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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
8. Practical strategies
8.1 Backsolve
8.2 Estimate
8.3 Eliminate
Wrapping up
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8.1 Backsolve
Achievable AMC
8. Practical strategies

Backsolve

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

Backsolving means testing the answer choices to see which one works, instead of solving the problem straight through. AMC problems usually have a clean “direct” solution, so backsolving is often a backup plan. Still, it can give you complete confidence in your answer, and sometimes it’s actually the quickest approach.

How backsolving works

Instead of building the answer from scratch, you start by assuming an answer choice is correct and check whether it satisfies the problem’s conditions.

When the answer choices are single numbers (not words, ordered pairs, or expressions), they’re usually listed in increasing order. That ordering lets you test efficiently:

  • Start by testing B.
    • If B works, you’re done.
    • If B is too large, then A must be correct.
    • If B is too small, test D next.
  • Then test D.
    • If D works, you’re done.
    • If D is too large, then C must be correct.
    • If D is too small, then E must be correct.

This “B then D” shortcut relies on the choices being ordered and on the output changing in a predictable direction as the input changes.

When the shortcut doesn’t apply

If the answer choices are ordered pairs, sentences, or variables/expressions, you can’t use the “B then D” logic. In those cases, backsolve by testing the choices that look most plausible first, and keep going until one satisfies the conditions.

Examples

For each of these AMC examples, find the solution by using the backsolving method exclusively.

Example: The question below is from 2018 AMC 10A

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that 65% of the votes cast on his video were like votes. How many votes had been cast on Sangho’s video at that point?
A. 200
B. 300
C. 400
D. 500
E. 600

(spoiler)

Answer: B. 300

Because the answers are single numbers in increasing order, the “B then D” approach fits well.

Test B: assume there were 300 votes.

  • Likes: 65% of 300 is 195.
  • Dislikes: 300−195=105.
  • Score: 195−105=90.

That matches the given score, so B is correct. If the score had come out larger than 90, you’d need fewer total votes; if it had come out smaller than 90, you’d need more total votes.

Example: The question below is from 2019 AMC 10B

All lines with equation ax+by=c such that a,b,c form an arithmetic progression pass through a common point. What are the coordinates of that point?
A. (−1,2)
B. (0,1)
C. (1,−2)
D. (1,0)
E. (1,2)

(spoiler)

Answer: A. (−1,2)

Here the answer choices are ordered pairs, so the “B then D” shortcut doesn’t apply. Instead, test choices directly.

Choices B and D don’t work because they force two of a,b,c to be equal, which prevents a,b,c from forming an arithmetic progression.

Now test A by plugging in x=−1 and y=2:

  • The equation becomes −a+2b=c.
  • This can happen when a,b,c are in arithmetic progression (for example, −1,0,1 or 100,101,102), so A is consistent with the condition.

So the common point is (−1,2).

Common themes

  • If you’re having trouble solving algebraically, try checking answer choices instead.
  • Pay attention to how changing the tested answer changes the output. If your result is too high or too low, that tells you whether to test a smaller or larger choice next. (Sometimes the relationship is inverse, so think about the direction before you commit.)
  • Backsolving isn’t only for finding an answer. It’s also a reliable way to double-check an answer you found by another method.

Backsolving overview\

  • Test answer choices to see which satisfies the problem
  • Useful as a backup or to confirm answers
  • Sometimes fastest approach, especially with clean answer choices

How backsolving works\

  • Assume an answer choice is correct, check if it fits problem conditions
  • For single-number, increasing order choices:
    • Test B first
      • If correct, done
      • If too large, A is correct
      • If too small, test D
    • Test D next
      • If correct, done
      • If too large, C is correct
      • If too small, E is correct
  • Shortcut relies on ordered choices and predictable output change

When shortcut doesn’t apply\

  • For ordered pairs, sentences, or variable/expression choices:
    • Can’t use “B then D” logic
    • Test most plausible choices first, continue until one fits

Example 1: Single-number choices (AMC 10A 2018)\

  • Apply “B then D” shortcut
  • Test B: 300 votes
    • 65% likes = 195 likes, 105 dislikes
    • Score: 195−105=90
    • Matches, so B is correct

Example 2: Ordered pair choices (AMC 10B 2019)\

  • Shortcut not applicable
  • Test choices directly
  • Plug (−1,2) into equation, check arithmetic progression
    • Satisfies condition, so A is correct

Common themes\

  • Use backsolving if stuck or to check algebraic answers
  • Observe how changing answer choice affects result
    • Adjust next guess based on whether result is too high/low
  • Backsolving is a reliable double-checking tool

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Backsolve

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

Backsolving means testing the answer choices to see which one works, instead of solving the problem straight through. AMC problems usually have a clean “direct” solution, so backsolving is often a backup plan. Still, it can give you complete confidence in your answer, and sometimes it’s actually the quickest approach.

How backsolving works

Instead of building the answer from scratch, you start by assuming an answer choice is correct and check whether it satisfies the problem’s conditions.

When the answer choices are single numbers (not words, ordered pairs, or expressions), they’re usually listed in increasing order. That ordering lets you test efficiently:

  • Start by testing B.
    • If B works, you’re done.
    • If B is too large, then A must be correct.
    • If B is too small, test D next.
  • Then test D.
    • If D works, you’re done.
    • If D is too large, then C must be correct.
    • If D is too small, then E must be correct.

This “B then D” shortcut relies on the choices being ordered and on the output changing in a predictable direction as the input changes.

When the shortcut doesn’t apply

If the answer choices are ordered pairs, sentences, or variables/expressions, you can’t use the “B then D” logic. In those cases, backsolve by testing the choices that look most plausible first, and keep going until one satisfies the conditions.

Examples

For each of these AMC examples, find the solution by using the backsolving method exclusively.

Example: The question below is from 2018 AMC 10A

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that 65% of the votes cast on his video were like votes. How many votes had been cast on Sangho’s video at that point?
A. 200
B. 300
C. 400
D. 500
E. 600

(spoiler)

Answer: B. 300

Because the answers are single numbers in increasing order, the “B then D” approach fits well.

Test B: assume there were 300 votes.

  • Likes: 65% of 300 is 195.
  • Dislikes: 300−195=105.
  • Score: 195−105=90.

That matches the given score, so B is correct. If the score had come out larger than 90, you’d need fewer total votes; if it had come out smaller than 90, you’d need more total votes.

Example: The question below is from 2019 AMC 10B

All lines with equation ax+by=c such that a,b,c form an arithmetic progression pass through a common point. What are the coordinates of that point?
A. (−1,2)
B. (0,1)
C. (1,−2)
D. (1,0)
E. (1,2)

(spoiler)

Answer: A. (−1,2)

Here the answer choices are ordered pairs, so the “B then D” shortcut doesn’t apply. Instead, test choices directly.

Choices B and D don’t work because they force two of a,b,c to be equal, which prevents a,b,c from forming an arithmetic progression.

Now test A by plugging in x=−1 and y=2:

  • The equation becomes −a+2b=c.
  • This can happen when a,b,c are in arithmetic progression (for example, −1,0,1 or 100,101,102), so A is consistent with the condition.

So the common point is (−1,2).

Common themes

  • If you’re having trouble solving algebraically, try checking answer choices instead.
  • Pay attention to how changing the tested answer changes the output. If your result is too high or too low, that tells you whether to test a smaller or larger choice next. (Sometimes the relationship is inverse, so think about the direction before you commit.)
  • Backsolving isn’t only for finding an answer. It’s also a reliable way to double-check an answer you found by another method.
Key points

Backsolving overview\

  • Test answer choices to see which satisfies the problem
  • Useful as a backup or to confirm answers
  • Sometimes fastest approach, especially with clean answer choices

How backsolving works\

  • Assume an answer choice is correct, check if it fits problem conditions
  • For single-number, increasing order choices:
    • Test B first
      • If correct, done
      • If too large, A is correct
      • If too small, test D
    • Test D next
      • If correct, done
      • If too large, C is correct
      • If too small, E is correct
  • Shortcut relies on ordered choices and predictable output change

When shortcut doesn’t apply\

  • For ordered pairs, sentences, or variable/expression choices:
    • Can’t use “B then D” logic
    • Test most plausible choices first, continue until one fits

Example 1: Single-number choices (AMC 10A 2018)\

  • Apply “B then D” shortcut
  • Test B: 300 votes
    • 65% likes = 195 likes, 105 dislikes
    • Score: 195−105=90
    • Matches, so B is correct

Example 2: Ordered pair choices (AMC 10B 2019)\

  • Shortcut not applicable
  • Test choices directly
  • Plug (−1,2) into equation, check arithmetic progression
    • Satisfies condition, so A is correct

Common themes\

  • Use backsolving if stuck or to check algebraic answers
  • Observe how changing answer choice affects result
    • Adjust next guess based on whether result is too high/low
  • Backsolving is a reliable double-checking tool

More from Practical strategies

  • Estimate
  • Eliminate