Achievable logoAchievable logo
AMC
Sign in
Sign up
Purchase
Textbook
Practice exams
Support
How it works
Exam catalog
Mountain with a flag at the peak
Textbook
Introduction
1. Algebra
2. Geometry
3. Number theory
4. Counting and probability
5. Intermediate topics (AMC 10/12)
6. Advanced topics (AMC 12)
7. General approaches
7.1 Creativity
7.2 Simplify
7.3 Direct solving
8. Practical strategies
Wrapping up
Achievable logoAchievable logo
7.2 Simplify
Achievable AMC
7. General approaches

Simplify

4 min read
Font
Discuss
Share
Feedback

This chapter applies to all AMC 8/10/12 test takers.

This chapter may feel a bit less concrete than some of the others. The main idea is simple, though: when you work the problems in this chapter (and the drills afterward), look for a way to simplify what you’re given into something basic and easy to compute.

A key AMC skill is turning a problem that looks long, abstract, or information-heavy into a small set of usable facts.

  • Sometimes a wordy problem hides a very short solution.
  • Sometimes the math isn’t stated directly, but the needed calculation is still precise.
  • Sometimes there’s a viewpoint or method that makes the whole problem straightforward.

In each case, the job is the same: translate the situation into a simpler form you can work with.

Here are two AMC problems that become much more accessible once you reduce them to a clean equation. After you try these, keep the same mindset for the quiz questions: if your current approach feels too slow or messy, there’s probably a simpler interpretation.

Example: The question below is from 2016 AMC 8

In an All-Area track meet, 216 sprinters enter a 100-meter dash competition. The track has 6 lanes, so only 6 sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?
A. 36
B. 42
C. 43
D. 60
E. 72

(spoiler)

Answer: C. 43

To end with 1 runner, we must eliminate 215 runners.

Each race eliminates 5 runners (everyone except the winner). So the number of races is the number of groups of 5 eliminations needed to reach 215:

This problem can be simplified into the equation: x=215/5=43.

Example: The question below is from 2020 AMC 10A

A point is chosen at random within the square in the coordinate plane whose vertices are (0,0),(2020,0),(2020,2020), and (0,2020). The probability that the point is within d units of a lattice point is 21​. (A point (x,y) is a lattice point if x and y are both integers.) What is d to the nearest tenth?
A. 0.3
B. 0.4
C. 0.5
D. 0.6
E. 0.7

(spoiler)

Answer: B. 0.4

This problem can be distilled down to the basic ratio (π)r2/1=1/2 where r is the solution.

Here’s the key observation behind that equation. The plane is tiled by 1×1 squares whose corners are lattice points. In each such square, the region within distance r of a lattice point consists of four quarter-circles (one at each corner). Those four quarter-circles add up to the area of one full circle of radius r.

So, within a typical 1×1 square:

  • area within distance r of a lattice point =(π)r2
  • total area of the square =1

The problem says this probability is 1/2, so we set

(π)r2/1=1/2.

This simplifies to just r2=1/(2π)≈.4.

Common themes

  • Remember that this is a relatively quick, timed test. If you are attempting a problem and noticing that your approach would take an exorbitant amount of time, it is likely the case that there is another approach, or way to interpret the question, that would make the problem easier to solve.
  • Series or sequence questions that would require too much calculation often follow a pattern that leads to simplification.

Simplifying AMC Problems

  • Focus on reducing complex, wordy, or abstract problems to basic, manageable facts
  • Look for hidden shortcuts or simple equations behind lengthy descriptions
  • Translate situations into forms that are easy to compute

Example: Track Meet Problem

  • Eliminate all but one competitor by grouping eliminations
  • Each race eliminates 5 runners (1 winner per race)
  • Total races needed: 215/5=43

Example: Lattice Point Probability

  • Probability region is area within distance r of lattice points in a 1×1 square
  • Area within r units: 4 quarter-circles =πr2
  • Set up equation: πr2=1/2; solve for r (r2=1/(2π)≈0.4)

Common Themes

  • Timed tests reward efficient, simple approaches
  • Problems that seem calculation-heavy often have patterns or shortcuts for simplification
  • Seek alternative interpretations if your method feels slow or complex

Sign up for free to take 6 quiz questions on this topic

Previous
Next  | 7.3 Direct solving
All rights reserved ©2016 - 2026 Achievable, Inc.

Simplify

This chapter applies to all AMC 8/10/12 test takers.

This chapter may feel a bit less concrete than some of the others. The main idea is simple, though: when you work the problems in this chapter (and the drills afterward), look for a way to simplify what you’re given into something basic and easy to compute.

A key AMC skill is turning a problem that looks long, abstract, or information-heavy into a small set of usable facts.

  • Sometimes a wordy problem hides a very short solution.
  • Sometimes the math isn’t stated directly, but the needed calculation is still precise.
  • Sometimes there’s a viewpoint or method that makes the whole problem straightforward.

In each case, the job is the same: translate the situation into a simpler form you can work with.

Here are two AMC problems that become much more accessible once you reduce them to a clean equation. After you try these, keep the same mindset for the quiz questions: if your current approach feels too slow or messy, there’s probably a simpler interpretation.

Example: The question below is from 2016 AMC 8

In an All-Area track meet, 216 sprinters enter a 100-meter dash competition. The track has 6 lanes, so only 6 sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?
A. 36
B. 42
C. 43
D. 60
E. 72

(spoiler)

Answer: C. 43

To end with 1 runner, we must eliminate 215 runners.

Each race eliminates 5 runners (everyone except the winner). So the number of races is the number of groups of 5 eliminations needed to reach 215:

This problem can be simplified into the equation: x=215/5=43.

Example: The question below is from 2020 AMC 10A

A point is chosen at random within the square in the coordinate plane whose vertices are (0,0),(2020,0),(2020,2020), and (0,2020). The probability that the point is within d units of a lattice point is 21​. (A point (x,y) is a lattice point if x and y are both integers.) What is d to the nearest tenth?
A. 0.3
B. 0.4
C. 0.5
D. 0.6
E. 0.7

(spoiler)

Answer: B. 0.4

This problem can be distilled down to the basic ratio (π)r2/1=1/2 where r is the solution.

Here’s the key observation behind that equation. The plane is tiled by 1×1 squares whose corners are lattice points. In each such square, the region within distance r of a lattice point consists of four quarter-circles (one at each corner). Those four quarter-circles add up to the area of one full circle of radius r.

So, within a typical 1×1 square:

  • area within distance r of a lattice point =(π)r2
  • total area of the square =1

The problem says this probability is 1/2, so we set

(π)r2/1=1/2.

This simplifies to just r2=1/(2π)≈.4.

Common themes

  • Remember that this is a relatively quick, timed test. If you are attempting a problem and noticing that your approach would take an exorbitant amount of time, it is likely the case that there is another approach, or way to interpret the question, that would make the problem easier to solve.
  • Series or sequence questions that would require too much calculation often follow a pattern that leads to simplification.
Key points

Simplifying AMC Problems

  • Focus on reducing complex, wordy, or abstract problems to basic, manageable facts
  • Look for hidden shortcuts or simple equations behind lengthy descriptions
  • Translate situations into forms that are easy to compute

Example: Track Meet Problem

  • Eliminate all but one competitor by grouping eliminations
  • Each race eliminates 5 runners (1 winner per race)
  • Total races needed: 215/5=43

Example: Lattice Point Probability

  • Probability region is area within distance r of lattice points in a 1×1 square
  • Area within r units: 4 quarter-circles =πr2
  • Set up equation: πr2=1/2; solve for r (r2=1/(2π)≈0.4)

Common Themes

  • Timed tests reward efficient, simple approaches
  • Problems that seem calculation-heavy often have patterns or shortcuts for simplification
  • Seek alternative interpretations if your method feels slow or complex

More from General approaches

  • Creativity
  • Direct solving