A number X is mistakenly divided by 10 instead of being multiplied by 10. What is the percentage error in the result, measured relative to the correct value?

Introduction / Context:
This question involves the concept of percentage error due to a calculation mistake. Instead of multiplying a number by 10, it is divided by 10. We need to compare the incorrect result with the correct result and find how large the error is in percentage terms relative to the correct value. Such questions test awareness of relative error and the difference between very large and very small results.

Given Data / Assumptions:

• The correct operation should be multiplication of X by 10.
• The mistaken operation is division of X by 10.
• Percentage error is measured as the difference between correct and obtained results, relative to the correct result, expressed in percent.
• We consider percentage error in magnitude, so we focus on the size of the error.

Concept / Approach:
Let the correct value be V_correct and the obtained or incorrect value be V_wrong. Percentage error is usually defined as (V_correct - V_wrong) / V_correct * 100 in magnitude. Here, V_correct equals 10X and V_wrong equals X / 10. By substituting these expressions and simplifying, we find the ratio of the error to the correct value, and from that the percentage error. The sign of the error shows whether the value is under or over the true value, but many exam questions focus on the absolute percentage error.

Step-by-Step Solution:
Step 1: Correct value: V_correct = 10X.Step 2: Wrong value due to division: V_wrong = X / 10.Step 3: Error in value = V_correct - V_wrong = 10X - (X / 10).Step 4: Simplify 10X - X / 10 by writing 10X as 100X / 10, giving (100X / 10) - (X / 10) = 99X / 10.Step 5: Percentage error (in magnitude) = (error / correct value) * 100 = (99X / 10) / (10X) * 100.Step 6: Simplify the fraction: (99X / 10) / (10X) = 99 / 100 = 0.99.Step 7: Multiply by 100 to get percentage error: 0.99 * 100 = 99 percent.

Verification / Alternative check:
Use a simple value for X, such as X = 1. The correct value should be 10 * 1 = 10. The wrong value is 1 / 10 = 0.1. The error is 10 - 0.1 = 9.9. Now, percentage error = 9.9 / 10 * 100 = 99 percent. This confirms the algebraic work. If we chose X = 2, we would get correct value 20 and wrong value 0.2, giving error 19.8 and percentage error 19.8 / 20 * 100 = 99 percent again, as expected.

Why Other Options Are Wrong:
The values plus or minus 100 percent would indicate that the wrong value is zero or twice the correct value, which is not the case. A zero percent error would mean the wrong and correct values are identical, which is clearly not true. A negative 99 percent would indicate direction but exam questions on percentage error usually use the magnitude. The best match with the usual definition is +99 percent, representing a 99 percent error in magnitude.

Common Pitfalls:
Some learners confuse relative error with absolute error and forget to divide by the correct value. Others may miscalculate 10X minus X / 10 or mishandle the division of fractions, leading to incorrect numerical answers like 90 percent or 100 percent. It is also easy to think that since division by 10 is the opposite of multiplication by 10, the error should be 100 percent, which is not correct. Careful simplification shows that the error is slightly less than the full value, at 99 percent.

Final Answer:
The result is off by 99% relative to the correct value, so the correct option is +99%.