📚 Core Concepts & Key Formulas
1. Arithmetic & Order of Operations
- PEMDAS: Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right)
- Divisibility: ÷2 (even last digit), ÷3 (digit sum divisible by 3), ÷5 (ends in 0 or 5), ÷9 (digit sum divisible by 9)
- GCD × LCM = product of the two numbers
2. Fractions, Decimals & Percents
- Percent change = (new − old) / old × 100%
- "X% of Y" means (X/100) × Y
- To compare fractions: cross-multiply or find common denominator
3. Algebra & Word Problems
- Translate English to equations step by step
- Consecutive even/odd integers: n, n+2, n+4, ...
- Average = Sum ÷ Count; rearrange to find Sum = Average × Count
- Rate × Time = Distance (or Work)
4. Plane Geometry
- Area: Rectangle = l × w; Triangle = ½bh; Circle = πr²; Trapezoid = ½(b₁+b₂)h
- Circumference = 2πr
- Pythagorean theorem: a² + b² = c²
- Classic triples: (3,4,5), (5,12,13), (8,15,17), (6,8,10)
- Triangle angles sum = 180°; Quadrilateral = 360°
5. Counting & Probability
- Multiplication principle: A ways then B ways ⇒ A × B total ways
- Combinations (order doesn't matter): C(n,r) = n! / (r! × (n−r)!)
- Permutations (order matters): P(n,r) = n! / (n−r)!
- Probability = favorable outcomes / total outcomes
6. Number Theory
- Prime factorization: write n as product of prime powers
- If n = pa × qb, then number of divisors = (a+1)(b+1)
- Modular arithmetic: find the pattern/cycle for remainders
- 21=2, 22=4, 23=8 ≡ 1 (mod 7) → period 3
7. Statistics & Data
- Mean = sum ÷ count; Median = middle value when sorted; Mode = most frequent
- Range = maximum − minimum
- Adding a new value: new sum = old sum + new value; new mean = new sum / new count
8. Solid Geometry
- Volume of rectangular box = l × w × h
- Volume of cube (side s) = s³
- Surface area of box = 2(lw + lh + wh)
9. Sequences & Patterns
- Arithmetic sequence: an = a1 + (n−1)d
- Sum of arithmetic sequence = n/2 × (first + last)
- Cyclic patterns: find the period, use n mod period
10. Logic & Reasoning
- Organize with tables, diagrams, or systematic lists
- Days-of-week problems: use mod 7 (Sunday=0 or anchor to a known day)
- Work backwards from a desired result when direct approach is complex
⚡ Must-Memorize Formulas
Percent Change
% change = (new − old) / old × 100%
Pythagorean Theorem
a² + b² = c² | Triples: (3,4,5) (5,12,13) (6,8,10) (8,15,17)
Area Formulas
Triangle: ½bh | Circle: πr² | Trapezoid: ½(b₁+b₂)h
Combinations
C(n,r) = n! / (r! × (n−r)!) | Probability = favorable / total
Number of Divisors
n = p^a × q^b ⇒ divisors = (a+1)(b+1)
Arithmetic Sequence
a_n = a_1 + (n−1)d | Sum = n/2 × (a_1 + a_n)
Days-of-Week Cycle
+k days from day D ⇒ (index of D + k) mod 7 [Sun=0, Mon=1, ..., Sat=6]
🔍 Worked Examples
Example A • Number Theory
How many positive divisors does 72 have?
Solution: 72 = 23 × 32. Number of divisors = (3+1)(2+1) = 4 × 3 = 12.
✓ Answer: 12 divisors
Example B • Geometry
A right triangle has legs 5 and 12. Find the hypotenuse.
Solution: c = √(5² + 12²) = √(25 + 144) = √169 = 13.
✓ Answer: 13 (the 5-12-13 Pythagorean triple)
Example C • Counting
In how many ways can 3 officers (president, VP, secretary) be chosen from 8 members?
Solution: Order matters (different roles), so: 8 × 7 × 6 = 336.
✓ Answer: 336 ways
📝 20 Practice Problems
Choose your answer for each problem. Immediate feedback and full explanation will appear after each selection.
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