x–4=0
✓ x–3=0

We solve for x in each equation
✓ x=4 (you add 4 to the left and right side of the equation .. x -4 +4 = 0 +4)
✓ x=3 (you add 3 to the left and right side of the equation .. x -3 +3 = 0 +3)
✓ These are the roots of the polynomial, or the values of x.

PEMDAS

Don’t forget PEMDAS, either. When performing a series of mathematical operations, follow this order:
parentheses, exponents or powers and roots, multiplication, division, addition then subtraction .

✓ 7+(8*62+7)----Look at what’s in parenthesis 1st and Calculate exponent operation 2nd
✓ 7+(8*36+7) --- Multiply
✓ 7+(288+7)---- Add within parenthesis
✓ 7+295=302--- Solve

An easy way to remember PEMDAS is to learn the phrase: Please Excuse My Dear Aunt Sally.

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ASVAB STUDY GUIDE – ARITHMETIC KNOWLEDGE

KNOWING WHEN TO USE THE RULES AND PROPERTIES ABOVE TAKE PRACTICE
✓ Remember to identify the type of problem
o Circle keywords
o Underline numbers

✓ Write rules that apply to that type of problem or equation
✓ Draw picture if necessary
✓ Set up your problem
✓ Solve

IF YOU APPLY THIS METHOD YOU WILL DO WELL….

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