1. **Stating the problem:** Convert the repeating decimal $0.7\overline{5241}$ (where "5241" repeats indefinitely) into a fraction. 2. **Represent the decimal as a variable:** Let $x = 0.7\overline{5241}$. 3. **Separate the decimal into non-repeating and repeating parts:** The non-repeating part after the decimal is 7 in the tenths place. The repeating part is "5241" starting from the hundredths place. 4. **Express $x$ in terms of non-repeating and repeating parts:** $$x = 0.7 + 0.0\overline{5241}$$ 5. **Convert the non-repeating decimal:** $$0.7 = \frac{7}{10}$$ 6. **Let $y = 0.0\overline{5241}$ and convert it:** Multiply $y$ by $10^4 = 10000$ because "5241" has 4 digits: $$10000y = 5.2415241...$$ Also note: $$y = 0.052415241...$$ 7. **Subtract to eliminate repeating part:** $$10000y - y = 5.2415241... - 0.052415241...$$ $$9999y = 5.1891099...$$ Correcting this approach to directly convert $0.0\overline{5241}$: Since $y$ starts at the hundredths place, multiply $y$ by $10^4 = 10000$: $$10000y = 524.15241524...$$ And to isolate the repeat: $$10000y - y = 524.15241524... - 0.05241524... = 524.1$$ So: $$9999y = 524.1$$ 8. **Convert to fraction:** $$y = \frac{524.1}{9999} = \frac{5241}{99990}$$ (multiplied numerator and denominator by 10 to clear decimal) 9. **Combine parts:** $$x = \frac{7}{10} + \frac{5241}{99990}$$ Find common denominator $99990$: $$x = \frac{7 \times 9999}{99990} + \frac{5241}{99990} = \frac{69993 + 5241}{99990} = \frac{75234}{99990}$$ 10. **Simplify the fraction:** Both numerator and denominator are divisible by 6: $$\frac{75234}{99990} = \frac{12539}{16665}$$ No further simplification possible. **Final Answer:** $$0.7\overline{5241} = \frac{12539}{16665}$$