Mastering Reverse Operations
Reverse operation questions (often called "I think of a number" problems) are the foundation of algebraic thinking. Once you master the art of working backwards, even the most complex problems become simple.
The Rules Every Reverse Operation Question Uses
To find a starting number, you have to "undo" everything that was done to it. Follow these three rules to never get stuck.
🔄 Rule 1: Use the Inverse
Every mathematical operation has an opposite (inverse). To undo a step, perform its opposite.
Inverses: Addition ↔ Subtraction, Multiplication ↔ Division.
⏪ Rule 2: Reverse the Order
You must work backwards. Start with the final answer and undo the very last step first, then the second-to-last, and so on.
Tip: Read the problem from bottom to top to keep your steps in order.
⚖️ Rule 3: Keep the Balance
Think of it like a set of scales. If you undo "multiply by 2" by "dividing by 2", you must apply that division to your current result.
Example: result is 10, last step was +2 → 10 − 2 = 8.
3 Worked Examples
One straightforward, one multi-step, and one classic mistake to avoid.
1. Simple Multi-Step Addition
Question: Cynthia thinks of a number. She adds 2. She then adds 6. The answer is 15. What was her original number?
Worked Method
Step 1: Start with the final result: 15.
Step 2: Undo the last step (+6) by subtracting 6: 15 − 6 = 9.
Step 3: Undo the previous step (+2) by subtracting 2: 9 − 2 = 7.
Answer: 7
Tip: For multiple additions, you can also add them together first (2 + 6 = 8) and subtract the total from the result (15 − 8 = 7).
2. Mixing Operations
Question: Nisha thinks of a number. She triples it. She then adds 4. The answer is 13. What was her original number?
Worked Method
Step 1: Start with the result: 13.
Step 2: Undo the last step (+4) by subtracting 4: 13 − 4 = 9.
Step 3: Undo the "triple" (×3) by dividing by 3: 9 ÷ 3 = 3.
Answer: 3
Tip: Always undo the addition/subtraction *before* undoing the multiplication/division in these types of problems.
3. Order of Operations Trap
Question: Nisha thinks of a number. She subtracts 6. She then multiplies it by 5. The answer is 5. What was her original number?
Worked Method
The Mistake: Many students try to undo the subtraction first (5 + 6 = 11) and then multiply (11 × 5 = 55). This is wrong because it follows the order of the story, not the reverse order of logic.
The Correct Way: Start at the end (5). Undo the last thing that happened (×5) by dividing by 5: 5 ÷ 5 = 1. Now undo the subtraction (−6) by adding 6: 1 + 6 = 7.
Answer: 7
The Rule: When working backwards, you must reverse the order of operations. Whatever happened last in the story is what you undo first.
Before You Buy
Want to check the level and layout first? Download the free 3-question sample. It uses the same question style, printable format, and answer-key approach as the full pack.
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The full pack contains 90 unique reverse operation word problems across 3 test sets. Each question comes with a full step-by-step worked solution.
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