A candidate was asked to find $\frac{7}{8}$ of a positive number. He found $\frac{7}{18}$ of the same by mistake. If his answer was 770 less than the correct one, then the original given number was

### Concept & Equation This is an algebraic fraction problem. The core logic is to set up a linear equation representing the difference between the correct mathematical operation and the mistaken operation. $$\text{Correct Calculation} - \text{Mistaken Calculation} = \text{Difference}$$ ### Step-by-Step Solution * **Define the variable:** Let the original positive number be $x$. * **Formulate the expressions:** The correct expected result: $\frac{7}{8}x$ The incorrect mistaken result: $\frac{7}{18}x$ * **Set up the equation based on the given condition:** The mistaken answer is 770 less than the correct answer. $$\frac{7}{8}x - \frac{7}{18}x = 770$$ * **Solve for x:** Find the Least Common Multiple (LCM) of 8 and 18, which is 72. Convert fractions to a common denominator: $$\frac{63}{72}x - \frac{28}{72}x = 770$$ $$\frac{35}{72}x = 770$$ $$x = \frac{770 \times 72}{35}$$ Divide 770 by 35 first: $\frac{770}{35} = 22$. $$x = 22 \times 72 = 1584$$ * **Check the options:** The calculated original number is 1584. Looking at the options: (a) 1260, (b) 1548, (c) 6160. 1584 is not listed among the numerical choices. ### Exam Strategy & Shortcut Factor out the common numerator (7) and the variable ($x$) to speed up fraction subtraction: $7x \times (\frac{1}{8} - \frac{1}{18}) = 770$ $x \times (\frac{18 - 8}{144}) = 110$ $x \times (\frac{10}{144}) = 110$ $x = 11 \times 144 = 1584$. This method handles smaller numbers and avoids finding the LCM of the denominators immediately. ### Common Pitfall A major trap here is option (b) 1548. It is extremely close visually to the correct answer (1584) due to a simple transposition of the middle digits (4 and 8). Careless reading under time pressure causes many students to select (b) instead of "None of these". ### Final Answer **Therefore, the correct answer is None of these.**