Essential Practice

20 Problems across 10 Core AMC 8 Topics

Arithmetic Geometry Algebra Probability Statistics Word Problems
0 / 20 answered
TOPIC 01 Arithmetic & Number Theory

Key Concepts

  • Divisibility rules, prime factorization, LCM and GCF
  • Order of operations (PEMDAS): Parentheses → Exponents → ×÷ → +−
  • Odd/Even rules: odd + odd = even, odd × odd = odd
★ Memorize
  • Primes ≤ 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
  • Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
  • LCM(a,b) = a × b ÷ GCF(a,b)
Example
Find the LCM of 12 and 18. 12 = 2² × 3, 18 = 2 × 3² → LCM = 2² × 3² = 36
Answer: 36

Practice Problems

Q1 What is the sum of all prime numbers between 10 and 20?
Q2 What is the greatest common factor (GCF) of 48 and 60?
TOPIC 02 Fractions, Decimals & Percents

Key Concepts

  • To add fractions: find a common denominator first
  • Percent means "per hundred": p% = p/100
  • Percent change = (New − Old) / Old × 100%
★ Memorize
  • 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 1/5 = 0.2 = 20%
  • 1/3 ≈ 33.3%, 2/3 ≈ 66.7%, 3/4 = 75%, 1/8 = 12.5%
Example
What is 15% of 80? 0.15 × 80 = 12
Answer: 12

Practice Problems

Q3 A jacket originally costs $60. It is on sale for 25% off. What is the sale price in dollars?
Q4 What is 3/8 + 5/12? Express your answer as a simplified fraction.
TOPIC 03 Ratios, Rates & Proportions

Key Concepts

  • A ratio compares two quantities: a : b or a/b
  • Unit rate: quantity per 1 unit (e.g., miles per hour)
  • Cross-multiplication: if a/b = c/d, then ad = bc
★ Memorize
  • If ratio a:b, parts are a/(a+b) and b/(a+b) of total
  • Distance = Rate × Time (D = R × T)
Example
A car travels 120 miles in 3 hours. What is its speed? Speed = 120 ÷ 3 = 40 mph
Answer: 40 mph

Practice Problems

Q5 A recipe calls for 2 cups of flour for every 3 cups of sugar. If you use 8 cups of flour, how many cups of sugar do you need?
Q6 A train travels at 60 mph. How many miles does it travel in 2 hours and 30 minutes?
TOPIC 04 Algebra & Equations

Key Concepts

  • To solve equations: apply inverse operations to isolate the variable
  • Distributive property: a(b + c) = ab + ac
  • Consecutive integers: n, n+1, n+2, …
★ Memorize
  • Sum of n consecutive integers: n/2 × (first + last)
  • If x + y = S and x − y = D: x = (S+D)/2, y = (S−D)/2
Example
Solve: 3x + 7 = 22 3x = 15 → x = 5
Answer: x = 5

Practice Problems

Q7 If 5x − 3 = 2x + 9, what is the value of x?
Q8 The sum of three consecutive even integers is 78. What is the largest of the three integers?
TOPIC 05 Geometry — Perimeter, Area & Angles

Key Concepts

  • Area of rectangle = length × width
  • Area of triangle = ½ × base × height
  • Area of circle = π r², Circumference = 2πr
  • Sum of interior angles of n-gon = (n − 2) × 180°
★ Memorize
  • Square: Area = s², Perimeter = 4s
  • Supplementary angles sum to 180°; Complementary angles sum to 90°
Example
A triangle has base 10 and height 6. What is its area? Area = ½ × 10 × 6 = 30
Answer: 30

Practice Problems

Q9 A rectangle has a length of 12 cm and a width of 7 cm. What is the area of the rectangle in square centimeters?
Q10 In a triangle, two angles measure 47° and 68°. What is the measure of the third angle in degrees?
TOPIC 06 Geometry — Pythagorean Theorem & Coordinate Plane

Key Concepts

  • Pythagorean Theorem: a² + b² = c² (c = hypotenuse)
  • Distance formula: √[(x₂−x₁)² + (y₂−y₁)²]
  • Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
★ Memorize
  • Pythagorean triples: (3,4,5) (5,12,13) (8,15,17) (7,24,25)
  • Multiples work too: (6,8,10), (9,12,15), …
Example
A right triangle has legs 6 and 8. Find the hypotenuse. c² = 36 + 64 = 100 → c = 10
Answer: 10

Practice Problems

Q11 A right triangle has one leg of length 5 and a hypotenuse of length 13. What is the length of the other leg?
Q12 What is the distance between the points (1, 2) and (4, 6)?
TOPIC 07 Statistics & Data Analysis

Key Concepts

  • Mean (average) = sum of values ÷ number of values
  • Median = middle value when data is sorted
  • Mode = most frequently occurring value
  • Range = maximum − minimum
★ Memorize
  • If mean of n numbers is M, their total sum = n × M
  • Missing value trick: missing = (desired total) − (known sum)
Example
Find the mean of 4, 7, 9, 12, 3. Sum = 35, Mean = 35 ÷ 5 = 7
Answer: 7

Practice Problems

Q13 The five test scores of a student are 82, 91, 78, 95, and 84. What is the median score?
Q14 The mean of six numbers is 15. Five of the numbers are 12, 18, 14, 16, and 10. What is the sixth number?
TOPIC 08 Probability & Counting

Key Concepts

  • Probability = favorable outcomes ÷ total outcomes
  • Counting Principle: event A × event B ways = m × n total ways
  • Combinations: C(n,r) = n! ÷ (r! × (n−r)!)
★ Memorize
  • P(A and B) = P(A) × P(B) [independent events]
  • P(not A) = 1 − P(A)
Example
A bag has 3 red and 5 blue marbles. P(red) = ? P = 3 ÷ (3 + 5) = 3/8
Answer: 3/8

Practice Problems

Q15 A fair coin is flipped 3 times. What is the probability of getting exactly 2 heads? Express as a fraction.
Q16 How many different 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if no digit is repeated?
TOPIC 09 Patterns, Sequences & Logic

Key Concepts

  • Arithmetic sequence: each term increases by constant d
  • n-th term: an = a1 + (n − 1) × d
  • Sum of arithmetic sequence: S = n/2 × (first + last)
★ Memorize
  • Sum of first n positive integers: n(n+1)/2
  • Sum of first n odd integers: n²
Example
Find the 10th term of 3, 7, 11, 15, … d = 4, a₁₀ = 3 + 9 × 4 = 3 + 36 = 39
Answer: 39

Practice Problems

Q17 What is the sum of the first 20 positive integers?
Q18 In a sequence, each term after the first is 3 more than twice the previous term. If the first term is 2, what is the fourth term?
TOPIC 10 Word Problems & Mixed Applications

Key Concepts

  • Work rate: if A finishes a job in a hours, rate = 1/a per hour
  • Combined rate: RateA + RateB = 1/T
  • Mixture problems: concentration × amount = total pure substance
★ Memorize
  • Read carefully: identify what is given vs. what is asked
  • Draw a diagram or table to organize information
  • Check your answer against the original conditions
Example
Alice paints a fence in 4 hrs, Bob in 6 hrs. Working together? Rate = 1/4 + 1/6 = 5/12 → Time = 12/5 = 2.4 hours
Answer: 2.4 hours (2 hrs 24 min)

Practice Problems

Q19 Pipe A fills a tank in 6 hours; pipe B fills it in 4 hours. With both pipes open, how many hours does it take to fill the tank? Express as a fraction.
Q20 Maria buys 3 notebooks and 2 pens for $11.50. Each notebook costs $2.50. What is the cost of each pen in dollars?

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