📐 Academic Workbook Series

Pre-Algebra
Master

20 Essential Problems · All Key Concepts

20Problems
30Minutes
10Topics
Integers Fractions Ratios & Proportions Percents Expressions Equations Inequalities Coordinate Plane Geometry Statistics

📚 Concept Review

Tap each topic to expand key formulas and examples

🔢
Integers & Absolute Value
Unit 1
📌 Key Rules
(+) × (+) = (+) (+) × (−) = (−)
(−) × (−) = (+) (−) × (+) = (−)

|x| = x if x ≥ 0, |x| = −x if x < 0

Integer Division: same rule as multiplication
Absolute value is always ≥ 0. |−7| = 7, |7| = 7
✏️ Example

Evaluate: (−3) × (−4) ÷ (−2)

= (+12) ÷ (−2) = −6 ✓

½
Fractions, Decimals & Mixed Numbers
Unit 2
📌 Operations
Add/Sub: Same denominator first (find LCD)
Multiply: (a/b) × (c/d) = (ac)/(bd)
Divide: (a/b) ÷ (c/d) = (a/b) × (d/c)

Mixed → Improper: a b/c = (a×c + b)/c
Always simplify: divide numerator & denominator by their GCF
✏️ Example

Simplify: 3/4 ÷ 1/2

= 3/4 × 2/1 = 6/4 = 3/2 = 1½ ✓

⚖️
Ratios & Proportions
Unit 3
📌 Core Formulas
Ratio: a : b = a/b
Proportion: a/b = c/d ⟹ ad = bc (cross-multiply)

Unit Rate: divide to find rate per 1 unit
Scale Factor: new size / original size
Cross-multiply to solve proportions: if a/b = c/d, then ad = bc
✏️ Example

If 3/5 = x/20, find x.

3 × 20 = 5x → 60 = 5x → x = 12 ✓

%
Percents
Unit 4
📌 Percent Formulas
Part = (Percent/100) × Whole
Percent = (Part/Whole) × 100

% Increase = (New − Old)/Old × 100
% Decrease = (Old − New)/Old × 100

Discount = Original × (discount%/100)
"percent" means "per hundred." 35% = 35/100 = 0.35
✏️ Example

A $80 shirt is 25% off. Final price?

Discount = 80 × 0.25 = $20 → Final = $60 ✓

📐
Algebraic Expressions & Properties
Unit 5
📌 Key Properties
Commutative: a + b = b + a
Associative: (a+b)+c = a+(b+c)
Distributive: a(b+c) = ab + ac
Identity: a + 0 = a, a × 1 = a
Like terms: 3x + 5x = 8x (same variable & exponent)
Only combine LIKE terms (same variable, same exponent)
✏️ Example

Simplify: 3(2x − 4) + 5x

= 6x − 12 + 5x = 11x − 12 ✓

🔤
Solving One- & Two-Step Equations
Unit 6
📌 Solving Strategy
Goal: isolate the variable
1. Distribute if needed
2. Combine like terms
3. Undo addition/subtraction first
4. Undo multiplication/division last
5. Check by substituting back
Whatever you do to one side, do the same to the other side
✏️ Example

Solve: 2x + 5 = 13

2x = 8 → x = 4 ✓ Check: 2(4)+5=13 ✓

<
Inequalities
Unit 7
📌 Inequality Rules
Solve like equations, BUT:
⚠ When multiplying or dividing by a NEGATIVE number,
FLIP the inequality sign!

Symbols: < (less than) > (greater than)
≤ (less/equal) ≥ (greater/equal)
Multiplying/dividing by negative → flip the inequality sign
✏️ Example

Solve: −3x > 12

x < −4 (flip! because dividing by −3) ✓

📍
Coordinate Plane & Graphing
Unit 8
📌 Key Formulas
Point: (x, y) — x is horizontal, y is vertical

Quadrants: I (+,+) II (−,+) III (−,−) IV (+,−)

Slope: m = (y₂−y₁)/(x₂−x₁)
Slope-intercept: y = mx + b (b = y-intercept)
Slope = rise/run. Positive slope goes up left-to-right; negative goes down.
✏️ Example

Slope between (1, 2) and (3, 8)?

m = (8−2)/(3−1) = 6/2 = 3 ✓

📏
Geometry: Area, Perimeter & Volume
Unit 9
📌 Essential Formulas
Rectangle: A = lw, P = 2(l+w)
Triangle: A = ½bh, P = a+b+c
Circle: A = πr², C = 2πr
Trapezoid: A = ½(b₁+b₂)h

Rectangular Prism: V = lwh
Cylinder: V = πr²h
Always check units: area in units², volume in units³
✏️ Example

Area of triangle: base = 8, height = 5

A = ½ × 8 × 5 = 20 sq units ✓

📊
Statistics & Data (Mean, Median, Mode)
Unit 10
📌 Measures of Central Tendency
Mean = sum of all values ÷ number of values
Median = middle value when sorted (average of two if even count)
Mode = most frequently occurring value
Range = max − min

Probability = favorable outcomes / total outcomes
Median is NOT affected by extreme values (outliers); mean IS.
✏️ Example

Find mean & median of: 3, 7, 7, 8, 10

Mean = 35/5 = 7; Median = 7 (middle value) ✓

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