# Important Math Formulas to Know for the ASVAB

## What Type of Math Is on the ASVAB?

Mathematics for the ASVAB test is much more than just numbers - it involves understanding the principles and methodologies that drive the solutions to problems. The breadth of math concepts that you may encounter in this test is wide. It begins with basic arithmetic operations like addition, subtraction, multiplication, and division, which form the foundation for many other math concepts. You should be proficient in performing these operations not just with whole numbers but also with fractions, decimals, and percentages, as these form the crux of many practical problems.

Moving up the complexity ladder, the test includes more advanced topics such as algebra, geometry, statistics, and probability. Algebra involves understanding how to solve equations, inequalities, and working with expressions and polynomials. It’s about finding unknown variables and establishing relationships between different quantities. Geometry involves a spatial understanding of shapes and figures, calculations of area, volume, and understanding key theorems like the Pythagorean theorem. Statistics and probability, while less common, deal with data interpretation, understanding the likelihood of events, and making predictions.

The ASVAB consists of two key sections that test your math skills - the Arithmetic Reasoning (AR) and Mathematics Knowledge (MK) sections. The AR section focuses more on your problem-solving skills. It measures your ability to interpret and solve word problems that often replicate real-life scenarios. These could include calculating distances if you were given speed and time, figuring out costs based on price and quantity, or working out times based on schedules. This section tests not just your raw mathematical ability but also your logical and reasoning skills.

Your performance on the MK section, together with your score in the AR section, contributes to your overall Armed Forces Qualification Test (AFQT) score. The MK section assesses your direct knowledge and application of high school math principles. The topics are more specific and detailed in this section, necessitating a strong grasp of high school level math.

While the AR section involves more practical, real-world applications, the MK section is about mastering more academic, theory-based mathematical knowledge. Understanding algebra, including how to solve equations and inequalities, interpreting algebraic expressions, and working with polynomials is crucial. Geometry too plays a significant part - you need to understand the basic properties of shapes and figures, calculate areas and volumes, and know the Pythagorean theorem.

## How To Prepare for the ASVAB Math Sections

Planning a study schedule well in advance of your test date can be very helpful. This gives you ample time to review these mathematical principles and concepts thoroughly. Make this schedule consistent and allocate regular time for each topic, ensuring you cover all bases.

To help you, we have compiled the essential arithmetic formulas you’ll need for both math parts of the ASVAB. You won’t have access to these formulas during the test, so it is important to master them before test day. The first two charts below pertain to the Arithmetic Reasoning section of the ASVAB, and the last two refer to concepts included in the Mathematics Knowledge section. We think you will find all of them extremely useful as you study.

Online practice tests are another great resource that can boost your performance. They help to evaluate your knowledge level and familiarize you with the test format of the ASVAB. Regular practice can help you identify areas where you might need additional review.

Finally, it’s important to maintain a positive mindset throughout your preparation. Math can indeed be challenging, but remember that every problem has a solution. Stay dedicated, practice regularly, and believe in your ability to learn and solve. A calm and focused mind can learn faster and perform better. We wish you the best of luck in your preparation and for your ASVAB test!

## ASVAB Arithmetic Reasoning Test

### Formulas for Arithmetic Questions

Formula Symbols Comment
Basic
Properties
of Numbers
$$a+b=b+a$$
$$a \cdot b=b \cdot a$$
a,b = any constant or variable Commutative
property
Basic
Properties
of Numbers
$$a+(b+c)=(a+b)+c$$
$$a \cdot (b \cdot c)=(a \cdot b) \cdot c$$
a,b,c = any constant or variable Associative
property
Basic
Properties
of Numbers
$$a \cdot(b+c) = a \cdot b + a \cdot c$$ a,b,c = any constant or variable Distributive
property
Basic
Properties
of Numbers
$$a+0=a$$ a = any constant or variable Identity
property of
Basic
Properties
of Numbers
$$a \cdot 1 = a$$ a = any constant or variable Identity
property of
multiplication
Rules of
Operations
$$e \pm e = e$$
$$e \pm o = o$$
$$o \pm o = e$$
e = even number
o = odd number
Rules of
Rules of
Operations
$$e \cdot e = e$$
$$e \cdot o = e$$
$$o \cdot o = o$$
$$p \cdot p = p$$
$$n \cdot n = p$$
$$p \cdot n = n$$
e = even number
o = odd number
p = positive number
n = negative number
Rules of
multiplication
Fractions $$\frac{a}{b} + \frac{c}{d} = \frac{(a \cdot d)+(c \cdot b)}{(b \cdot d)}$$ a,b,c,d = any real number Remember to
simplify the
fraction
(if possible).
Fractions $$\frac{a}{b} \cdot \frac{(a \cdot c)}{(b \cdot d)}$$ a,b,c,d = any real number Remember to
simplify the
fraction
(if possible).
Fractions $$\frac{a}{b} \div \frac{(a \cdot d)}{(b \cdot c)}$$ a,b,c,d = any real number Remember to
simplify the
fraction
(if possible).
Fractions $$a\frac{b}{c} = \frac{(a \cdot c) + b}{c}$$ a,b,c = any real number Remember to
simplify the
fraction
(if possible).
Exponents $$x^a \times x^b = x^{a+b}$$ a,b,x = any real number
Exponents $$\frac{x^a}{x^b} = x^{a-b}$$ a,b,x = any real number
Exponents $$(x^a)^b =x ^{a \cdot b}$$ a,b,x = any real number
Exponents $$(x \cdot y)^a = x^a \cdot y^a$$ a,x,y = any real number
Exponents $$x^1=x$$ x = any real number
Exponents $$x^0=1$$ x = any real number
Exponents $$x^{-a} = \frac{1}{x^a}$$ a,x = any real number
Percents % a in b = $$\frac{a}{b} \cdot 100$$ a, b = any real number
Percents $$a \cdot b \% = a \cdot \frac{b}{100}$$ a = any real number
b % = any precent
Remember to
simplify the
fraction
(if possible).
Percents $$\% = \frac{|b-a|}{b} \cdot 100 = \frac{c}{b} \cdot 100$$ % = % increase or decrease
a = new value
b = original value
c = amount of change

### Formulas for Probability and Problem-Solving Questions

Formula Symbols Comment
$$\overline{x} = \dfrac{\Sigma x_i}{n}$$ $$\overline{x}$$ = sample mean
$$x_i$$ = value of each measurement
$$n$$ = number of measurements

$$Md = (\frac{n+1}{2})^{th} \ term$$ $$Md$$ = median
$$n$$ = number of measurements (odd)

$$Md = \dfrac{(\frac{n}{2})^{th} \ term + (\frac{n}{2} +1)^{th} \ term}{2}$$ $$Md$$ = median
$$n$$ = number of measurements (even)

$$s = \sqrt{\Sigma(x_i- \overline{x})^2 / (n-1)}$$ $$s$$ = standard deviation
$$\overline{x}$$ = mean
$$x_i$$ = value of each measurement
$$n$$ = number of measurements

$$V = s^2$$ $$V$$ = variance
$$s$$ = standard deviation

$$CV = RSD = 100 \cdot \frac{s}{\overline{x}}$$ $$CV$$ = coefficient of variation
$$RSD$$ = relative standard deviation
$$s$$ = standard deviation
$$\overline{x}$$ = Mean

$$p= \frac{d}{t}$$ $$p$$ = probability of an event
$$d$$ = number of ways desired event can occur
$$t$$ = total number of possible events

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ $$P(A \cup B)$$ = probability of A or B
$$P(A)$$ = probability of A
$$P(B)$$ = probability of B
$$P(A \cap B)$$ = probability of A and B
$$P(A \cap B) = P(A) \cdot P(B)$$ $$P(A \cap B)$$ = probability of A and B
$$P(A)$$ = probability of A
$$P(B)$$ = probability of B
Independent events
$$P(A \cap B) = 0$$ $$P(A \cap B)$$ = probability of A and B Mutually exclusive Events
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$ $$P(A|B)$$ = probability of A given B
$$P(A \cap B)$$ probability of A and B
$$P(B)$$ = probability of B
Conditional probability
$$SI= P \cdot IR \cdot t$$ $$SI$$ = simple interest
$$P$$ = principal (amount borrowed
$$IR$$ = interest rate
$$t$$ = time (same units as in IR

$$A_{SI} = P + SI = P \cdot (1+IR \cdot t)$$ $$A_{SI}$$ = future value to be paid (for $$SI$$)
$$P$$ = principal (amount to be paid)
$$SI$$ = simple interest
$$IR$$ = interest rate
$$t$$ = time (same units as in $$IR$$)

$$A_{CI} = P \cdot (1 + \frac{IR}{n})^{n \cdot t}$$ $$A_{CI}$$ = future value to be paid (for $$CI$$)
$$P$$ = principal (amount borrowed)
$$IR$$ = interest rate
$$n$$ = number of times interest is compounded
per time unit
$$t$$ = time (same units as $$IR$$)

$$d= r \cdot t$$
$$w = r \cdot t$$
$$W = r \cdot t$$
$$d$$ = distance
$$w$$ = wage
$$W$$ = work done
$$r$$ = rate
$$t$$ = time

$$TW = WA + WB$$ $$TW$$ = total work done
$$WA$$ = work done by A
$$WB$$ = work done B
Combined work
$$\frac{1}{t} = \frac{1}{t_A} + \frac{1}{t_B}$$ $$t$$= total time
$$t_A$$= time consumed by A
$$t_B$$ = time consumed by B
Combined work
$$D_u=S_u \cdot \frac{D_u}{S_u} = S_u \cdot CF$$ $$D_u$$ = desired unit
$$S_u$$ = starting unit
$$CF$$ = conversion factor
Multiple steps may
be needed.
$$a \cdot b\% = a \cdot \frac{b}{100}$$ $$a$$ = any real number
$$b\%$$ = any percent
Remember to simplify
(if possible)
$$\% = \frac{|b-a|}{b} \cdot 100 = \frac{c}{b} \cdot 100$$ $$\%$$ = % increase or decrease
$$a$$ = new value
$$b$$ = original value

$$x_n = x_1 + d(n-1)$$ $$x_n = n^{th}$$ element of an arithmetic
sequence
$$x_1 = 1^{st}$$ element of an arithmetic
sequence
$$d$$ = common difference
$$n$$ = the number of terms in the sequence
Arithmetic sequence
$$x_n = x_1 \cdot r^{n-1}$$ $$x_n = n^{th}$$ element of a geometric
sequence
$$x_1 = 1^{st}$$ element of a geometric
sequence
$$r$$ = common ratio
$$n$$ = the number of terms in the sequence
Geometric sequence

## ASVAB Mathematics Knowledge Test

### Formulas for Algebra Questions

Category Formula Symbols Comment
General
Algebra
$$x+a=b \Rightarrow x=b-a$$
$$x-a=b \Rightarrow x=b-a$$
$$x\cdot a = b \Rightarrow x=b \div a$$
$$x \div a = b \Rightarrow x = b \cdot a$$
$$x^a=v \Rightarrow x=b^a$$
$$\sqrt[a]{x} = b \Rightarrow x= b^a$$
$$a^x=b \Rightarrow x = \frac{log b}{log a}$$
a, b = constants
x = variable

General
Algebra
$$x^a+x^a=2x^a$$ a, b, x = any real number
General
Algebra
$$x^a \cdot x^b=x^{a+b}$$ a, b, x = any real number
General
Algebra
$$\frac{x^a}{x^b}=x^{a-b}$$ a, b, x = any real number
General
Algebra
$$(x^a)^b = x^{a \cdot b}$$ a, b, x = any real number
General
Algebra
$$(x \cdot y)^a = x^a \cdot y^a$$ a, b, y = any real number
General
Algebra
$$x^1 = x$$ x = any real number
General
Algebra
$$x^0 = 1$$ x = any real number
General
Algebra
$$x^{-a} = \frac {1}{x^a}$$ a, x = any real number
General
Algebra
$$x^{\frac {a}{b}} = \sqrt[b]{x^a} = (\sqrt[b]{x})^a$$ a, b, x = any real number
General
Algebra
$$\frac{x}{\sqrt{y}} \cdot \frac {\sqrt{y}}{\sqrt{y}} = \frac{x \sqrt{y}}{y}$$ x, y = any real number
Equations
$$(a \pm b)^2 = a^2 \pm 2 \cdot a \cdot b + b^2$$ a, b = constants or variables Perfect Squares
Equations
$$a^2 - b^2 = (a-b) \cdot (a+b)$$ a, b = constants or variables Difference of
squares
Cubic
Equations
$$a^3 - b^3 = (a-b) \cdot (a^2+ab+b^2)$$ a, b = constants or variables Difference of cubes
Cubic
Equations
$$a^3 + b^3 = (a+b) \cdot (a^2-ab+b^2)$$ a, b = constants or variables Sum of cubes
Linear
Equations
$$A \cdot x + B \cdot y = C$$ A, B, C = any real number
y = dependent variable
x = independent variable
Standard form
Linear
Equations
$$y= m \cdot x +b$$ y = dependent variable
m = slope
x = independent variable
b = y-axis intercept
Slope intercept
form. Try to convert any
given linear equation to
this format.
Linear
Equations
$$m = \frac{(y_2 - y_1)}{(x_2 - x_1)}$$ m = slope
$$y_n$$ = independent variable (point n)
$$x_n$$ = dependent variable (point n) |
This is a rearrangement
of the point-slope form
Linear
Equations
$$y-y_1 = m(x-x_1)$$ $$(x_1,y_1)$$ = point on the line
m = slope
y = independent variable
x = dependent variable
Point-slope form
Equations
$$y = ax^2 + bx + c$$ a, b, c = constants
c = y axis intercept
x = variable
Standard form
Equations
$$y = a(x-h)^2 + k$$ k = y coordinate of the vertex
a = constant
h= x coordinate of vertex
x = variable
Vertex form
Equations
$$x= \frac{-b}{2a}$$ a, b= constants
x = x-coordinate of axis of symmetry
Axis of symmetry for an
equation in the form
$$ax^2+bx+c=0$$
Equations
$$x= \frac{-b \pm \sqrt{b^-4ac}}{2a}$$ a, b= constants
c = constant (y-axis intercept)
x = variable
equation in the form
$$ax^2+bx+c=0$$

### Formulas for Geometry Questions

Category Formula Symbols Comment
Coordinate
Geometry
$$Ax+By=C$$ A, B, C = any real number
y = dependent variable
x = independent variable
Standard form
Coordinate
Geometry
$$y=m \cdot x+b$$ y = dependent variable
m = slope
x = independent variable
b = y-axis intercept
Slope intercept form:
Try to convert any given
linear equation to this
form.
Coordinate
Geometry
$$m= \frac{(y_2-y_1)}{(x_2-x_1)}$$ m = slope
$$y_n$$ = dependent variable (point n)
$$x_n$$ = independent variable (point n)
This is a rearranged
version of the point-
slope form
Coordinate
Geometry
$$y-y_1=m(x-x_1)$$ $$(x_1, y_1)$$ = point on the line
y = dependent variable
x = independent variable
m = slope
Point-slope form
Coordinate
Geometry
$$d= \sqrt{(y_2-y_1)^2 + (x_2-x_1)^2}$$ d = distance between two points
$$(x_1, y_1)$$ = first point
$$(x_2, y_2)$$ = second point

Regular
Polygons
$$\Sigma \theta = 180 \cdot (n-2)$$ $$\Sigma \theta$$ = sum of the interior angles
n = number of sides

Regular
Polygons
$$\theta =\frac{180 \cdot (n-2)}{n}$$ $$\theta$$ = measure of interior angle
n = number of sides

Triangles $$P=s_1+s_2+s_3$$ P =perimeter of a triangle
$$s_n$$ = side length

Triangles $$A= \frac{1}{2} b \cdot h$$ A = area of a triangle
b = base
h = height (altitude)

Triangles $$\Sigma \theta = 180^\circ$$ $$\Sigma \theta$$ = sum of the interior angles
Triangles $$a^2+b^2=c^2$$ a, b = legs of a right triangle
c = hypotenuse of a right triangle
Pythagorean
Theorem
Quadrilaterals $$P=4 \cdot s$$ P = perimeter of a square
s = side length

Quadrilaterals $$P = (2 \cdot l)+(2 \cdot w)$$ P = perimeter of a rectangle
l - length
w = width

Quadrilaterals $$A= s^2$$ A - area of a square
s = side length

Quadrilaterals $$A = l \cdot w$$ A = area of a rectangle
l = length
w = width

Quadrilaterals $$A = h \cdot \frac{(b_1+b_2)}{2}$$ A = area of a trapezoid
$$b_n$$ = base n
h = height

Circles $$(x-h)^2 + (y-k)^2 = r^2$$ (h, k) = center of a circle
x, y = variables
Standard form of
a circle
Circles $$x^2+y^2+Ax+By+C = 0$$ x, y = variables
A, B, C = constants
General form of
a circle
Circles $$C = 2 \cdot \pi \cdot r$$
$$C = \pi d$$
C = circumference (perimeter) of a circle
d = diameter
$$\pi$$ = 3.14

Circles $$s=r \cdot \theta$$ s = arc length
$$\theta$$ = central angle in radians

Circles $$A=\pi \cdot r^2$$ A = area of a circle
$$\pi$$ = 3.14

Prisms $$V = l \cdot w \cdot h$$ V = volume of a rectangular prism
l = length
w = width
h = height

Prisms $$SA = \Sigma A_{fi}$$ SA = surface area of a prism
$$A_{fi}$$ = area of face i

Pyramids $$V= \frac{1}{3} b \cdot h$$ V = volume of a pyramid
b = area of the base
h = height

Pyramids $$SA = \Sigma A_{fi} +b$$ SA - surface area of a pyramid
$$A_{fi} =$$ area of face i
b = area of the base

Cones $$V= \frac{1}{3} \pi \cdot r^2 \cdot h$$ V = volume of a cone
r = radius of the cone
h = height of the cone
$$\pi$$ = 3.14

Cylinders $$V= \pi \cdot r^2 \cdot h$$ V = volume of a cylinder
Cylinders $$SA = 2b + (c \cdot h)$$ SA = surface area of a cylinder