1. Algebra Fundamentals

Algebra makes up about 40% of PUPCET Math! Master these foundational concepts.

Essential Algebraic Skills

Topic	Key Formula/Concept
Linear Equations	ax + b = c; solve for x = (c-b)/a
Quadratic Formula	x = (-b ± √(b²-4ac)) / 2a
Factoring	FOIL, difference of squares, perfect square trinomial
Exponents	aᵐ × aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ
Radicals	√(ab) = √a × √b; √(a/b) = √a / √b

Systems of Equations

• Substitution method
• Elimination method
• Graphical method
• Consistent vs inconsistent systems

Inequalities

• Linear inequalities
• Compound inequalities (and/or)
• Absolute value inequalities
• Graphing on number line

2. Geometry Essentials

Geometry is about 25-30% of PUPCET Math. Memorize formulas and properties!

Area and Perimeter Formulas

Shape	Area	Perimeter
Rectangle	A = lw	P = 2l + 2w
Triangle	A = ½bh	P = a + b + c
Circle	A = πr²	C = 2πr
Trapezoid	A = ½(b₁+b₂)h	P = sum of sides
Parallelogram	A = bh	P = 2(a+b)

Pythagorean Theorem:

a² + b² = c² (for right triangles). Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.

3. Basic Trigonometry

Know SOH-CAH-TOA and special angle values!

SOH

sin θ = Opposite/Hypotenuse

CAH

cos θ = Adjacent/Hypotenuse

TOA

tan θ = Opposite/Adjacent

4. Word Problems

PUPCET includes practical math problems. Know these common types:

Age Problems

Setup: Present age = x, Past = x - n, Future = x + n

Distance-Rate-Time

D = R × T | Remember: same direction (subtract rates), opposite (add rates)

Work Problems

Combined rate: 1/T = 1/T₁ + 1/T₂

Percentage Problems

Part = Percentage × Whole; Increase/Decrease = Original × Rate

5. Number Theory & Sequences

Arithmetic Sequence

• aₙ = a₁ + (n-1)d
• Sum: Sₙ = n/2(a₁ + aₙ)

Geometric Sequence

• aₙ = a₁ × rⁿ⁻¹
• Sum: Sₙ = a₁(1-rⁿ)/(1-r)

6. Practice Questions

1. If 3x - 7 = 14, what is x?

Show Answer

x = 7. Solve: 3x = 14 + 7 = 21, then x = 21/3 = 7.

2. A rectangle has length 12 and width 5. What is its diagonal?