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2016

2016 AMC8 Paper & Solutions Pick

25 questions with solutions, balanced algebra and geometry, inclusion-exclusion in combinatorics. Moderate difficulty.

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12 Sample Questions
40 Minutes
Max Score 25
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Exam Overview

Exam Overview

This exam emphasizes algebra, geometry, combinatorics. Overall difficulty is moderate.

DDifficulty

  • EasyQ1-10
  • MediumQ11-20
  • HardQ21-25

TTopics

  • Algebra35%
  • Geometry30%
  • Number Theory15%
  • Combinatorics20%

AAwards

Distinguished Honor Roll
Distinguished Honor Roll (Top 1%)
21+
Honor Roll
Honor Roll (Top 5%)
17+
Achievement Roll
Grade 6 and below · 15+ points
15+
Sample Problems

2016 AMC8 Sample Problems (12 questions)

Sample reference problems by difficulty — click an option to check your answer

Q1EasyArithmetic
Calculate the value of 456 + 789 - 245.
A) 1000
B) 900
C) 950
D) 1050
E) 1100

Steps

456+789=1245
1245-245=1000
Answer: ARound to convenient numbers to simplify the calculation
Q3EasyKey concept: fractions
Calculate the value of 1/2 + 1/3 + 1/6.
A) 1/2
B) 1
C) 2/3
D) 5/6
E) 7/6

Steps

Find a common denominator: 3/6+2/6+1/6=6/6=1
Answer: BThe denominator 6 is the least common multiple of 2, 3, and 6
Q5EasyConvert percentages to fractions
After a 15% markdown, an item sells for 170 yuan. What was the original price?
A) 185
B) 190
C) 200
D) 195
E) 210

Steps

0.85x=170 → x=200
Answer: CA 15% markdown means paying 85% of the original price
Q9EasyArea
A square has a perimeter of 36 cm. What is its area?
A) 64
B) 72
C) 81
D) 100
E) 144

Steps

side length=36/4=9, area=81
Answer: CPerimeter of a square = 4 × side length
Q11MediumKey concept: equations
Two numbers have a sum of 45 and a difference of 9. What is the larger number?
A) 24
B) 27
C) 25
D) 30
E) 36

Steps

a+b=45, a-b=9
2a=54 → a=27
Answer: BSum-and-difference problem: larger number = (sum + difference)/2
Q13MediumDivisibility
How many numbers from 1 to 40 are divisible by 3 but not by 5?
A) 8
B) 9
C) 11
D) 10
E) 12

Steps

Divisible by 3: ⌊40/3⌋ = 13 numbers
Divisible by 15 (by both 3 and 5): ⌊40/15⌋ = 2 numbers
13 − 2 = 11
Answer: CInclusion-exclusion: divisible by 3 but not 5 = (divisible by 3) − (divisible by 15)
Q15MediumArea
A trapezoid has parallel sides of length 5 and 11 and a height of 6. What is its area?
A) 36
B) 42
C) 54
D) 66
E) 48

Steps

area=1/2×(5+11)×6=48
Answer: EArea of a trapezoid = 1/2 × (sum of the two parallel sides) × height
Q18MediumProbability
A bag contains 5 red balls and 3 white balls. What is the probability of drawing a red ball?
A) 3/8
B) 5/8
C) 1/2
D) 3/5
E) 2/3

Steps

P=5/(5+3)=5/8
Answer: BBasic probability = favorable outcomes / total outcomes
Q21HardInclusion-exclusion
Among 50 people, 35 like math and 30 like Chinese, while 5 like neither. How many people like both subjects?
A) 20
B) 15
C) 18
D) 22
E) 25

Steps

Like at least one subject = 50-5=45
35+30-x=45 → x=20
Answer: AInclusion-exclusion principle: |A∪B|=|A|+|B|-|A∩B|
Q22HardNumber Theory
A three-digit number has digits that sum to 18. The hundreds digit is twice the units digit, and the tens digit is 2 more than the units digit. What is the number?
A) 576
B) 648
C) 864
D) 657
E) 756

Steps

Let the units digit be a; then the hundreds digit is 2a and the tens digit is a+2
2a + (a+2) + a = 18 → 4a + 2 = 18 → a = 4
units digit 4, hundreds digit 8, tens digit 6 → 864
Answer: CUse algebra to represent each digit
Q24HardKey concept: permutations
Five people stand in a row. A and B are not adjacent. How many arrangements are there?
A) 36
B) 48
C) 60
D) 84
E) 72

Steps

Total arrangements = 5! = 120
Arrangements with A and B adjacent = 4! × 2 = 48
Not adjacent = 120 - 48 = 72
Answer: EFor non-adjacency problems, use "total − adjacent"
Q25HardGeometry
A circle has an area of 154 cm² (π ≈ 22/7). What is its diameter?
A) 7
B) 10
C) 14
D) 12
E) 21

Steps

πr²=154 → r²=154×7/22=49
r=7, diameter=14
Answer: CArea of a circle = πr²; work backward to find the radius

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