Operation mistake on a fraction: Gopal had to find 7/9 of a fraction x. By mistake, he divided x by 7/9 (i.e., multiplied by 9/7). His answer exceeded the correct answer by 8/21. What is the correct answer (i.e., 7/9 of x)?

Introduction / Context:
This problem is a direct application of turning a word statement into an equation comparing a mistaken operation with the correct one. It highlights the reversal of division by a fraction, and finding the exact amount by which the wrong answer exceeds the right answer.

Given Data / Assumptions:

• Correct result C = (7/9) * x.
• Mistaken result M = x ÷ (7/9) = x * (9/7).
• Excess M − C = 8/21.

Concept / Approach:
Express the excess as a multiple of x, solve for x, and finally compute C = (7/9) * x. Keep all steps exact with fractions to avoid rounding.

Step-by-Step Solution:
M − C = x*(9/7) − x*(7/9) = x * ( (81 − 49)/63 ) = x * (32/63).So x * (32/63) = 8/21.Solve for x: x = (8/21) * (63/32) = 24/32 = 3/4.Correct answer C = (7/9) * (3/4) = 21/36 = 7/12.

Verification / Alternative check:
Compute M: (3/4) * (9/7) = 27/28. Then M − C = 27/28 − 7/12 = (81 − 49)/84 = 32/84 = 8/21, confirming consistency.

Why Other Options Are Wrong:
3/7, 2/21, and 1/3 do not match the derived value; 5/12 is a common but incorrect simplification slipped from intermediate steps.

Common Pitfalls:
Forgetting to invert while dividing by 7/9; arithmetic error in (81 − 49); simplifying 21/36 incorrectly.

Final Answer:
7/12