· Math Explorers Club · Competition Prep · 11 min read
AMC 8 Practice Problems for Beginners (with Solutions)
Ten original AMC 8 practice problems at the level a first-year student can reach, covering every topic area on MAA's list, with solutions that explain the trap answers too.
Printable PDF: every problem with room to work, and the answer key on a separate page.
Most AMC 8 practice problems online come straight from past papers, and a full past paper is a tough place for a 5th or 6th grader to start. The hardest questions on a real AMC 8 stump plenty of 8th graders. A first-year student doesn’t need those yet. What pays off is getting reliably good at the problems they can reach.
That’s what this set is for: ten original AMC 8 practice problems at a beginner’s level, with worked solutions that explain why each trap answer is tempting. None of them comes from a real AMC 8. We wrote them to cover every topic area the Mathematical Association of America (MAA) lists for the contest.
The contest itself is 25 multiple-choice questions in 40 minutes, with no calculator. The score is the number of correct answers, and there is no penalty for guessing. For a student in 6th grade or below, 15 correct earns a place on MAA’s Achievement Roll, which makes it a good first-year target. Our parent’s guide to preparing for AMC 8 has the full plan.
What these problems cover
MAA describes the AMC 8 as covering the middle school curriculum, including counting and probability, estimation, proportional reasoning, elementary geometry (including the Pythagorean Theorem), spatial visualization, and interpreting graphs and tables. The ten problems below touch each of those, plus the number sense that runs through all of them:
| Problem | Topic (MAA’s wording where it applies) |
|---|---|
| 1 | Arithmetic and number sense |
| 2 | Proportional reasoning |
| 3 | Estimation |
| 4 | Percentages |
| 5 | Elementary geometry (Pythagorean Theorem) |
| 6 | Counting |
| 7 | Probability |
| 8 | Spatial visualization |
| 9 | Interpreting graphs and tables; averages |
| 10 | Number theory: factors |
The table’s grouping is ours, not MAA’s; the official list doesn’t assign topics to question numbers.
How to use these practice problems
- Round 1: Problems 1-5 with a 10-minute timer. Round 2: Problems 6-10, another 10 minutes, on a different day if your child prefers. That’s a bit more time per problem than the real contest gives (40 minutes ÷ 25 questions is 1.6 minutes each, our arithmetic), which is right for a first try.
- Pencil, eraser, and blank scratch paper only. Those are allowed on contest day; calculators, compasses, protractors, and graph paper are not.
- Always pick an answer. With no penalty for guessing, a blank is a wasted chance. Practice eliminating choices that can’t be right and guessing among the rest.
- Review every problem, right or wrong, with the solutions below. The trap answers are the most useful part.
The problems
Problem 1
What is (2 + 4 + 6 + … + 20) − (1 + 3 + 5 + … + 19)?
(A) 1 (B) 5 (C) 10 (D) 20 (E) 110
Problem 2
A car uses 3 gallons of gas to drive 90 miles. At the same rate, how many gallons does it need to drive 210 miles?
(A) 5 (B) 6 (C) 7 (D) 8 (E) 9
Problem 3
Which of these is closest to (0.49 × 401) ÷ 0.2?
(A) 10 (B) 100 (C) 400 (D) 1,000 (E) 10,000
Problem 4
A class has 25 students, and 60% of them have a pet. Of the students who have a pet, one third have a cat. How many students in the class have a cat?
(A) 3 (B) 5 (C) 8 (D) 10 (E) 15
Problem 5
A 13-foot ladder leans against a vertical wall. The bottom of the ladder is 5 feet from the base of the wall. How high up the wall does the ladder reach, in feet?
(A) 8 (B) 10 (C) 12 (D) 13 (E) 18
Problem 6
Four friends, Amy, Ben, Cal, and Dev, sit in a row of four chairs. Amy and Ben insist on sitting next to each other. In how many different orders can the four friends sit?
(A) 6 (B) 8 (C) 12 (D) 18 (E) 24
Problem 7
Two fair coins are flipped and a fair six-sided die is rolled. What is the probability that both coins land heads and the die shows an even number?
(A) 1/12 (B) 1/8 (C) 1/6 (D) 1/4 (E) 3/8
Problem 8
A closed box is shaped like a rectangular prism that is 2 inches by 3 inches by 4 inches. What is the total area of its six faces, in square inches?
(A) 24 (B) 26 (C) 48 (D) 52 (E) 104
Problem 9
The table shows how many laps Maya swam each day last week. One number is missing.
| Day | Mon | Tue | Wed | Thu | Fri |
|---|---|---|---|---|---|
| Laps | 12 | 15 | ? | 10 | 18 |
Maya’s average (mean) for the five days was 14 laps. How many laps did she swim on Wednesday?
(A) 12 (B) 14 (C) 15 (D) 16 (E) 18
Problem 10
How many positive whole numbers divide 36 evenly? (Count 1 and 36.)
(A) 6 (B) 8 (C) 9 (D) 10 (E) 12
Solutions, with the trap answers explained
Problem 1: Answer (C) 10
There’s no need to add up either list. Pair the terms instead: (2 − 1) + (4 − 3) + (6 − 5) + … + (20 − 19). Each pair is 1, and there are 10 pairs (the even numbers 2 to 20 are ten numbers), so the answer is 10.
Why (E) 110 is there: 110 is the sum of the even numbers alone. A student who starts adding and loses track of the question lands on it. Contest answer choices often include the number you’d get halfway through a problem.
Problem 2: Answer (C) 7
3 gallons for 90 miles means 1 gallon for every 30 miles. 210 ÷ 30 = 7 gallons.
Another way: 210 miles is 90 + 90 + 30, which needs 3 + 3 + 1 = 7 gallons.
The habit: find the “one unit” rate first (miles per gallon here). Proportional reasoning problems on the AMC 8 are usually one short step once the unit rate is on paper.
Problem 3: Answer (D) 1,000
Round to friendly numbers: 0.49 is about ½, and 401 is about 400. So the top is about ½ × 400 = 200. Dividing by 0.2 is the same as multiplying by 5 (because 0.2 is 1/5), so 200 × 5 = 1,000.
The exact value is 982.45, and 1,000 is clearly the closest choice.
Why (B) and (E) are there: they’re what you get if a decimal point slips one place (about 98 or about 9,800). Estimation problems reward checking the size of the answer: “about half of 400, then five times that” is easy to hold in your head.
Problem 4: Answer (B) 5
60% of 25 students is 0.6 × 25 = 15 students with a pet. One third of those have a cat: 15 ÷ 3 = 5.
Why (E) 15 is there: it’s the number of pet owners, the answer to the first step. Stopping one step early is one of the most common beginner mistakes, and answer choices are often built to catch it.
Problem 5: Answer (C) 12
The wall, the ground, and the ladder make a right triangle, and the ladder is the longest side (the hypotenuse). By the Pythagorean Theorem:
5² + height² = 13² 25 + height² = 169 height² = 144 height = 12 feet
Worth memorizing: 5-12-13 is one of a handful of whole-number right triangles (along with 3-4-5 and 6-8-10) that show up again and again in contest problems. Recognizing one saves a whole calculation.
Why (E) 18 is there: 13 + 5. Adding the two given lengths skips the geometry entirely. And (D) 13 can’t be right, because the ladder is slanted: it can’t reach as high as its own length.
Problem 6: Answer (C) 12
Glue Amy and Ben together into one “block.” Now there are three things to arrange: the AB block, Cal, and Dev. Three things can be put in order in 3 × 2 × 1 = 6 ways.
Inside the block, Amy can sit on the left or on the right: 2 ways. So the total is 6 × 2 = 12.
Why (A) 6 is there: forgetting that the block can flip (AB or BA). Why (E) 24 is there: 4 × 3 × 2 × 1 = 24 is the number of orders with no rule at all.
Problem 7: Answer (B) 1/8
The three events don’t affect each other, so multiply their probabilities:
- First coin heads: 1/2
- Second coin heads: 1/2
- Die even (2, 4, or 6): 3/6 = 1/2
1/2 × 1/2 × 1/2 = 1/8.
Check by counting: there are 2 × 2 × 6 = 24 equally likely outcomes. The ones that work are H, H, and then 2, 4, or 6: that’s 3 outcomes, and 3/24 = 1/8. ✓
Why (D) 1/4 is there: it’s the chance of two heads alone, forgetting the die.
Problem 8: Answer (D) 52
A box has three pairs of matching faces:
- Two faces are 2 × 3 = 6 square inches each.
- Two faces are 2 × 4 = 8 each.
- Two faces are 3 × 4 = 12 each.
One of each is 6 + 8 + 12 = 26, and there are two of each, so the total is 52 square inches.
Why (A) 24 and (B) 26 are there: 24 is the box’s volume (2 × 3 × 4), a different measurement. 26 counts only three faces. Picturing the box, or sketching its unfolded net, shows there are six faces in three matching pairs.
Problem 9: Answer (C) 15
Turn the average into a total. An average of 14 laps over 5 days means 14 × 5 = 70 laps in all.
The four days in the table add to 12 + 15 + 10 + 18 = 55. Wednesday is the rest: 70 − 55 = 15 laps.
Check: 12 + 15 + 15 + 10 + 18 = 70, and 70 ÷ 5 = 14. ✓
Why (B) 14 is there: it’s the average itself. A student who reasons “the missing day is probably average” picks it without doing the arithmetic. No algebra needed, either: with multiple choice, your child can also test each answer by adding up the five days and dividing by 5.
Problem 10: Answer (C) 9
List the divisors in pairs that multiply to 36:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
That’s 1, 2, 3, 4, 6, 9, 12, 18, 36: 9 divisors. The pair 6 × 6 contributes only one number, which is why the count is odd.
Why (D) 10 is there: counting 6 twice. Perfect squares (like 36) are the only numbers with an odd number of divisors, a fact that shows up in contest problems surprisingly often.
What a score on this set tells you
This is our own rough guide, not an MAA benchmark. These ten problems are pitched at the level a first-year student can reach, so:
- 8-10 correct: your child is ready for a full official sample paper under real timing. The Achievement Roll target of 15 on the real contest is realistic.
- 5-7 correct: a normal starting point. Look at which misses were trap answers (fixed by rereading the question) and which were topics your child hasn’t met yet (fixed by learning them).
- 0-4 correct: fine for a 4th or 5th grader, and a sign to build up gradually. Our weekly worksheets for grades 5-6 cover the same problem-solving habits at a gentler level.
Where to find more AMC 8 practice problems
- Official MAA sample competition (free): MAA posts the 2023 AMC 8 free as a sample competition, with solutions, and sells older papers in its store. After this set, a full official paper under the 40-minute timer is the natural next step.
- Our AMC 8 prep guide: a first-year course with 10 chapters, 90+ original practice problems with step-by-step solutions, and a full 25-question practice test.
- Our MOEMS practice problems: two free practice olympiads, Division E (grades 4-6) and Division M (grades 6-8), with the same kind of non-routine thinking in a free-answer format.
- Our free Problem of the Week: one original problem a week by email.
Registration goes through a school or local host, not directly through MAA. For this season’s competition window and registration deadlines, see dates and deadlines in our AMC 8 parent’s guide.
This guide is not affiliated with or endorsed by the Mathematical Association of America (MAA) or the American Mathematics Competitions. “AMC 8” is a trademark of its respective owners. This guide is not affiliated with or endorsed by Math Olympiads for Elementary and Middle Schools (MOEMS®). “MOEMS” is a trademark of its respective owners.