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AMC 8 Comprehensive Review

Master Practice Exam

20 carefully verified problems across all AMC 8 topics — with concept review, examples, and detailed explanations.

20
Problems
10
Topics
40
Minutes
5
Choices Each
📚 Key Concepts & Formulas to Know

Master these concepts before attempting the problems below.

Number Theory

GCD, LCM & Primes

The GCD is the largest divisor common to two numbers. The LCM is the smallest common multiple.

GCD(a,b) × LCM(a,b) = a × b

To find GCD: use prime factorization — take the lowest powers. For LCM: take the highest powers.

Tip: GCD divides, LCM multiplies — "Go Down Carefully, Later Come Multiply"
Quick Example

GCD(12, 18)? Factors: 12 = 2²×3, 18 = 2×3². GCD = 2¹×3¹ = 6. LCM = 2²×3² = 36.

✓ GCD = 6, LCM = 36
Arithmetic

Fractions & Decimals

To add/subtract fractions: find common denominator. To multiply: multiply tops and bottoms. To divide: multiply by the reciprocal.

a/b ÷ c/d = a/b × d/c = ad/bc
Tip: "Keep, Change, Flip" for division — keep first fraction, change ÷ to ×, flip second.
Quick Example

(3/4) ÷ (1/2) = (3/4) × (2/1) = 6/4 = 3/2

✓ Answer: 3/2
Percent & Ratio

Percent Change

Percent change measures the relative increase or decrease from an original value.

% Change = (New − Old) / Old × 100

Successive percent changes: a 20% increase then 20% decrease does NOT return to the original! Multiply factors: 1.2 × 0.8 = 0.96 (4% net decrease).

Tip: Successive changes — always multiply the decimal factors, not add percentages.
Quick Example

Price $80 rises to $100. % Change = (100−80)/80 × 100 = 20/80 × 100 = 25%

✓ 25% increase
Algebra

Linear Equations & Inequalities

Solve by isolating the variable. When multiplying/dividing by a negative, flip the inequality sign.

ax + b = c → x = (c − b) / a

Word problems: translate "is" → =, "more than" → +, "less than" → −, "times" → ×.

Tip: Flip inequality when multiplying/dividing by a negative number.
Quick Example

3x + 5 = 20. Subtract 5: 3x = 15. Divide by 3: x = 5.

✓ x = 5
Geometry

Area & Perimeter

Critical formulas for common shapes:

Rectangle: A = l × w, P = 2(l+w)
Triangle: A = ½ × b × h
Circle: A = πr², C = 2πr
Tip: For a triangle in a grid, use A = ½|base||height| or the Shoelace formula.
Quick Example

Triangle with base 8 and height 5: A = ½ × 8 × 5 = 20

✓ Area = 20
Probability

Basic Probability

Probability = (favorable outcomes) ÷ (total outcomes). For independent events, multiply probabilities.

P(A and B) = P(A) × P(B) (if independent)
P(A or B) = P(A) + P(B) − P(A and B)
Tip: "And" means multiply; "Or" means add (minus overlap).
Quick Example

Bag has 3 red, 2 blue balls. P(red) = 3/5. Draw 2 with replacement: P(both red) = (3/5)×(3/5) = 9/25.

✓ P = 9/25
Statistics

Mean, Median, Mode, Range

Mean = sum ÷ count. Median = middle value when sorted. Mode = most frequent. Range = max − min.

Mean = (Σ values) / n
Tip: To find a missing value given the mean: (sum needed) − (sum of known values).
Quick Example

Data: {3, 7, 7, 9, 14}. Mean = 40/5 = 8. Median = 7. Mode = 7. Range = 14−3 = 11.

✓ Mean=8, Median=7, Mode=7, Range=11
Counting

Counting & Combinations

Fundamental Counting Principle: multiply choices at each step. Combinations (order doesn't matter): C(n,k) = n! / (k!(n−k)!)

C(n,k) = n! / (k! × (n−k)!)
Tip: "Choose" = Combinations. C(5,2) = 5×4/(2×1) = 10.
Quick Example

How many ways to choose 2 from 5? C(5,2) = 5!/(2!×3!) = (5×4)/(2×1) = 10.

✓ 10 ways
Ratio & Proportion

Ratios & Proportions

A ratio a:b means for every a of one thing, there are b of another. Cross-multiply to solve proportions.

a/b = c/d → ad = bc
Tip: Scale ratios by multiplying both parts. Ratio 2:3 with total 20 → 2k+3k=20 → k=4 → 8:12.
Quick Example

Ratio of boys:girls = 3:4 in a class of 35. Boys = 3/(3+4) × 35 = 3/7 × 35 = 15.

✓ 15 boys
Patterns & Sequences

Arithmetic & Geometric Sequences

Arithmetic: constant difference d. nth term = a + (n−1)d. Sum of n terms = n/2 × (first + last).

Arith: a_n = a₁ + (n−1)d
Sum = n(a₁ + aₙ)/2
Tip: Sum of 1+2+...+n = n(n+1)/2. Sum of first n odd numbers = n².
Quick Example

1+2+3+...+10 = 10×11/2 = 55. Sum of first 5 odd numbers = 5² = 25.

✓ 55 and 25
✏️ Practice Problems

AMC 8 style — 5 choices per question. Select your answer and check immediately, or submit all at the end.

1
Number Theory
⬛⬜⬜ Easy–Medium
What is the least common multiple (LCM) of 8 and 12?

2
Arithmetic — Fractions
⬛⬜⬜ Easy–Medium
What is the value of 3/4 + 5/6?

3
Percent & Ratio
⬛⬛⬜ Medium
A store reduces the price of a jacket by 25%, then later increases that reduced price by 20%. What is the overall percent change from the original price?

4
Algebra
⬛⬜⬜ Easy–Medium
If 5x − 3 = 2x + 9, what is the value of x?

5
Geometry — Area
⬛⬛⬜ Medium
A right triangle has legs of length 6 and 8. What is the area of this triangle?

6
Probability
⬛⬛⬜ Medium
A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. If one marble is drawn at random, what is the probability it is NOT green?

7
Statistics
⬛⬛⬜ Medium
The mean of five numbers is 14. Four of the numbers are 8, 12, 18, and 20. What is the fifth number?

8
Counting & Combinations
⬛⬛⬜ Medium
In how many ways can a student choose 3 books from a shelf of 6 different books?

9
Ratio & Proportion
⬛⬛⬜ Medium
If 3 pens cost $4.50, how much do 8 pens cost?

10
Patterns & Sequences
⬛⬛⬜ Medium
What is the sum of the arithmetic sequence 3, 7, 11, 15, …, 39?

11
Number Theory — Divisibility
⬛⬛⬜ Medium
How many positive integers less than 50 are divisible by both 3 and 4?

12
Algebra — Word Problem
⬛⬛⬜ Medium
Maya has twice as many stickers as Noah. Together they have 48 stickers. How many stickers does Maya have?

13
Geometry — Pythagorean Theorem
⬛⬛⬜ Medium
A ladder 10 feet long leans against a wall. The base of the ladder is 6 feet from the wall. How high up the wall does the ladder reach?

14
Probability — Two Events
⬛⬛⬛ Hard
Two fair six-sided dice are rolled. What is the probability that the sum equals 7?

15
Statistics — Median & Range
⬛⬛⬜ Medium
The data set is: {5, 13, 8, 3, 11, 7, 9}. What is the median of this data set?

16
Number Theory — Prime Factorization
⬛⬛⬜ Medium
How many positive divisors does 72 have?

17
Geometry — Circles
⬛⬛⬜ Medium
A circle has a circumference of 20π. What is the area of the circle?

18
Counting — Paths on Grid
⬛⬛⬛ Hard
How many different 4-digit numbers can be formed using the digits 1, 2, 3, 4 if each digit is used exactly once?

19
Algebra — System of Equations
⬛⬛⬛ Hard
If x + y = 10 and x − y = 4, what is the value of x × y?

20
Geometry — Perimeter & Area Challenge
⬛⬛⬛ Hard
A rectangle has a perimeter of 36 and a length that is twice its width. What is the area of the rectangle?

Or check individual questions using the Check buttons above.

📖 Complete Solutions & Explanations

Detailed step-by-step solutions for all 20 problems