AMC 8 Math Competition Test — Questions and Answers
Question 1: In a class of 30 students, 18 play soccer and 14 play basketball, and 6 play both. How many students play neither sport?
Correct answer: 4
By inclusion-exclusion, 18 + 14 − 6 = 26 play at least one sport, so 30 − 26 = 4 play neither.
Question 2: From a group of 5 students, in how many ways can 2 students be chosen for a committee?
Correct answer: 10
C(5,2) = 5!/(2! × 3!) = 10.
Question 3: What is the prime factorization of 120?
- 2³ × 3² × 5
- 2 × 3 × 5²
- 2³ × 3 × 5 (Correct answer)
- 2² × 3 × 5
Correct answer: 2³ × 3 × 5
120 = 8 × 15 = 2³ × 3 × 5.
Question 4: Solve for y: 4y/3 = 20.
- 15 (Correct answer)
- 12
- 20
- 18
Correct answer: 15
Multiplying both sides by 3: 4y = 60, so y = 15.
Question 5: How many degrees are there in the sum of the interior angles of a pentagon?
- 360°
- 720°
- 540° (Correct answer)
- 450°
Correct answer: 540°
The interior angle sum of a pentagon = (5−2) × 180° = 3 × 180° = 540°.
Question 6: A student scored 18 out of 24 on a test. What percentage did they score?
- 75% (Correct answer)
- 65%
- 70%
- 72%
Correct answer: 75%
18/24 = 3/4 = 75%.
Question 7: Which correctly orders 3/5, 7/10, and 0.65 from least to greatest?
- 3/5, 7/10, 0.65
- 3/5, 0.65, 7/10 (Correct answer)
- 0.65, 7/10, 3/5
- 7/10, 0.65, 3/5
Correct answer: 3/5, 0.65, 7/10
Converting to decimals: 3/5 = 0.60, 0.65, 7/10 = 0.70, so the order is 3/5 < 0.65 < 7/10.
Question 8: What is the greatest common factor (GCF) of 48 and 36?
Correct answer: 12
The GCF of 48 and 36 is 12, since 12 divides both and no larger number does.
Question 9: Carlo's car consumes a gallon of gas every 35 miles and has a gas tank capacity of 14 gallons when full. Karl started with a full tank of gas, traveled 350 miles, stopped to buy 8 gallons of gas, and then drove the rest of the way to his destination. His gas tank was half full when he arrived. Karl drove how many miles that day?
- 500
- 300
- 525 (Correct answer)
- 650
- 500
Correct answer: 525
The correct answer: <br>
He had consumed 350/35 = 10 gallons of gas after 350 miles since he uses a gallon of gas every 35 miles. As a result, he had 14 – 10 = 4 gallons of gas left after the first leg of his journey. Then he went out and bought another 8 gallons of gas, bringing his total gas tank capacity to 12 gallons. He had 1/2 14 = 7 gallons of gas when he arrived. As a result, on the second part of his journey, he used 5 gallons of gas. As a result, the second leg of his journey covered 5 35 = 175 miles. When we add this to the 350 miles, we get 350 + 175 = 525 miles.
Question 10: How many ways can you arrange the letters in the word CAT?
Correct answer: 6
Three distinct letters can be arranged in 3! = 3 × 2 × 1 = 6 different orders.
Question 11: A rectangle's length is increased by 20% and its width is decreased by 10%. By what percent does the area change?
- 8% decrease
- 10% increase
- 8% increase (Correct answer)
- 10% decrease
Correct answer: 8% increase
New area = 1.2 × l × 0.9 × w = 1.08lw, which is an 8% increase.
Question 12: Scores of 60, 70, 80, and 90 in a frequency table have frequencies of 2, 5, 8, and 5. What is the mode?
- 80 (Correct answer)
- 90
- 70
- 75
Correct answer: 80
The mode is 80 because it has the highest frequency of 8 occurrences.
Question 13: A rectangle has perimeter 24 and length 8. What is its area?
- 32 (Correct answer)
- 48
- 16
- 64
Correct answer: 32
Using P = 2(l + w): 24 = 2(8 + w), so w = 4, and area = 8 × 4 = 32.
Question 14: What is the sum of the first 10 positive integers?
- 50
- 45
- 60
- 55 (Correct answer)
Correct answer: 55
Using Gauss's formula: n(n+1)/2 = 10 × 11/2 = 55.
Question 15: 10,000 is the sum of 25 consecutive even integers. Which of these 25 consecutive integers is the largest?
- 415
- 400
- 330
- 424 (Correct answer)
- 350
Correct answer: 424
The correct answer: <br>
Assume that n is the 13th consecutive even integer to be added. Because (n – 2k) +... + (n – 4) + (n – 2) + (n) + (n + 2) + (n + 4) +... +(n + 2k) = 25n, we can see that the sum of all 25 even numbers will reduce to 25n. 25n = 10000, which equals n=400. We want to discover the 25th integer, which is 400 + 2(25 – 13) = 424, keeping in mind that this is the 13th integer.
Question 16: Solve for x: 5(x − 2) = 3x + 4.
Correct answer: 7
Expanding: 5x − 10 = 3x + 4, so 2x = 14 and x = 7.
Question 17: A tank is 1/3 full. After adding 20 gallons it is 1/2 full. What is the tank's total capacity?
- 90
- 100
- 60
- 120 (Correct answer)
Correct answer: 120
The 20 gallons fills 1/2 − 1/3 = 1/6 of the tank, so the total capacity = 20 × 6 = 120 gallons.
Question 18: How many 3-digit even numbers can be formed from {2, 4, 5, 6} with no repetition?
- 24
- 30
- 18 (Correct answer)
- 12
Correct answer: 18
Choose an even units digit (3 options: 2, 4, 6), then arrange the remaining 3 digits in the first two positions (3 × 2 = 6 ways): 3 × 6 = 18.
Question 19: What is the remainder when 2²⁵ is divided by 3?
Correct answer: 2
Powers of 2 mod 3 alternate: 2¹≡2, 2²≡1, 2³≡2, …; odd powers give remainder 2, so 2²⁵ ≡ 2 (mod 3).
Question 20: A card is drawn from a standard 52-card deck. What is the probability it is a face card (J, Q, or K)?
- 1/4
- 3/13 (Correct answer)
- 1/13
- 3/52
Correct answer: 3/13
There are 12 face cards (4 Jacks + 4 Queens + 4 Kings), so P = 12/52 = 3/13.
Question 21: A number increased by 20% equals 60. What is the original number?
- 40
- 52
- 48
- 50 (Correct answer)
Correct answer: 50
1.2 × n = 60, so n = 60 ÷ 1.2 = 50.
Question 22: How many three-digit numbers have digits that are strictly increasing from left to right (each digit greater than the one before)?
- 84 (Correct answer)
- 56
- 210
- 120
Correct answer: 84
Each set of 3 distinct digits from 1–9 arranged in increasing order forms exactly one such number, giving C(9,3) = 84.
Question 23: How many positive factors does 36 have?
Correct answer: 9
36 = 2² × 3² has (2+1)(2+1) = 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Question 24: What is 3.6 ÷ 0.4?
- 0.9
- 9 (Correct answer)
- 0.09
- 90
Correct answer: 9
3.6 ÷ 0.4 = 36 ÷ 4 = 9.
