1
Substitution & Elimination
Plug in each option and rule out wrong choices
Substitute the answer choices into the problem one by one to verify which one satisfies all conditions.
When: Questions ask "which is correct" or when solving the equation directly is difficult.
1Read the question and all options carefully.
2Start with the simplest option.
3Plug it in and check each condition.
4Pick the option that works.
Example: A two-digit number has a tens digit twice its units digit. Options: A) 21, B) 42, C) 63, D) 84. Checking B: tens digit 4 = 2 × units digit 2 ✓/div>
2
Drawing & Visualization
Use diagrams to turn abstract problems into pictures
Draw a figure or diagram to represent the unknowns and relationships in the problem.
When: Geometry, motion, or set problems that need visual reasoning.
1Identify the spatial or logical relationship.
2Sketch a diagram or line segment.
3Mark all known quantities.
4Use the diagram to find the answer.
Example: Two cars start 120 km apart and drive toward each other. Draw a line segment, mark the starting points, directions, and speeds to see when they meet.
3
Systematic Enumeration
List all possible cases in order
List every possible case in a fixed order so nothing is missed or counted twice.
When: Permutation, combination, counting, or pattern-finding problems.
1Decide the range and order of enumeration.
2List cases from smallest to largest.
3Use a table or tree diagram to record results.
4Count and check for omissions.
Example: How many three-digit numbers can be formed from 1, 2, 3 without repetition? 123, 132, 213, 231, 312, 321 —total 6.
4
Working Backwards
Start from the final state and reverse each step
Begin with the known final result and undo each operation step by step until you reach the initial state.
When: The final state is known but the initial state is unknown, or forward reasoning is difficult.
1Identify the final state.
2Figure out the inverse of each step.
3Work from the end to the beginning.
4Verify by running the result forward.
Example: A bucket of water: after pouring out half and adding 3 L, there are 10 L left. Reverse: 10 −3 = 7, then 7 × 2 = 14 L originally.
5
Special Values
Pick simple numbers to test or eliminate choices
Choose simple values such as 0, 1, or 2 to substitute and quickly narrow down or identify the answer.
When: Problems contain variables or unknowns, or when you need a quick check among multiple choices.
1Choose a simple special value (e.g., 0, 1, 2).
2Substitute and compute.
3Compare the result with the choices.
4If needed, try another value to confirm.
Example: f(x) = ax + b passes through (1, 3) and (2, 5). Find f(0). At x = 1: a + b = 3; at x = 2: 2a + b = 5. Solving gives a = 2, b = 1, so f(0) = 1.
