A student was supposed to multiply a number by 13/6 but instead multiplied it by 6/13. What is the percentage error in the result obtained, compared to the correct value?

Introduction / Context:
This question tests understanding of percentage error when the wrong multiplicative factor is used in a calculation. Instead of multiplying by a fraction greater than one, the student uses the reciprocal, resulting in a significantly smaller value. The task is to measure how large this error is in percentage terms relative to the correct result.

Given Data / Assumptions:

• The correct operation is to multiply a number N by 13/6.
• The student instead multiplies N by 6/13.
• We must find the percentage error in the student result compared to the correct result.
• We assume N is positive so that the direction of error is clear.

Concept / Approach:
Let N be the original number. Correct result = N * (13/6). Obtained result = N * (6/13). Percentage error is usually measured as: Error percent = [(Correct value - Obtained value) / Correct value] * 100. Since the student uses a much smaller fraction, the obtained value is less than the correct value, giving a positive error percentage.

Step-by-Step Solution:
Step 1: Correct result = N * (13/6). Step 2: Obtained result = N * (6/13). Step 3: Compute the ratio of obtained to correct result: (Obtained / Correct) = [N * (6/13)] / [N * (13/6)]. Step 4: Cancel N to get (6/13) / (13/6) = (6/13) * (6/13) = 36 / 169. Step 5: Thus, obtained value = (36/169) * correct value. Step 6: Error fraction = 1 - 36/169 = (169 - 36) / 169 = 133 / 169. Step 7: Error percent = (133 / 169) * 100. Step 8: Compute 133 / 169 approximately = 0.787, so error percent ≈ 78.7 percent.

Verification / Alternative check:
Take N = 1 for a simple numerical check. Correct result = 13/6 ≈ 2.1667. Obtained result = 6/13 ≈ 0.4615. Difference = 2.1667 - 0.4615 ≈ 1.7052. Error percent = (1.7052 / 2.1667) * 100 ≈ 78.7 percent. This numerical verification matches the algebraic computation.

Why Other Options Are Wrong:
39.35 percent: This is about half of the correct error and arises if the fraction is mishandled or if the wrong reference value is used. 184.72 percent and 369.44 percent: These drastically overstate the error and may come from reversing the ratio or using obtained instead of correct value as the denominator. 21.3 percent: This is far too small and not consistent with the large difference between 13/6 and 6/13.

Common Pitfalls:
A common mistake is to compute error relative to the obtained value instead of the correct value. Others may simply take the numerical difference between 13/6 and 6/13 as a percentage without proper scaling. It is also easy to forget that when a fraction is inverted, its value changes drastically if it is not equal to one. Always form the ratio of obtained to correct, simplify it, and then use the standard percentage error formula relative to the correct value.

Final Answer:
The percentage error in the student's result is approximately 78.7 percent less than the correct value.