A student was supposed to multiply a number by 5/3, but by mistake the student multiplied the number by 3/5 instead. What is the percentage error in the result of this incorrect calculation, compared to the correct value?

Introduction / Context:
This question involves percentage error resulting from an incorrect multiplication. The student was expected to multiply a number by a certain fraction but instead used the reciprocal or a different fraction. We must compare the incorrect result to the correct result and express the difference as a percentage of the correct value. Such questions help build understanding of relative change and percentage error, which are useful in quantitative aptitude and data interpretation topics.

Given Data / Assumptions:
- Let the original number be x.- Correct operation: multiply x by 5/3.- Incorrect operation: multiply x by 3/5.- We need the percentage error in the result due to using 3/5 instead of 5/3.- Percentage error is usually measured as (correct value - incorrect value) / correct value * 100%, taking magnitude as required.

Concept / Approach:
The idea is to compute the correct result and the incorrect result in terms of x, then measure how much the incorrect result deviates from the correct result as a fraction of the correct result. The percentage error is this fraction multiplied by 100. Because the incorrect multiplier 3/5 is smaller than 1 and the correct multiplier 5/3 is larger than 1, the incorrect result will be much smaller than the correct result, leading to a large percentage error.

Step-by-Step Solution:
Step 1: Let the number be x.Step 2: Correct result = x * (5/3) = (5x / 3).Step 3: Incorrect result = x * (3/5) = (3x / 5).Step 4: Difference between correct and incorrect results:Difference = correct - incorrect = (5x / 3) - (3x / 5).Step 5: Find a common denominator of 15.(5x / 3) = (25x / 15) and (3x / 5) = (9x / 15).Step 6: Difference = (25x / 15) - (9x / 15) = (16x / 15).Step 7: Percentage error = (Difference / Correct) * 100%.Step 8: Correct value = 5x / 3.Step 9: Percentage error = ( (16x / 15) / (5x / 3) ) * 100%.Step 10: Simplify the fraction:(16x / 15) * (3 / 5x) = (16 * 3) / (15 * 5) = 48 / 75.Step 11: 48 / 75 reduces by dividing by 3: 16 / 25.Step 12: Convert 16 / 25 to a percentage: (16 / 25) * 100% = 64%.Therefore, the percentage error in the result is 64%.

Verification / Alternative check:
Take a specific value for x to see the effect.Let x = 15.Correct result = 15 * (5/3) = 15 * 5 / 3 = 25.Incorrect result = 15 * (3/5) = 15 * 3 / 5 = 9.Difference = 25 - 9 = 16.Percentage error = (16 / 25) * 100% = 64%.This matches our algebraic calculation.

Why Other Options Are Wrong:
- 54 %: This arises if one miscalculates the fractions or performs incorrect simplification.- 74 %: This would require a difference of 74/100 of the correct result, which is not supported by the derived ratio.- 84 %: This is far too large and would imply the incorrect result is extremely small compared to the correct one beyond what 3/5 vs 5/3 yields.

Common Pitfalls:
- Using the incorrect number as denominator when calculating percentage error instead of using the correct result as the base.- Confusing 3/5 and 5/3 numerically and not recognizing how far apart the resulting values are.- Failing to simplify the fractions properly and getting a wrong ratio.- Interpreting error as 100% - (incorrect / correct * 100) without understanding the underlying formula.

Final Answer:
The percentage error in the calculation is 64%.