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2015

2015 AMC8 Paper & Solutions Pick

25 questions with solutions, geometry focuses on triangle and quadrilateral areas, algebra on systems of equations.

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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, equations. 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%)
20+
Honor Roll
Honor Roll (Top 5%)
16+
Achievement Roll
Grade 6 and below · 15+ points
15+
Sample Problems

2015 AMC8 Sample Problems (12 questions)

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

Q1EasyArithmetic
Calculate the value of 2015 - 1999.
A) 14
B) 15
C) 17
D) 18
E) 16

Steps

2015-1999=16
Answer: EUse simple subtraction.
Q4EasyKey concept: ratios
The ratio of sugar to water is 2:7, and the sugar-water mixture weighs 450 grams in total. How many grams of sugar are there?
A) 80
B) 90
C) 110
D) 100
E) 120

Steps

Each part = 450/9 = 50, so sugar = 2 × 50 = 100
Answer: DDivide proportionally.
Q6EasyArea
A triangle has a base of 12 and a height of 9. What is its area?
A) 54
B) 48
C) 60
D) 72
E) 108

Steps

Area = 1/2 × 12 × 9 = 54
Answer: AArea of a triangle = base × height ÷ 2.
Q8EasyApply the relevant mathematical concept
What is the median of the data 2, 3, 5, 5, 7, 8?
A) 4
B) 5.5
C) 6
D) 7
E) 5

Steps

After sorting, the middle two values are 5 and 5, so the median = 5
Answer: EWith an even number of data values, take the average of the middle two.
Q12MediumSystem of Equations
Buying 3 apples and 2 oranges costs 21 yuan, and buying 1 apple and 4 oranges costs 17 yuan. What is the price of one orange?
A) 2
B) 4
C) 5
D) 3
E) 6

Steps

Let an apple cost a yuan and an orange cost o yuan: 3a+2o=21, a+4o=17
From the second equation, a = 17 − 4o; substitute into the first equation
3(17−4o)+2o = 21 → 51 − 10o = 21 → o = 3
Answer: DSolve by substitution.
Q14MediumPrime Numbers
Among the numbers up to 50, what is the difference between the largest prime and the smallest prime?
A) 45
B) 42
C) 43
D) 44
E) 46

Steps

The largest prime up to 50 = 47, and the smallest prime = 2
Difference = 47 - 2 = 45
Answer: A1 is not a prime number.
Q16MediumArea
The diagonals of a rhombus are 10 and 24. What is its area?
A) 60
B) 100
C) 200
D) 240
E) 120

Steps

Area = 1/2 × 10 × 24 = 120
Answer: EArea of a rhombus = product of the diagonals ÷ 2.
Q19MediumProbability
A die is rolled twice. What is the probability that the sum of the two rolls is 10?
A) 1/9
B) 1/12
C) 1/6
D) 1/4
E) 1/3

Steps

Sum of 10: (4,6), (5,5), (6,4) = 3 ways
Probability = 3/36 = 1/12
Answer: BList all the combinations that satisfy the condition in order.
Q21HardKey concept: permutations
How many three-digit numbers with no repeated digits can be formed using 1, 2, 3, and 4?
A) 24
B) 12
C) 18
D) 36
E) 48

Steps

The hundreds digit has 4 choices, the tens digit has 3 choices, and the units digit has 2 choices
4 × 3 × 2 = 24
Answer: AThe number of permutations is P(4,3) = 24.
Q22HardNumber Theory
Find the value of 1² + 2² + 3² + ... + 10².
A) 385
B) 330
C) 355
D) 405
E) 435

Steps

Formula: n(n+1)(2n+1)/6
= 10 × 11 × 21 / 6 = 385
Answer: AYou need to memorize the formula for the sum of squares.
Q24HardGeometry
A right triangle has an area of 30, and one of its legs is 5. What is the hypotenuse?
A) 10
B) 12
C) 15
D) 17
E) 13

Steps

The other leg = 2 × 30 / 5 = 12
Hypotenuse = √(25+144) = 13
Answer: E5-12-13 is a classic Pythagorean triple.
Q25HardCombinatorics
How many ways are there to choose 3 people from 10 to form a team?
A) 90
B) 100
C) 150
D) 120
E) 180

Steps

C(10,3) = 10!/(3! × 7!) = 120
Answer: DCombination formula: C(n,k) = n!/(k!(n-k)!).

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