A student multiplies a number by 3/10 instead of multiplying it by 10/3.\nWhat is the percentage error in the student's result compared to the correct result?

Introduction / Context:
This problem assesses understanding of percentage error when an incorrect multiplier is used instead of the correct one. It is a classic example from quantitative aptitude where a student uses the reciprocal of the intended factor, leading to a significant mistake. Calculating the relative error as a percentage is an essential skill in many exam and practical contexts.

Given Data / Assumptions:

• Let the original number be N.
• The correct operation should be multiplication by 10/3.
• The student mistakenly multiplies the number by 3/10 instead.
• We must find the percentage error of the student's result relative to the correct result.

Concept / Approach:
First, compute the correct result, which is N * (10/3). Then compute the wrong result, which is N * (3/10). The absolute error is the difference between the correct result and the wrong result. The percentage error is given by (absolute error / correct result) * 100. We can treat N as a symbolic variable and simplify the ratio using algebra, so that N cancels out and we get a simple percentage.

Step-by-Step Solution:
Step 1: Correct result = N * (10/3).Step 2: Wrong result = N * (3/10).Step 3: Absolute error = correct result - wrong result = N * (10/3) - N * (3/10).Step 4: Factor out N: error = N * (10/3 - 3/10).Step 5: Compute the bracket: 10/3 - 3/10 = (100/30 - 9/30) = 91/30.Step 6: So error = N * (91/30).Step 7: Percentage error = (error / correct result) * 100.Step 8: Substitute: percentage error = [N * (91/30)] / [N * (10/3)] * 100.Step 9: Simplify the fraction: (91/30) / (10/3) = (91/30) * (3/10) = 273 / 300 = 0.91.Step 10: Therefore percentage error = 0.91 * 100 = 91 percent.

Verification / Alternative check:
We can also pick a simple value for N, such as N = 10. The correct result is 10 * (10/3) = 100/3 ≈ 33.33. The wrong result is 10 * (3/10) = 3. The error is 33.33 - 3 = 30.33. The percentage error is (30.33 / 33.33) * 100 ≈ 91 percent. This numeric check aligns with the algebraic calculation and confirms the result.

Why Other Options Are Wrong:
1011.11 percent and 505.56 percent are extremely large and would imply a wrong result many times larger than the correct value, which is not the case here.
45.5 percent is roughly half of the actual error and might come from an incorrect partial simplification of the fractions.
75 percent is another distractor that is not supported by either algebraic or numeric checks.

Common Pitfalls:
A common mistake is to compare the difference between 10/3 and 3/10 directly as a percentage of the original number rather than as a percentage of the correct result. Another frequent error is performing fraction arithmetic incorrectly or forgetting that N cancels out when computing relative error. To avoid such issues, always write down the formula for percentage error clearly, simplify the algebra carefully, and, if needed, verify the answer with a simple numerical example.

Final Answer:
The percentage error in the student's calculation is 91 percent.