PERT math formulas and algebra rules.

Q: Which rules should I learn first?

Start with the group where your practice misses cluster. If you have no results yet, the arithmetic and exponent rules come first, because every later group, from equations to quadratics, depends on them.

Q: Can I use a calculator on PERT Math?

You can’t bring your own calculator unless an approved disability accommodation allows one. An on-screen calculator appears only on some math questions, and the study guide describes it as a standard four-function calculator, so you still need to do algebra and most arithmetic by hand.

Q: What math is on the PERT?

Our course groups PERT Mathematics into expressions, equations, polynomials and coordinate relationships, which is why this sheet runs from arithmetic and exponents through lines, factoring and quadratics. The PERT Math guide covers the subject in detail.

Q: Is memorizing these rules enough?

No. Each rule needs practice in context, because PERT questions often hide it inside a word problem or a multistep expression. Use the lessons and question-type practice sets alongside the sheet.

These 40 rules cover the PERT Mathematics topics, from signed numbers to the quadratic formula. Each comes with a short worked example and a link to the free lesson that teaches it.

Test Ninjas Research · Updated November 5, 2026 · 8 min read

How to use this sheet

On PERT Mathematics you can’t bring your own calculator unless an approved disability accommodation allows one, and an on-screen calculator appears only on some questions, so these rules need to be automatic. Use the sheet in three passes:

Cover the middle column. Read the rule’s name and say how it works, then check yourself.
Rework the mini example on paper without looking at the answer. If you get stuck, open the lesson: each one explains the rule with several worked examples and a practice question.
Drill what you missed with the flashcards, then try the rule in practice with the algebra and mathematical models guide or the expressions, polynomials and factoring guide.

The groups follow the order of the free PERT course, so the sheet also works as a checklist for PERT Math: if every example here is easy, your time is better spent on mixed practice.

Arithmetic, fractions and percents

Signed numbers, fractions and percents

Rule	How it works	Mini example	Lesson
Order of operations	Grouping and exponents first, then multiplication and division left to right, then addition and subtraction left to right	−4 + 3(2 − 7) = −4 + 3(−5) = −19	Signed numbers and order of operations
Negative base	An exponent applies only to what it touches; parentheses put the sign under the exponent	−3² = −9, but (−3)² = 9	Signed numbers and order of operations
Signed numbers	Subtracting is adding the opposite; same signs multiply or divide to a positive, different signs to a negative	5 − (−2) = 7; (−6)(−2) = 12; −12 ÷ 3 = −4	Signed numbers and order of operations
Adding fractions	Rewrite over a common denominator, then add the numerators	1/4 + 2/3 = 3/12 + 8/12 = 11/12	Fractions and rational arithmetic
Dividing fractions	Multiply by the reciprocal of the nonzero divisor	(3/4) ÷ (2/5) = (3/4)(5/2) = 15/8	Fractions and rational arithmetic
Percent of a quantity	Write the percent as a decimal and multiply; a discount leaves 100% minus the discount	25% off $72: 0.75 × 72 = $54	Percents and proportional models
Percent change	Divide the change by the original amount, then multiply by 100	80 to 92: 12 ÷ 80 = 0.15 = 15%	Percents and proportional models
Successive percents	Each percent applies to the amount after the previous change	100 up 20% is 120; 120 down 20% is 96, a 4% net drop	Percents and proportional models
Proportion	Two equal ratios; cross-multiply and solve	x/6 = 10/15, so 15x = 60 and x = 4	Percents and proportional models

Exponents and expressions

Exponent rules, substitution and like terms

Rule	How it works	Mini example	Lesson
Product and quotient of powers	Same base: multiply by adding exponents, divide by subtracting them (base not 0)	x²·x³ = x⁵; x⁵/x² = x³	Powers and exponent structure
Power of a power or product	Multiply exponents; a power of a product applies to every factor	(x²)³ = x⁶; (2x)³ = 8x³	Powers and exponent structure
Zero and negative exponents	a⁰ = 1 and a⁻ⁿ = 1/aⁿ for any nonzero a	7⁰ = 1; 2⁻³ = 1/8	Powers and exponent structure
Monomial products and quotients	Multiply or divide the coefficients, then apply the exponent rules to each variable	(3x²y)(4xy³) = 12x³y⁴; 12x⁵/(4x²) = 3x³	Products and quotients of monomials
Substitution	Replace each variable with its value inside parentheses, then follow the order of operations	If x = −2: x² − 3x = (−2)² − 3(−2) = 4 + 6 = 10	Substituting into algebraic expressions
Distributive property	a(b + c) = ab + ac; a minus sign before a group changes the sign of every term inside	2(x − 3) − (x + 4) = 2x − 6 − x − 4 = x − 10	Distribution and like terms
Like terms	Combine only terms with the same variables raised to the same powers	4x + 3y − x + 2y = 3x + 5y; x² + x stays as it is	Distribution and like terms

Equations and inequalities

Linear equations, inequalities and models

Rule	How it works	Mini example	Lesson
Linear equation	Do the same operation to both sides, then substitute to check	3x + 4 = 19, so 3x = 15 and x = 5; check 3(5) + 4 = 19	Multistep linear equations
Fractions and decimals	Multiply every term by the common denominator (or a power of 10) to clear them	x/3 + 1/2 = 5/6; times 6: 2x + 3 = 5, so x = 1	Linear equations with fractions and decimals
Identity or no solution	If the variable cancels, a true statement means every real number works; a false one means none does	3(x + 2) = 3x + 6: all real numbers. 2(x − 4) = 2x + 1: no solution	Identities, contradictions and parameters
Literal equation	Isolate the requested variable with the same steps; note any divisor that cannot be 0	A = bh/2 gives h = 2A/b, for b ≠ 0	Solving literal equations
Inequality sign flip	Multiplying or dividing both sides by a negative number reverses the inequality	−2x < 6, so x > −3	Linear inequalities
Compound inequalities	“And”: apply each step to all three parts. “Or”: solve each part; the answer is everything that satisfies either	−1 < 2x + 3 ≤ 9 gives −4 < 2x ≤ 6, so −2 < x ≤ 3	Compound inequalities
Linear model	Total = starting amount + rate × units; the rate is the slope	$25 plus $0.40 a mile: C = 25 + 0.40m; 100 miles costs $65	Building and comparing linear models

Lines and systems

Slope, line equations and systems

Rule	How it works	Mini example	Lesson
Slope	m = (y₂ − y₁)/(x₂ − x₁), rise over run	(1, 2) and (4, 8): m = 6/3 = 2	Slope and coordinate relationships
Slope-intercept form	y = mx + b, where m is the slope and b the y-intercept	y = 2x + b through (1, 2): 2 = 2 + b, so b = 0	Equations of lines
Point-slope form	y − y₁ = m(x − x₁) for slope m through (x₁, y₁)	Slope 3 through (2, 5): y − 5 = 3(x − 2), so y = 3x − 1	Equations of lines
Horizontal and vertical lines	y = c is horizontal with slope 0; x = c is vertical with undefined slope	y = −2 is horizontal; x = −2 is vertical	Equations of lines
Parallel and perpendicular	Parallel lines share a slope; perpendicular slopes are negative reciprocals (product −1)	Parallel to y = −4x + 9: slope −4. Perpendicular to y = (2/3)x − 5: slope −3/2	Parallel and perpendicular lines
Solving a system	The solution satisfies both equations; substitute or add equations to eliminate a variable	x + y = 7 and x − y = 1: adding gives 2x = 8, so (4, 3)	Solving simultaneous linear equations
Number of solutions	Different slopes: one solution. Same slope, different intercepts: none. Same line: infinitely many	y = 2x + 1 and y = 2x − 3 are parallel: no solution	Interpreting and modeling systems

Polynomials, quadratics and rational expressions

Polynomial, quadratic and rational rules

Rule	How it works	Mini example	Lesson
Adding and subtracting polynomials	Combine like terms; when subtracting, change the sign of every term in the second polynomial	(3x² + 2x − 5) − (x² − 4x + 1) = 2x² + 6x − 6	Adding and subtracting polynomials
Multiplying binomials	Multiply every term by every term, then combine like terms	(x + 2)(x − 3) = x² − 3x + 2x − 6 = x² − x − 6	Multiplying binomials
Factoring	Take out the greatest common factor first; for x² + bx + c, find two numbers that multiply to c and add to b	6x² + 9x = 3x(2x + 3); x² + 7x + 12 = (x + 3)(x + 4)	Factoring polynomials
Difference of squares	a² − b² = (a − b)(a + b)	x² − 9 = (x − 3)(x + 3)	Factoring polynomials
Polynomial division	Divide leading terms step by step; check that quotient × divisor + remainder gives the original	(2x² + 5x − 3) ÷ (x + 3) = 2x − 1, since (2x − 1)(x + 3) = 2x² + 5x − 3	Dividing polynomials
Zero-product property	Move every term to one side, factor, then set each factor equal to 0	x² − 7x + 10 = 0: (x − 2)(x − 5) = 0, so x = 2 or 5	Solving quadratic equations
Square roots of both signs	If a square equals a positive number, use the positive and negative roots	(x − 3)² = 16: x − 3 = 4 or −4, so x = 7 or −1	Solving quadratic equations
Quadratic formula	For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a)	2x² + 3x − 2 = 0: x = (−3 ± √25)/4, so x = 1/2 or −2	Solving quadratic equations
Rational domain	Exclude every value that makes an original denominator 0, even after canceling	(x² − 9)/(x + 3) = x − 3, with x ≠ −3	Rational expressions, equations and domains
Checking rational equations	Clear denominators, solve, then reject any answer that is excluded	x/(x − 2) = 2/(x − 2) gives x = 2, which is excluded: no solution	Rational expressions, equations and domains

Flashcards

Fifteen core rules as flashcards. Say the rule out loud before you flip the card, and note any you hesitate on so you can rework their examples above.

1 of 15 · Select the card to reveal the meaning and example.

Put the rules into practice

Knowing a rule and using it under test conditions are different skills. After a pass through the sheet, build a 10-question Math set in the question bank and, for every miss, find the row above that would have prevented it. When the sheet feels easy, take the Mathematics practice test, or follow the PERT study plan to fit the rules into a schedule.

Use the rules on free practice questions.

Build a free PERT Math set with an explanation for every question.

Build a Math set

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