In an examination X secures $58\%$ marks and Y secures $105$ marks less than X. If the maximum marks were $700$, then what percent of marks did Y secure?

### Concept & Logic When working with percentages of a common base (the maximum marks), you can convert absolute differences directly into percentage differences to save calculation time. ### Step-by-Step Solution **Given:** * Maximum marks = $700$ * X's score = $58\%$ of maximum marks * Y's score = X's score $- 105$ marks **Calculation:** * First, find X's absolute marks: $$X = \frac{58}{100} \times 700$$ $$X = 58 \times 7 = 406$$ * Next, calculate Y's absolute marks: $$Y = 406 - 105 = 301$$ * Finally, convert Y's marks back to a percentage: $$Y_{\%} = \frac{301}{700} \times 100$$ $$Y_{\%} = \frac{301}{7} = 43\%$$ ### Exam Strategy & Shortcut Instead of calculating the absolute marks for both students, convert the difference in marks into a percentage difference! * Difference in marks = $105$ * Percentage equivalent of this difference = $\frac{105}{700} \times 100 = 15\%$ * Y's percentage = X's percentage $- 15\%$ * Y's percentage = $58\% - 15\% = 43\%$ This method completely bypasses the need to multiply $58$ by $7$. ### Common Pitfall A common mistake is calculating X's marks ($406$) and stopping there, or incorrectly subtracting the $105$ marks directly from the $58\%$ figure without converting it. Always ensure you are adding or subtracting like units (marks with marks, percentages with percentages). ### Final Answer **Therefore, the correct answer is 43 percent.**