A student multiplied a number by $ \frac{3}{5} $ instead of $ \frac{5}{3} $. What is the percentage error in the calculation?

### Concept & Formula This problem requires finding the percentage error caused by using an incorrect multiplier. The formula for percentage error is: $$ \text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100 $$ Alternatively, it can be written as: $$ \text{Percentage Error} = \left( \frac{\text{Error}}{\text{True Value}} \right) \times 100 $$ ### Step-by-step Solution **Given:** The correct multiplier should have been $ \frac{5}{3} $. The incorrect multiplier used was $ \frac{3}{5} $. Let the number to be multiplied be $x$. The true (correct) value = $ \frac{5}{3}x $ The measured (incorrect) value = $ \frac{3}{5}x $ Calculate the error (difference between the true value and the measured value): $$ \text{Error} = \frac{5}{3}x - \frac{3}{5}x $$ Find a common denominator to subtract the fractions: $$ \text{Error} = \frac{25x - 9x}{15} = \frac{16x}{15} $$ Now, calculate the percentage error using the formula: $$ \text{Percentage Error} = \frac{\frac{16x}{15}}{\frac{5}{3}x} \times 100 $$ The $x$ variables cancel out. Simplify the complex fraction: $$ \text{Percentage Error} = \left( \frac{16}{15} \times \frac{3}{5} \right) \times 100 $$ $$ \text{Percentage Error} = \frac{16}{25} \times 100 $$ Calculate the final percentage: $$ 16 \times 4 = 64\% $$ ### Exam Strategy & Shortcut Use the **LCM Assumption Method** to avoid variables entirely. Look at the denominators of the two fractions: 3 and 5. Assume the original number is their Least Common Multiple (LCM), which is 15. Correct calculation: $ 15 \times \frac{5}{3} = 25 $. Incorrect calculation: $ 15 \times \frac{3}{5} = 9 $. Error: $ 25 - 9 = 16 $. Percentage error: $ \frac{16}{25} \times 100 = 64\% $. This method is significantly faster and less prone to algebraic mistakes. ### Common Pitfall A very common mistake is calculating the percentage error against the *incorrect* value (the measured value) instead of the *true* value. If you divide by 9 instead of 25, you get $ \frac{16}{9} \times 100 \approx 177.7\% $, which is totally wrong. Always use the intended correct result as the base denominator. ### Final Answer **Therefore, the correct answer is 64%.**