Pre-Algebra Master Quiz | 20 Essential Problems

📌 Topic 1: Integers & Absolute Value

Number Theory Foundation

Core Concept

Integers include all whole numbers and their negatives: …−3, −2, −1, 0, 1, 2, 3… Absolute value |x| is the distance from zero — always non-negative.

|x| = x if x ≥ 0 | |x| = −x if x < 0

Absolute value removes the negative sign

🧠 Must Memorize

Negative × Negative = Positive
Negative × Positive = Negative
|−7| = 7, |7| = 7, |0| = 0
Adding integers with same sign: add & keep sign
Adding integers with different signs: subtract, keep sign of larger absolute value

📝 Example

Evaluate: (−3) × (−4) − |−5|

Step 1: (−3) × (−4) = +12

Step 2: |−5| = 5

Step 3: 12 − 5 = 7 ✓

📌 Topic 2: Order of Operations

PEMDAS / BODMAS

Core Concept

Always evaluate expressions in this strict order to get correct results.

P → E → MD → AS

Parentheses · Exponents · Multiply/Divide (L→R) · Add/Subtract (L→R)

🧠 Must Memorize

Please Excuse My Dear Aunt Sally
Multiply and Divide are EQUAL priority (left to right)
Add and Subtract are EQUAL priority (left to right)

📝 Example

Evaluate: 3 + 2² × (8 − 6) ÷ 4

Step 1 (Parentheses): 8 − 6 = 2

Step 2 (Exponents): 2² = 4

Step 3 (Multiply): 4 × 2 = 8

Step 4 (Divide): 8 ÷ 4 = 2

Step 5 (Add): 3 + 2 = 5 ✓

📌 Topic 3: Fractions & Decimals

Rational Number Operations

Core Concept

Fractions represent parts of a whole. Operations require a common denominator for addition/subtraction; multiplication multiplies numerators and denominators directly.

a/b + c/d = (ad+bc)/(bd) | a/b × c/d = ac/bd

Fraction addition needs LCD; multiplication is direct

🧠 Must Memorize

To divide fractions: Keep · Change · Flip (KCF)
Converting: 3/4 = 0.75 = 75%
Mixed number: 2½ = 5/2
LCD = Least Common Denominator

📝 Example

Calculate: 2/3 + 3/4

LCD of 3 and 4 = 12

2/3 = 8/12, 3/4 = 9/12

8/12 + 9/12 = 17/12 = 1 5/12 ✓

📌 Topic 4: Ratios, Proportions & Percentages

Proportional Reasoning

Core Concept

A ratio compares two quantities. A proportion states two ratios are equal. Cross-multiplication solves proportions.

a/b = c/d ⟹ ad = bc

Cross-multiply to solve proportions

🧠 Must Memorize

% means "per hundred": 45% = 45/100 = 0.45
Part = Percent × Whole
% Change = (New − Old)/Old × 100
% Discount: multiply by (1 − rate)

📝 Example

What is 35% of 80?

0.35 × 80 = 28 ✓

📌 Topic 5: Exponents & Square Roots

Powers and Radicals

Core Concept

An exponent tells how many times to multiply the base by itself. Square root asks: what number times itself gives this value?

aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ/aⁿ = aᵐ⁻ⁿ | (aᵐ)ⁿ = aᵐⁿ

Laws of Exponents — same base only!

🧠 Must Memorize

a⁰ = 1 (any nonzero base)
a⁻ⁿ = 1/aⁿ
Perfect squares: 1,4,9,16,25,36,49,64,81,100,121,144
√144 = 12, √169 = 13, √225 = 15

📝 Example

Simplify: 2³ × 2⁴

= 2³⁺⁴ = 2⁷ = 128 ✓

📌 Topic 6: Variables & Expressions

Algebraic Thinking

Core Concept

A variable is a letter representing an unknown value. An expression combines variables, numbers, and operations. Like terms have identical variable parts and can be combined.

Distributive: a(b + c) = ab + ac

Key to expanding and simplifying expressions

🧠 Must Memorize

Like terms: 3x and 7x (can combine → 10x)
Unlike terms: 3x and 7y (cannot combine)
Coefficient: the number in front of a variable
3(x + 4) = 3x + 12

📝 Example

Simplify: 4x + 3y − 2x + y

Combine x-terms: 4x − 2x = 2x

Combine y-terms: 3y + y = 4y

Result: 2x + 4y ✓

📌 Topic 7: Solving One-Step & Two-Step Equations

Linear Equations

Core Concept

Solve equations by isolating the variable using inverse operations. Whatever you do to one side, do to the other.

ax + b = c ⟹ x = (c − b) / a

Subtract b first, then divide by a

🧠 Must Memorize

Inverse of + is −, inverse of × is ÷
Undo operations in reverse PEMDAS order
Always check your answer by substituting back!

📝 Example

Solve: 3x − 7 = 14

Step 1: 3x = 14 + 7 = 21

Step 2: x = 21 ÷ 3 = 7

Check: 3(7) − 7 = 21 − 7 = 14 ✓

📌 Topic 8: Inequalities

Comparing & Graphing

Core Concept

Inequalities show relationships where values are not necessarily equal. Solving follows same rules as equations with one key exception.

⚠️ When multiplying or dividing by a NEGATIVE — FLIP the inequality sign!

This is the most common error in inequalities

🧠 Must Memorize

< (less than): open dot on number line
≤ (less than or equal): closed dot on number line
−2x < 6 → divide by −2 → x > −3 (FLIP!)

📝 Example

Solve: −4x + 2 > 10

Step 1: −4x > 8

Step 2: x < −2 (divide by −4, flip!)

📌 Topic 9: Geometry Basics

Area, Perimeter, Volume

Core Concept

Essential geometric formulas for 2D and 3D shapes used throughout algebra and beyond.

Rectangle: A = lw, P = 2(l+w)

Area = length × width; Perimeter = sum of all sides

Triangle: A = ½bh | Circle: A = πr², C = 2πr

Triangle area needs base and height; Circle uses radius

🧠 Must Memorize

π ≈ 3.14
Pythagorean Theorem: a² + b² = c²
Rectangle: A = lw
Rectangular Prism Volume: V = lwh

📝 Example

Rectangle: l = 8 cm, w = 5 cm

Area = 8 × 5 = 40 cm²

Perimeter = 2(8 + 5) = 2 × 13 = 26 cm ✓

📌 Topic 10: Statistics & Probability

Data Analysis

Core Concept

Statistics describes data sets. The three averages (mean, median, mode) each measure the center differently. Probability measures likelihood of events.

Mean = Sum ÷ Count | P(event) = favorable/total

Mean is the arithmetic average; Probability is always between 0 and 1

🧠 Must Memorize

Mean: add all, divide by count
Median: middle value of ordered list (average of two middles if even count)
Mode: most frequently occurring value
Range = Maximum − Minimum
Probability always: 0 ≤ P ≤ 1

📝 Example

Data: 3, 7, 7, 10, 13

Mean: (3+7+7+10+13)/5 = 40/5 = 8

Median: 7 (middle value)

Mode: 7 (appears twice) ✓

📌 Topic 1: Integers & Absolute Value

📌 Topic 2: Order of Operations

📌 Topic 3: Fractions & Decimals

📌 Topic 4: Ratios, Proportions & Percentages

📌 Topic 5: Exponents & Square Roots

📌 Topic 6: Variables & Expressions

📌 Topic 7: Solving One-Step & Two-Step Equations

📌 Topic 8: Inequalities

📌 Topic 9: Geometry Basics

📌 Topic 10: Statistics & Probability

📋 Complete Answer Key & Explanations