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AMC 8 trail

Number sense, ratios, geometry, data and creative counting.

Original AMC-style practice

New reasoning quests

20 new problems inspired by competition skills, with new settings, guided hints and step-by-step explanations.

40 problems with answer checking, guided hints and worked explanations. Sign in as a student to save your attempts and share progress with your parent.

Foundation, Developing and Challenge are Numerive practice levels. These original exercises build competition skills; they are not official AMC questions or a timed mock exam.

Foundation

Original practice · Foundation

Badge colors

A club has 64 badges. The ratio of red badges to blue badges is $\displaystyle 3 : 5$ . How many blue badges are there?

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Original practice · Foundation

Sale, then tax

A backpack is priced at $40. A store takes $\displaystyle 25\%$ off, then adds $\displaystyle 10\%$ sales tax to the discounted price. What is the final price in dollars?

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Original practice · Foundation

Filling the tank

A tank is $\displaystyle \frac{3}{8}$ full. Adding 15 liters makes it $\displaystyle \frac{3}{4}$ full. What is the capacity of the tank in liters?

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Original practice · Foundation

Three shuttle clocks

Three shuttles leave a station together. They then leave every 12, 18 and 30 minutes, respectively. After how many minutes will all three next leave together?

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Original practice · Foundation

A one-tile border

An 8-by-6 rectangle is made from unit squares. A border one square wide runs all the way around its inside edge. How many unit squares are in the border?

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Original practice · Foundation

A sum of nine

Two fair six-sided dice are rolled. What is the probability that their sum is 9? Give a fraction.

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New reasoning quest · Foundation

Workshop sticker reserve

An art workshop prints some stickers and sets aside $\displaystyle \frac{1}{6}$ of them. All remaining stickers are shared among 5 teams of 6 children, with each child receiving 2 stickers. How many stickers were printed?

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New reasoning quest · Foundation

The repeating robot route

A robot repeats this route six times: 3 meters east, 2 meters north, 1 meter west, and 2 meters south. How many meters from its starting point is it after all six repetitions?

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New reasoning quest · Foundation

Balancing the bead tray

A tray contains 45 beads, all red or blue, in a red-to-blue ratio of $\displaystyle 2 : 3$ . Six blue beads are removed. How many red beads must then be added so that the two colors have equal counts?

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New reasoning quest · Foundation

Two water stops

A hiking group uses $\displaystyle \frac{1}{4}$ of its water at the first stop and $\displaystyle \frac{2}{5}$ of the remaining water at the second stop. It then has 9 liters left. How many liters did the group start with?

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Developing

Original practice · Developing

The removed score

Six numbers have mean 14. One number is removed, and the remaining five numbers have mean 12. What number was removed?

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Original practice · Developing

Meeting cyclists

Two cyclists start 81 kilometers apart and ride toward each other at constant speeds of 12 and 15 kilometers per hour. After how many hours do they meet?

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Original practice · Developing

Palindromes ending in five

How many three-digit palindromes are divisible by 5? A palindrome reads the same in both directions.

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Original practice · Developing

The missing digit

The three-digit number 4a2 is divisible by 9, where a is a digit. What is a?

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Original practice · Developing

Largest fenced rectangle

A rectangle has integer side lengths and perimeter 34. What is its largest possible area?

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Original practice · Developing

Neither empty nor full

A design uses a selection from four available colors. At least one color must be used, but using all four is forbidden. How many different selections are possible?

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Original practice · Developing

Avoiding an intersection

A path from (0,0) to (3,2) uses only one-unit steps east or north. How many such paths avoid the point (1,1)?

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Original practice · Developing

Passing coins around

Box A gives 6 coins to B, then B gives 4 coins to C, then C gives 2 coins to A. All three boxes finish with 20 coins. How many coins did A have at the start?

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Original practice · Developing

Two quiz groups

Twelve students average 70 on a quiz, and eight other students average 85. What is the mean score of all twenty students?

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