# Math Formula Chart for the ASVAB

The ASVAB test is designed to discover your strengths and weaknesses. It can help you get the job you want in the military, or enable your school to give you proper career advice. There are two sections of the ASVAB that are related to math: Mathematics Knowledge, which will test your algebra and geometry skills, and Arithmetic Reasoning, which focuses on word problems.

Since the ASVAB test is designed to measure your skills, you will not be allowed to use any supporting materials or calculators when taking it. To show what you actually know and can do, you have to prepare well. The following formula chart can help with this preparation. It lists the essential formulas you’ll need if you want to improve your score on the ASVAB Math sections. Try using it to solve the math practice tests we have at Union Test Prep!

## Fractions

Formula Symbols Comment
$$\dfrac{a}{b}+\dfrac{c}{d} = \dfrac{(a \cdot d)+(c \cdot b)}{b \cdot d}$$ $$a, b, c, d = \text{ any real number}$$ Simplify the fraction (if possible)
$$\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{a \cdot c}{b \cdot d}$$ $$a, b, c, d = \text{ any real number}$$ Simplify the fraction (if possible)
$$\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a \cdot d}{b \cdot c}$$ $$a, b, c, d = \text{ any real number}$$ Simplify the fraction (if possible)
$$a \frac{b}{c} = \dfrac{(a\cdot c)+b}{c}$$ $$a, b, c=\text{ any real number}$$ Simplify the fraction (if possible)

## Percents

Formula Symbols Comment
$$a \cdot b\% = a \cdot \dfrac{b}{100}$$ $$a = \text{ any real number}$$
$$b \%= \text{ any percent}$$
Simplify (if possible).
$$\%=\dfrac{\lvert b-a \rvert}{b} \cdot 100 = \dfrac{c}{b} \cdot 100$$ $$\% = \% \text{ increase or decrease}$$
$$a= \text{ new value}$$
$$b= \text{ original value}$$
$$c= \text{ amount of change}$$

## Exponents

Formula Symbols
$$x^a \cdot x^b = x^{a+b}$$ $$a, b, x =\text{any real number}$$
$$\dfrac{x^a}{x^b}=x^{a-b}$$ $$a, b, x =\text{any real number}$$
$$(x^a)^b=x^{a \cdot b}$$ $$a, b, x =\text{any real number}$$
$$(x \cdot y)^a=x^a \cdot y^a$$ $$a, x, y =\text{any real number}$$
$$x^1=x$$ $$x= \text{any real number}$$
$$x^0 = 1$$ $$x = \text{any real number}$$
$$x^{-a} = \dfrac{1}{x^a}$$ $$a,x = \text{any real number}$$

## Factorials

Formula Symbols
$$n \! = 1 \cdot 2 \cdot 3 \cdot \; ... \; \cdot n$$ $$n = \text{any real number}$$

## Logarithms

Formula Symbols
$$a^x = b \Rightarrow \log_a{b}=x$$ $$a, b, x = \text{any real number}$$
$$\ln(x) = \log_e x$$ $$x = \text{any real number}$$
$$e = \text{Euler's number} \approx 2.718$$
$$a^{\log_a x}=x$$ $$a, x = \text{any real number}$$
$$\log(a \cdot b) = \log(a) + \log(b)$$ $$a, b = \text{any real number}$$
$$\log(a \div b) = \log(a) - \log(b)$$ $$a, b = \text{any real number}$$
$$\log(a^b) = b \cdot \log(a)$$ $$a, b = \text{any real number}$$
$$\log_a x = \log_b x \cdot \log_a b$$ $$a, b, x = \text{any real number}$$
$$\log_a b = \dfrac{\log_x b}{\log_x a}$$ $$a, b,x = \text{any real number}$$
$$\log_a a = 1$$ $$a = \text{any real number}$$
$$\log(1) = 0$$

## Statistics

Formula Symbols
$$p = \dfrac{d}{t}$$ $$p = \text{probability of an event}$$
$$d = \text{desired event}$$
$$t= \text{total number of possible events}$$
$$\bar{x} = \dfrac{\sum{x_i}}{n}$$ $$\bar{x} = \text{mean}$$
$$x_i = \text{value of each measurement}$$
$$n = \text{number of measurements}$$
$$s = \sqrt{\mathstrut \dfrac{\sum(x_i - \bar{x})^2}{n-1}}$$ $$s = \text{standard deviation}$$
$$\bar{x}=\text{mean}$$
$$x_i = \text{value of each measurement}$$
$$n = \text{number of measurements}$$
$$V = s^2$$ $$V = \text{variance}$$
$$s = \text{standard deviation}$$
$$CV = RSD = 100 \cdot \dfrac{s}{\bar{x}}$$ $$CV = \text{coefficient of variation}$$
$$RSD = \text{relative standard deviation}$$
$$s = \text{standard deviation}$$
$$\bar{x} = \text{mean}$$

## Algebra

Formula Symbols Comment
$$x+a = b \Rightarrow x = b-a$$
$$x-a = b \Rightarrow x = b+a$$
$$x \cdot a = b \Rightarrow x = b \div a$$
$$x \div a = b \Rightarrow x = b \cdot a$$
$$x^a = b \Rightarrow x = \sqrt[a]{b}$$
$$\sqrt[a]{x} = b \Rightarrow x = b^a$$
$$a^x = b \Rightarrow x = \dfrac{\log b}{\log a}$$
$$a, b = \text{constants}$$
$$x = \text{variable}$$

$$(a \pm b)^2 = a^2 \pm 2ab + b^2$$ $$a,b = \text{constants or variables}$$ Square of a sum or difference
$$a^2-b^2 = (a+b)\cdot (a-b)$$ $$a,b = \text{constants or variables}$$ Difference of squares
$$a^3-b^3 = (a-b) \cdot (a^2+ab + b^2)$$ $$a,b = \text{constants or variables}$$ Difference of cubes
$$a^3+b^3 = (a+b) \cdot (a^2-ab + b^2)$$ $$a,b = \text{constants or variables}$$ Sum of cubes

## Geometry

Formula Symbols Comments
$$A = s^2$$ $$A = \text{area of a square}$$
$$s = \text{side length}$$

$$A = l \cdot w$$ $$A = \text{area of a rectangle}$$
$$l = \text{length}$$
$$w = \text{width}$$

$$A = \dfrac{1}{2} b \cdot h$$ $$A = \text{area of a triangle}$$
$$b= \text{base}$$
$$h = \text{height}$$

$$A = \pi \cdot r^2$$ $$A = \text{area of a circle}$$
$$r = \text{radius}$$

$$A = h \cdot \dfrac{b_1+b_2}{2}$$ $$A = \text{area of a trapezoid}$$
$$b_n = \text{base }n$$
$$h = \text{height}$$

$$C= 2 \pi r = \pi d$$ $$C = \text{perimeter of a circle}$$
$$r = \text{radius}$$
$$d = \text{diameter}$$

$$V = s^3$$ $$V = \text{volume of a cube}$$
$$s = \text{side length}$$

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

$$V = \dfrac{4}{3} \pi r^3$$ $$V = \text{volume of a sphere}$$
$$r = \text{radius}$$

$$V = \pi r^2 h$$ $$V = \text{volume of a cylinder}$$
$$r = \text{radius of base}$$
$$h = \text{height}$$

$$V = \dfrac{1}{3} \pi r^2 h$$ $$V = \text{volume of a cone}$$
$$r = \text{radius}$$
$$h = \text{height}$$

$$V = \dfrac{1}{3} l \cdot w \cdot h$$ $$V = \text{volume of a pyramid}$$
$$l = \text{length}$$
$$w = \text{width}$$
$$h = \text{height}$$

$$d= \sqrt{\mathstrut (y_2 - y_1)^2 + (x_2-x_1)^2}$$ $$d = \text{distance between two points}$$
$$y_n = y \text{ value at point n}$$
$$x_n = x \text{ value at point n}$$

$$a^2 + b^ 2 = c^ 2$$ $$a,b = \text{legs of a right triangle}$$
$$c = \text{hypotenuse of a right triangle}$$
Pythagorean theorem
$$(x-h)^2 + (y-k)^2 = r^2$$ $$(h,k) = \text{center of a circle}$$
$$r = \text{radius}$$
Standard form of a circle
$$x^2 + y^2 + Ax + By + C = 0$$ $$x, y = \text{variables}$$
$$A,B,C = \text{constants}$$
General form of a circle