AMC 8 Master Practice Test

Key Concepts to Master

Topic 01 · Arithmetic

Order of Operations & Fractions

Always evaluate: Parentheses → Exponents → Multiply & Divide (left to right) → Add & Subtract (left to right). For fractions, find the LCM of denominators before adding or subtracting.

PEMDAS: P → E → MD (L→R) → AS (L→R)
a/b + c/d = (ad + bc) / bd

Topic 02 · Number Theory

Primes, GCF & LCM

A prime number has exactly 2 factors: 1 and itself. Use prime factorization to find GCF (take minimum exponents of shared primes) and LCM (take maximum exponents of all primes).

GCF × LCM = a × b (for two numbers)
Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 …

Topic 03 · Percentages

Percent, Ratio & Proportion

Percent means "per 100." A p% increase multiplies by (1 + p/100); a p% decrease multiplies by (1 − p/100). Use inclusion-exclusion for overlapping groups: |A ∪ B| = |A| + |B| − |A ∩ B|.

Percent change = (new − old) / old × 100%
Proportion: a/b = c/d → ad = bc

Topic 04 · Geometry

Area, Perimeter & Angles

Memorize area formulas for triangles, rectangles, circles, and trapezoids. Triangle angle sum = 180°. Polygon interior angle sum = (n − 2) × 180°. Exterior angle = sum of two non-adjacent interior angles.

Triangle: ½bh | Circle: πr² | Trapezoid: ½(b₁+b₂)h
Pythagorean theorem: a² + b² = c²

Topic 05 · Algebra

Equations & Word Problems

Define a variable for the unknown, translate the problem into an equation, solve, and verify your answer makes sense. Consecutive odd/even integers: n, n+2, n+4, … Ratio a:b with total T → each part = T/(a+b).

Consecutive odds/evens: n, n+2, n+4 …
Rate × Time = Distance (or Work)

Topic 06 · Counting

Permutations & Combinations

Fundamental counting principle: multiply the number of choices at each step. Use permutations when order matters (arrangements), combinations when it doesn't (selections).

Permutation: P(n,r) = n!/(n−r)!
Combination: C(n,r) = n! / (r!(n−r)!)

Topic 07 · Probability & Statistics

Probability, Mean, Median & Mode

P(event) = favorable outcomes / total equally-likely outcomes. Mean = sum ÷ count. Median = middle value when sorted. Mode = most frequent. Range = max − min.

P(event) = favorable / total
Mean = Σx / n | Range = max − min

Topic 08 · Sequences & Patterns

Arithmetic Sequences & Unit Digits

Arithmetic sequence: each term adds constant d. The nth term = a₁ + (n−1)d. For unit-digit patterns of powers, find the cycle length of the last digit and use the remainder when dividing the exponent by the cycle length.

nth term: aₙ = a₁ + (n−1)d
Sum of n terms: Sₙ = n(a₁ + aₙ) / 2

Worked Examples

Example · Order of Operations

What is the value of (2 + 3) × 4 − 6 ÷ 2?

Step 1 — Parentheses: (2+3) = 5
Step 2 — Left to right: 5×4 = 20, then 6÷2 = 3
Step 3 — Subtraction: 20 − 3 = 17 ✓

Example · LCM

Find the LCM of 12 and 18.

12 = 2² × 3 | 18 = 2 × 3²
LCM = 2² × 3² = 4 × 9 = 36 ✓

Example · Fractions

What is 1/3 + 1/4 + 1/6?

LCM(3,4,6) = 12
4/12 + 3/12 + 2/12 = 9/12 = 3/4 ✓

Example · Percent Discount

A jacket costs $80. After 15% off, what is the sale price?

Discount = 0.15 × 80 = $12
Sale price = $80 − $12 = $68 ✓

Example · Pythagorean Theorem

A right triangle has legs 5 and 12. Find the hypotenuse.

c² = 5² + 12² = 25 + 144 = 169
c = √169 = 13 ✓ (5-12-13 triple)

Example · Probability

A bag has 3 red and 5 blue marbles. What is P(red)?

Total = 3 + 5 = 8 marbles
P(red) = 3/8 ✓

Example · Arithmetic Sequence

What is the 20th term of 7, 11, 15, 19, …?

d = 4, a₁ = 7
a₂₀ = 7 + (20−1) × 4 = 7 + 76 = 83 ✓

Example · Unit Digit Pattern

What is the units digit of 7¹⁰⁰?

Units digits of 7ⁿ cycle: 7, 9, 3, 1 (period 4)
100 ÷ 4 = 25 remainder 0 → same as 7⁴ → units digit = 1 ✓