A student accidentally multiplies a number by 4/7 instead of 7/4. What is the percentage error in the result compared with the correct value?

Introduction / Context:
This problem tests understanding of percentage error when the wrong multiplier is used. It is a classic exam question where the candidate must compare an incorrect result with a correct one and then express the difference as a percentage of the correct value.

Given Data / Assumptions:

• Correct operation: multiply a number by 7/4.
• Incorrect operation: multiply the same number by 4/7.
• We must find the percentage error relative to the correct result.

Concept / Approach:
Let the original number be N. The correct result is N * 7/4. The wrong result is N * 4/7. The ratio of wrong result to correct result simplifies the calculation. Percentage error is (correct - wrong) / correct * 100, assuming the wrong result is smaller than the correct one in this case.

Step-by-Step Solution:
Correct result = N * 7/4.Wrong result = N * 4/7.Ratio wrong / correct = (N * 4/7) / (N * 7/4) = (4/7) * (4/7) = 16/49.So wrong result is 16/49 of correct result.Percentage error = (correct - wrong) / correct * 100.This equals (1 - 16/49) * 100 = (33/49) * 100.Compute 33/49 * 100 ≈ 67.35 percent.

Verification / Alternative check:
Take N = 28 for easy computation. Correct result = 28 * 7/4 = 28 * 1.75 = 49. Wrong result = 28 * 4/7 = 16. Difference = 49 - 16 = 33. Percentage error = 33 / 49 * 100 ≈ 67.35 percent. This matches the algebraic method.

Why Other Options Are Wrong:
Values 206.25 percent and 103.13 percent correspond to other incorrect comparisons such as wrong over correct or using inverse fractions wrongly. The value 33.67 percent is roughly half of the correct error, again from using wrong ratios.

Common Pitfalls:
Many candidates compute the ratio correct / wrong or subtract fractions wrongly, leading to absurdly large or very small percentages. Some also forget that percentage error must be based on the correct value as the denominator.

Final Answer:
The percentage error in the calculation is approximately 67.35 percent.