Mastering Reverse Operations Practice Test Worksheets
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.
For students who can follow a calculation forward, but lose marks when they have to reverse the operations in the right order.
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What Do Reverse Operations Questions Look Like?
Reverse operation questions ask you to work backwards from a final answer to find an unknown starting number. These problems test whether you can identify inverse operations and apply them in reverse order without losing track. Here is a worked example.
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: C. 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).
Rules to Help You Solve Reverse Operations Questions
To solve reverse operation questions like the example above, you have to "undo" each calculation in reverse chronological order. Keep these key rules in mind:
🔄 Rule 1: Use the Inverse
For the operations used in these puzzles, undo addition with subtraction and subtraction with addition. Undo multiplication with division and division with multiplication, using the same non-zero number.
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: Apply each inverse to the current result
After undoing one step, use the number you have just found as the starting point for undoing the next step.
Example: If the original steps were +2 then ×3 and the final result is 18, first calculate 18 ÷ 3 = 6, then use that 6: 6 − 2 = 4.
Two More Worked Examples
Here are two more examples showing how to handle mixed operations and avoid common reverse-order traps.
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: B. 3
Tip: Undo the last stated operation first, whatever its type. Here, adding 4 happened last, so subtract 4 before dividing by 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: Using the correct inverses in the original order gives (5 + 6) ÷ 5 = 2.2. This is wrong because it undoes the subtraction before the multiplication, even though the multiplication happened last.
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: B. 7
The Rule: When working backwards, you must reverse the order of operations. Whatever happened last in the story is what you undo first.
