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

The Preliminary SAT/National Merit Scholarship Qualifying Test (PSAT/NMSQT) plays a pivotal role in preparing students for the SAT and opening doors to National Merit Scholarships. While the test covers various areas, including Reading, Writing, and Math, today’s spotlight is on the critical math formulas that students often need to recall to tackle the Mathematics section effectively. Let’s explore how to apply these formulas in your test prep journey, using our comprehensive formula chart as a guide. ## Understanding the PSAT Math Section

Before delving into the intricacies of the PSAT Math Test, it’s essential to grasp its structure. The PSAT/NMSQT Math Test is divided into two portions:

• No Calculator Section: Comprising 17 questions to be tackled in 25 minutes, this section assesses your mathematical proficiency without the aid of a calculator. It involves multiple-choice and grid-in questions, evaluating your fundamental arithmetic, algebraic, and some geometric skills.

• Calculator Section: This segment allows the use of a calculator to navigate through its 31 questions within a 45-minute timeframe. It’s designed to evaluate your ability to solve problems with a computational tool, challenging your skills across various mathematical domains.

Key Content Areas on the PSAT Math Section

Navigating the intricate landscape of the PSAT Math Test involves a deep dive into its main content areas. Each area targets specific mathematical concepts and skills. Let’s break down these key areas to better understand what’s expected and how best to prepare.

• Heart of Algebra: This area delves into the core algebraic concepts focusing on linear equations and systems. Understanding how to manipulate algebraic expressions, solve linear equations and inequalities, and interpret linear functions is pivotal.

• Problem Solving and Data Analysis: This section is rooted in statistical understanding, involving concepts of ratios, percentages, and proportional reasoning. It assesses your ability to make quantitatively informed decisions and apply arithmetic skills.

• Passport to Advanced Math: Addressing more complex algebraic concepts, this area encompasses quadratic and nonlinear equations. Your proficiency in understanding and manipulating complex expressions and functions is evaluated here.

• Additional Topics: This broad category encapsulates various domains like geometry, trigonometry, and complex numbers, often involving 2D/3D geometric shapes, trigonometric applications, and understanding of complex number arithmetic.

Essential PSAT Formulas

To effectively navigate the challenges of the PSAT Math section, arming yourself with key formulas is crucial.The following table consolidates these formulas, serving as a quick review tool to refresh your memory and enhance your problem-solving toolkit.

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 Quadratic
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 Quadratic
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	Quadratic formula
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 \(\pi \approx 3.14\)
Advanced Math	\((a \pm b)^2 = a^2 \pm 2ab + b^2\)	\(a,b\) = constants or variables	Square of a sum or difference
Advanced Math	\(a^2-b^2 = (a+b)\cdot(a-b)\)	\(a,b\) = constants or variable	Difference of squares
Advanced Math	\(a^3 - b^3 = (a-b)\cdot(a^2 + ab + b^2)\)	\(a,b\) = constants or variables	Difference of cubes
Advanced Math	\(a^3+b^3 = (a+b) \cdot (a^2 - ab + b^2)\)	\(a,b\) = constants or variables	Difference of cubes
Advanced 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)
Advanced 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)
Advanced 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)
Advanced Math	\(a \frac{b}{c} = \frac{ac+b}{c}\)	\(a,b,c\) = any real number	Remember to simplify the fraction (if possible)
Advanced Math	\(x^a \cdot x^b = x^{a+b}\)	\(a,b,x\) = any real number
Advanced Math	\(\dfrac{x^a}{x^b} = x^{a-b}\)	\(a,b,x\) = any real number
Advanced Math	\((x^a)^b = x^{a\cdot b}\)	\(a,b,x\) = any real number
Advanced Math	\((xy)^a = x^a \cdot y^a\)	\(a,x,y\) = any real number
Advanced Math	\(x^1 = x\)	\(x\) = any real number
Advanced Math	\(x^0 = 1\)	\(x\) = any real number (\(x\ne 0\))
Advanced Math	\(x^{-a} = \frac{1}{x^a}\)	\(a,x\) = any real number (\(x\ne 0\))
Advanced Math	\(x^{\frac{a}{b}} = \sqrt[b]{x^a} = (\sqrt[b]{x})^a\)	\(a,b,x\) = any real number
Advanced Math	\(a^x = b \rightarrow \log _a b = x\)	\(a,b,x\) = any real number
Advanced Math	\(\ln (x) = \log _e x\)	\(x\) = any real number \(e \approx 2.718\) = Euler’s number
Advanced Math	\(a^{\log_a x} = x\)	\(a,x\) = any real number
Advanced Math	\(\log (a\cdot b) = \log (a) + \log (b)\)	\(a,b\) = any real number
Advanced Math	\(\log (a\div b) = \log (a) - \log (b)\)	\(a,b\) = any real number
Advanced Math	\(\log (a^b) = b \cdot \log (a)\)	\(a,b\) = any real number
Advanced Math	\(\log_a x = \log_b x \cdot \log_a b\)	\(a,b,x\) = any real number
Advanced Math	\(\log_a b = \dfrac{\log_x b}{\log_x a}\)	\(a,b,x\) = any real number
Advanced Math	\(\log_a a = 1\)	\(a\) = any real number (\(a\ne 0\))
Advanced 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

Applying PSAT Math Formulas: Insights and Examples

While having a list of formulas at your fingertips is essential, truly understanding how to use them on the test is even more important. In this section, we’ll break down a selection of these formulas, showcasing their use through examples.

Linear Equations and Inequalities

Formulas like “y = mx + b” (slope-intercept form) are pivotal in this segment. Let’s decode it:

• y and x are the coordinates of a point on the line
• m is the slope
• b is the y-intercept

If given two points on a line, remember to use the slope formula \(m = (y2 - y1) / (x2 - x1)\) to find m and then leverage that in your slope-intercept formula to find your equation of the line.

Example Problem: Given the two points \(A(2, 3)\) and \(B(4, 7)\), find the equation of the line passing through these points in slope-intercept form.

Solution:

1. Identify the given points: \(x_1 = 2\), \(y_1 = 3\) \(x_2 = 4\), \(y_2 = 7\)

2. Find the slope (m) using the given formula: \(m = \frac{y2 - y1}{x2 - x1}\)

   Substituting in the given values: \(m = \frac{7 - 3}{4 - 2}\) \(m = \frac{4}{2}\) \(m = 2\)

3. Use the slope (m) and one of the given points to find the y-intercept (b):

We’ll use point \(A(2, 3)\) in the slope-intercept formula:

\[y = mx + b\] \[3 = 2(2) + b\] \[3 = 4 + b\] \[b = -1\]

1. Write the equation:

\[y = 2x - 1\]

So, the equation of the line passing through the points \(A(2, 3)\) and \(B(4, 7)\) is \(y = 2x - 1\).

Quadratics

You’ll encounter quadratic equations in the form \(ax^2 + bx + c = 0\). Utilize the quadratic formula to find x:

\[x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\]

= If a = 1, b = -3, and c = 2, insert these values into the quadratic formula to find the roots of the equation.

Example Problem: Given the quadratic equation \(x^2 - 5x + 6 = 0\), determine the values of \(x\) using the quadratic formula.

Solution:

1. Identify the coefficients:

\[a = 1\] \[b = -5\] \[c = 6\]

1. Plug the coefficients into the quadratic formula:

\[x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\]

Using the given coefficients:

\[x = \frac{5 \pm \sqrt{25-24}}{2}\] \[x = \frac{5 \pm 1}{2}\]

1. Calculate the two possible values for \(x\):

\[x_1 = \frac{5 + 1}{2} = \frac{6}{2} = 3\] \[x_2 = \frac{5 - 1}{2} = \frac{4}{2} = 2\]

So, the quadratic equation \(x^2 - 5x + 6 = 0\) has roots \(x = 3\) and \(x = 2\).

Geometry: Circles

The standard equation of a circle is \((x - h)^2\) + \((y - k)^2 = r^2\), where (h, k) represents the center, and r is the radius.

If tasked with finding whether a point (p, q) is inside, outside, or on the circle, substitute the point into the equation and evaluate. If the left side is equal to \(r^2\), it’s on the circle; if less, it’s inside; if more, it’s outside.

Example Problem: Given the equation of a circle as \((x - 3)^2 + (y + 2)^2 = 25\), determine the position of the point \(A(5, 0)\) relative to the circle.

Solution:

1. Identify the given values:

Center of the circle (h, k) = (3, -2)

Radius squared \(r^2\) = 25

Point \(A(p, q)\) = (5, 0)

1. Substitute the point \(A\) into the circle’s equation:

   \[(5 - 3)^2 + (0 + 2)^2\]

   = \(2^2 + 2^2\)

   = 4 + 4

   = 8

2. Evaluate the position based on the result:

The value obtained (8) is less than \(r^2\) (25).

Therefore, the point \(A(5, 0)\) is inside the circle.

Test Day Tips for the PSAT Math Section

Now that you have your formulas and know how to use them, what else can you do to ensure success on the math section of the PSAT? Here are some strategies and recommendations:

• Practice Makes Perfect: Dedicate time to taking a full-length PSAT practice test. This helps you become familiar with the nuances of the PSAT format, reinforcing your application of formulas.

• Good Night’s Sleep: A well-rested mind can think more clearly, solve problems more efficiently, and recall formulas and concepts effectively.

• Equip Yourself: Ensure you have an approved calculator, extra batteries, pencils, and a good eraser. Given that the PSAT allows calculator use in certain sections, practice with the one you intend to bring to be comfortable with its functions.

• Time Management: The math section provides 70 minutes, so pace yourself. If a question seems challenging, move on and return to it if time permits. Remember, every question carries the same weight, so allocate your time wisely.

• Review Your Work: Should you finish a section ahead of time, revisit your answers. Double-check for any oversight or potential errors.

• Read Carefully: The PSAT often includes tough questions or presents information in a manner that requires careful reading. Always ensure you grasp what’s being asked before formulating an answer.

• Arrive Early: The morning of the PSAT can be hectic. By arriving early, you can acclimate to the testing environment, locate your assigned room, and mentally prepare without feeling rushed.

Remember, the PSAT is a stepping stone and a practice round for the SAT. Use it as an opportunity to gauge your strengths and areas needing improvement. Best of luck!