Instead of multiplying a number by 25, a student multiplies the number by 52 and obtains a result that is 324 more than the correct answer. What is the original number?

Introduction / Context:
This is a typical error detection and correction problem involving multiplication. A student is supposed to multiply a number by 25 but incorrectly multiplies it by 52. The difference between the wrong result and the correct result is given, and you must determine the original number. These problems test the ability to translate verbal descriptions into algebraic equations and reason about multiplicative errors.

Given Data / Assumptions:
- The original number is some unknown value, say x.
- The correct operation is to multiply x by 25, giving 25x.
- The student mistakenly multiplies x by 52, giving 52x.
- The wrong answer is 324 more than the correct answer, so 52x = 25x + 324.
- We need to find the value of x.

Concept / Approach:
The key idea is to use the difference between the wrong and correct results to form an equation in x. Since both expressions are linear in x, subtracting one from the other will give a simple equation. Solving this linear equation provides the original number. This method avoids guessing and directly captures the size of the error as an extra multiple of x.

Step-by-Step Solution:
Step 1: Let the original number be x.Step 2: The correct result should be 25x.Step 3: Due to the mistake, the student computes 52x instead.Step 4: According to the problem, the wrong result 52x is 324 more than the correct result 25x. So, 52x = 25x + 324.Step 5: Subtract 25x from both sides to isolate the error term: 52x - 25x = 324.Step 6: Simplify the left side: 27x = 324.Step 7: Divide both sides by 27 to solve for x: x = 324 / 27.Step 8: Compute 324 / 27. Since 27 * 12 = 324, we have x = 12.Step 9: Therefore, the original number is 12.

Verification / Alternative check:
Check by computing both the correct and incorrect results for x = 12. The correct result is 25 * 12 = 300. The incorrect result is 52 * 12 = 624. The difference between these two results is 624 - 300 = 324, exactly matching the problem statement. This confirms that the original number is indeed 12 and that the algebraic steps were accurate.

Why Other Options Are Wrong:
If x were 15, the difference between 52 * 15 and 25 * 15 would be (52 - 25) * 15 = 27 * 15 = 405, not 324. For x = 25, the difference would be 27 * 25 = 675, and for x = 32, it would be 27 * 32 = 864. None of these differences match 324. Similarly, x = 18 would give 27 * 18 = 486. Thus, only x = 12 satisfies the condition that the wrong answer is exactly 324 more than the correct answer.

Common Pitfalls:
A common mistake is to think the student divided instead of multiplied and set up the wrong equation. Another error is to forget that the difference between the wrong and correct results is (52x - 25x), leading to miscalculated differences. Always express both the correct and incorrect operations explicitly and then use the given difference to form a linear equation in x.

Final Answer:
The original number is 12.