 # Formulas You’ll Need for the PSAT/NMSQT® Test

## What do you need to succeed on the math section of the PSAT/NMSQT® Test?

Everything can be summarized in two sentences:

### Why?

Well, you are going to be able to use a calculator during the PSAT/NMSQT® Math Test-Calculator, but if you don’t know how to use it to get the results you want accurately and quickly, you probably won’t have the time to solve all the questions in that section.

### Do I Need to Know Formulas?

For both the Calculator and the No Calculator sections, you’ll need to know the formulas. Most of them are going to be given to you, but you need to know when to use them, and how to manipulate them to get the results you need.

### What’s the Best Way to Prepare?

We are here to help you! Below, you’ll find a formula chart with the essential formulas in math that you are probably going to use during your PSAT/NMSQT® Math Test.

### Why don’t you start becoming familiar with them by solving the free sample exercises we have for you here?

Category Formula Symbols Comment
Heart of
Algebra
$$x+a = b \rightarrow x = b-a$$
$$x-a = b \rightarrow x= b+a$$
$$x \cdot a = b \rightarrow x = b \div a$$
$$x\div a = b \rightarrow x = b \cdot a$$
$$x^a = b \rightarrow x = \sqrt[a]{b}$$
$$\sqrt[a]{x}=b \rightarrow x = b^a$$
$$a^x = b \rightarrow \dfrac{\log{b}}{\log{a}}$$
$$a,b$$ = constants
$$x$$ = variable

Heart of
Algebra
$$Ax+By=C$$ $$A, B, C$$= Constants
$$y$$ = dependent variable
$$x$$ = independent variable
Standard Form
Heart of
Algebra
$$y=mx+b$$ $$m$$ = slope
$$b$$ = intercept
$$y$$ = dependent variable
$$x$$ = independent variable
Slope-Intercept Form
Heart of
Algebra
$$y-y_1 = m(x-x_1)$$ $$m$$ = slope of a line
$$y_n$$ = dependent variable (point n)
$$x_n$$ = independent variable (point n)
Point-Slope Form
Heart of
Algebra
$$m = \dfrac{y_2-y_1}{x_2-x_1}$$ $$m$$ = slope of a line
$$y_n$$ = dependent variable (point n)
$$x_n$$ = independent variable (point n)
Slope of a line
Heart of
Algebra
$$f(x) = ax^2 + bx + c$$ $$a,b$$ = constants
$$c$$ = constant (y-axis intercept)
$$x$$ = variable
Standard form of
Heart of
Algebra
$$f(x) = a(x-h)^2 + k$$ $$a$$ = constant
$$h$$ = constant (horizontal shift)
$$k$$ = constant (vertical shift)
$$x$$ = variable
Vertex form of
Heart of
Algebra
$$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$$ $$a,b$$ = constants from Standard form
$$c$$ = constant (y-axis intercept)
$$x$$ = x-intercept
Heart of
Algebra
$$x = \frac{-b}{2a}$$ $$a,b$$ = constants from Standard form
$$x$$ = axis of symmetry
Axis of symmetry
Heart of
Algebra
$$P=4s$$ $$P$$ = Perimeter of a square
$$s$$ = side length

Heart of
Algebra
$$P = 2l + 2w$$ $$P$$ = Perimeter of a rectangle
$$l$$ = length
$$w$$ = width

Heart of
Algebra
$$P = s_1 + s_2 + s_3$$ $$P$$ = Perimeter of a triangle
$$s_n$$ = side length

Heart of
Algebra
$$C = 2 \pi r = \pi d$$ $$C$$ = Perimeter of a circle
$$r$$ = radius
$$d$$ = diameter
$$\pi \approx 3.14$$

Heart of
Algebra
$$A = s^2$$ $$A$$ = Area of a square
$$s$$ = side length

Heart of
Algebra
$$A = lw$$ $$A$$ = Area of a rectangle
$$l$$ = length
$$w$$ = width

Heart of
Algebra
$$A = bh$$ $$A$$ = Area of a parallelogram
$$b$$ = base
$$h$$ = height

Heart of
Algebra
$$A = \frac{1}{2}bh$$ $$A$$ = Area of a triangle
$$b$$ = base
$$h$$ = height

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

Heart of
Algebra
$$A = \pi r^2$$ $$A$$ = Area of a circle
$$r$$ = radius
$$\pi \approx 3.14$$

Heart of
Algebra
$$V=lwh$$ $$V$$ = Volume of a rectangular prism
$$l$$ = length
$$w$$ = width
$$h$$ = height

Heart of
Algebra
$$V=Bh$$ $$V$$ = Volume of a right prism
$$B$$ = area of the base
$$h$$ = height

Heart of
Algebra
$$V = \pi r^2 h$$ $$V$$ = Volume of a cylinder
$$r$$ = radius
$$h$$ = height
$$\pi \approx 3.14$$

Heart of
Algebra
$$V = \frac{1}{3}Bh$$ $$V$$ = Volume of a pyramid
$$B$$ = area of the base
$$h$$ = height

Heart of
Algebra
$$V = \frac{1}{3}\pi r^2 h$$ $$V$$ = Volume of a cone
$$r$$ = radius
$$h$$ = height
$$\pi \approx 3.14$$

Heart of
Algebra
$$V = \frac{4}{3} \pi r^3$$ $$V$$ = Volume of a sphere
$$r$$ = radius
$$\pi \approx 3.14$$

Math
$$(a \pm b)^2 = a^2 \pm 2ab + b^2$$ $$a,b$$ = constants or variables Square of a sum or
difference
Math
$$a^2-b^2 = (a+b)\cdot(a-b)$$ $$a,b$$ = constants or variable Difference of
squares
Math
$$a^3 - b^3 = (a-b)\cdot(a^2 + ab + b^2)$$ $$a,b$$ = constants or variables Difference of cubes
Math
$$a^3+b^3 = (a+b) \cdot (a^2 - ab + b^2)$$ $$a,b$$ = constants or variables Difference of cubes
Math
$$\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}$$ $$a,b,c,d$$ = any real number Remember to
simplify the fraction
(if possible)
Math
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$ $$a,b,c,d$$ = any real number Remember to
simplify the fraction
(if possible)
Math
$$\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$$ $$a,b,c,d$$ = any real number Remember to
simplify the fraction
(if possible)
Math
$$a \frac{b}{c} = \frac{ac+b}{c}$$ $$a,b,c$$ = any real number Remember to
simplify the fraction
(if possible)
Math
$$x^a \cdot x^b = x^{a+b}$$ $$a,b,x$$ = any real number
Math
$$\dfrac{x^a}{x^b} = x^{a-b}$$ $$a,b,x$$ = any real number
Math
$$(x^a)^b = x^{a\cdot b}$$ $$a,b,x$$ = any real number
Math
$$(xy)^a = x^a \cdot y^a$$ $$a,x,y$$ = any real number
Math
$$x^1 = x$$ $$x$$ = any real number
Math
$$x^0 = 1$$ $$x$$ = any real number ($$x\ne 0$$)
Math
$$x^{-a} = \frac{1}{x^a}$$ $$a,x$$ = any real number ($$x\ne 0$$)
Math
$$x^{\frac{a}{b}} = \sqrt[b]{x^a} = (\sqrt[b]{x})^a$$ $$a,b,x$$ = any real number
Math
$$a^x = b \rightarrow \log _a b = x$$ $$a,b,x$$ = any real number
Math
$$\ln (x) = \log _e x$$ $$x$$ = any real number
$$e \approx 2.718$$ = Euler’s number

Math
$$a^{\log_a x} = x$$ $$a,x$$ = any real number
Math
$$\log (a\cdot b) = \log (a) + \log (b)$$ $$a,b$$ = any real number
Math
$$\log (a\div b) = \log (a) - \log (b)$$ $$a,b$$ = any real number
Math
$$\log (a^b) = b \cdot \log (a)$$ $$a,b$$ = any real number
Math
$$\log_a x = \log_b x \cdot \log_a b$$ $$a,b,x$$ = any real number
Math
$$\log_a b = \dfrac{\log_x b}{\log_x a}$$ $$a,b,x$$ = any real number
Math
$$\log_a a = 1$$ $$a$$ = any real number ($$a\ne 0$$)
Math
$$\log (1) = 0$$
Problem
Solving and
Data Analytics
$$a\cdot b \% = a \cdot \frac{b}{100}$$ $$a$$ = any real number
$$b \%$$ = any percent
Remember to
simplify (if possible)
Problem
Solving and
Data Analytics
$$\% = \dfrac{\lvert b - a \rvert}{b} \cdot 100 = \frac{c}{b} \cdot 100$$ $$\% = \%$$ increase or decrease
$$a$$ = new value
$$b$$ = original value
$$c$$ = amount of change

Problem
Solving and
Data Analytics
$$\overline{x} = \dfrac{\Sigma x_i}{n}$$ $$\overline{x}$$ = mean
$$x_i$$ = value of each measurement
$$n$$ = number of measurements

Problem
Solving and
Data Analytics
$$Md = (\frac{n+1}{2})^{th}$$ term $$Md$$ = Median
$$n$$ = number of measurements (odd)

Problem
Solving and
Data Analytics
$$Md = \dfrac{(\frac{n}{2})^{th} \text{ term } + (\frac{n}{2} + 1)^{th} \text{ term}}{2}$$ $$Md$$ = Median
$$n$$ = number of measurements (even)

Problem
Solving and
Data Analytics
$$s = \sqrt{\dfrac{\Sigma (x_i - \overline{x})^2}{n-1}}$$ $$s$$ = standard deviation
$$\overline{x}$$ = mean
$$x_i$$ = value of each measurement
$$n$$ = number of measurements

Problem
Solving and
Data Analytics
$$V = s^2$$ $$V$$ = Variance
$$s$$ = standard deviation

Problem
Solving and
Data Analytics
$$CV = RSD = 100 \cdot s \div \overline{x}$$ $$CV$$ = Coefficient of variation
$$RSD$$ = Relative standard deviation
$$s$$ = standard deviation

Problem
Solving and
Data Analytics
$$p=\frac{d}{t}$$ $$p$$ = probability of an event
$$d$$ = desired event
$$t$$ = total number of possible events