The AMC 8 rewards flexible problem solving, careful reading, and smart strategy more than long calculations. This cheat sheet summarizes the most useful tools for middle school contest math, including arithmetic shortcuts, algebra patterns, geometry formulas, and counting ideas. Students need it to recognize common problem types quickly and choose an efficient path under time pressure.
The core ideas include using variables wisely, drawing clean diagrams, checking units, and estimating before doing exact work. Key formulas include area, volume, averages, ratios, probability, and special triangle relationships. Strong AMC 8 work also depends on number theory facts such as divisibility, factors, multiples, and remainders.
Key Facts
- The average of numbers is , so the sum is .
- A percent means parts per , so of is .
- For a circle, the circumference is and the area is .
- For a triangle, the area is , where is the base and is the perpendicular height.
- The Pythagorean theorem for a right triangle is , where is the hypotenuse.
- The probability of an event is .
- The number of ways to make sequential choices is found by multiplying the number of options at each step, such as .
- If , then and have the same remainder when divided by .
Vocabulary
- Estimation
- Estimation is using nearby values or size comparisons to predict a reasonable answer before calculating exactly.
- Ratio
- A ratio compares two quantities by division, such as or .
- Remainder
- A remainder is the amount left after dividing one integer by another, as in .
- Factor
- A factor of a number is an integer that divides it evenly with no remainder.
- Complement
- The complement of an event is everything outside that event, and its probability is .
- Invariant
- An invariant is a quantity or property that stays the same while other parts of a problem change.
Common Mistakes to Avoid
- Rushing into computation before reading all answer choices is wrong because AMC 8 questions often have shortcuts based on estimation, symmetry, or elimination.
- Using the slanted side as a triangle height is wrong because in must be perpendicular to the chosen base.
- Adding probabilities for dependent choices is wrong because sequential choices usually require multiplication, such as .
- Forgetting to convert percents to fractions or decimals is wrong because means , not .
- Assuming a diagram is drawn to scale is wrong because contest diagrams may be misleading unless equal lengths, right angles, or parallel lines are stated or marked.
Practice Questions
- 1 The average of numbers is . If one number is removed, the average of the remaining numbers is . What number was removed?
- 2 A rectangle has perimeter and length . What is its area?
- 3 A bag contains red marbles, blue marbles, and green marbles. What is the probability of choosing a marble that is not green?
- 4 In an AMC 8 problem, when is it better to test answer choices instead of solving directly with equations? Explain using a general strategy, not a specific calculation.
Understanding AMC 8 Strategy & Reference
Many contest problems become easier when you choose the right representation before calculating. A table can reveal a repeating pattern. A number line can show which values are possible.
A sketch can expose equal lengths or hidden right angles. For word problems, write down what each quantity means and give it a unit when possible. This prevents a common error where a student combines quantities that cannot be combined.
In a multiple choice problem, the answer choices are useful information. Estimation can remove choices that are too large, too small, negative, or not reasonable. Working backward from a choice is often faster than building an answer from the beginning.
Arithmetic questions often test structure rather than speed. Break awkward numbers into friendly parts, such as a nearby multiple of ten or one hundred. Keep track of whether a percent change is based on the original amount or the new amount.
An increase of twenty percent followed by a decrease of twenty percent does not return to the starting value, because the two changes use different bases. In algebra, a variable stands for a number with conditions, not just a blank to fill. Substitute simple test values when checking a pattern.
When an equation contains fractions, think about what values would make a denominator zero. Those values are not allowed, even if later steps seem to produce them.
Geometry rewards careful attention to what a diagram proves and what it merely suggests. A picture may not be drawn to scale. Do not assume two segments are equal because they look equal.
Look instead for markings, stated facts, symmetry, parallel lines, or angle rules. Complex shapes can often be split into rectangles, triangles, or circles. Sometimes it is quicker to find the area of a large enclosing shape and subtract missing pieces.
For three dimensional figures, separate surface area from volume. Surface area measures the outside covering, while volume measures the space inside. Labeling every known length on a drawing helps students notice which measurements are actually needed.
Counting and probability require a clear definition of one outcome. A two digit number, an arrangement of letters, and a path through a grid each have different kinds of outcomes. List small cases to check that no case is missed or counted twice.
For larger cases, organize choices in stages. Be alert when order matters. Choosing Ava then Ben differs from choosing Ben then Ava in some problems, but not in a team selection.
Probability questions need equally likely outcomes before a simple favorable over total method works. Number theory problems often become manageable by testing small remainders, making factor pairs, or looking for cycles in final digits.
Practice should include reviewing wrong answers. Identify the exact assumption or step that failed, then solve the problem again without looking at the solution.