Error in operating on a fraction: A boy was asked to find 13/14 of a certain fraction. Instead, he mistakenly divided the fraction by 13/14 and obtained a result that exceeded the correct value by 3/65. What is the correct original fraction?

Introduction / Context:
This question checks understanding of operations with fractions and how a procedural mistake affects the result. The intended operation was to multiply the unknown fraction by 13/14, but the boy instead divided by 13/14, which is equivalent to multiplying by 14/13. We are told the error increased the answer by 3/65 over the correct value. We must translate this into an equation and solve for the original fraction.

Given Data / Assumptions:

• The unknown fraction is x.
• Correct result should be (13/14) * x.
• Mistaken result is x ÷ (13/14) = (14/13) * x.
• Excess due to the mistake is 3/65 over the correct result.

Concept / Approach:
Express the stated excess as the difference between the mistaken result and the correct result. This gives a linear equation in x. Solving that equation yields the value of x directly. This is standard equation setup using fraction arithmetic and difference-of-results reasoning.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 12/65: Does not satisfy the formed equation; difference is not 3/65.
• 13/45: This equals the correct result, not the original fraction.
• 2/7: Produces a different excess; arithmetic check fails.

Common Pitfalls:

• Confusing multiplying by 13/14 with dividing by 13/14.
• Not aligning denominators correctly when subtracting the two expressions.

Final Answer: