Three rational numbers between $\frac{1}{5}$ and $\frac{1}{6}$ are

### Concept & Logic To easily find or verify rational numbers between two given fractions, convert the boundaries and the given options into their decimal forms. This bypasses the need for complex fraction homogenization. $$ \text{Fraction} \rightarrow \text{Decimal} $$ ### Step-by-Step Solution * Convert the given range boundaries to decimals: * $\frac{1}{5} = 0.20$ * $\frac{1}{6} \approx 0.1666...$ * The required numbers must fall strictly between these two values (i.e., $> 0.1666...$ and $< 0.20$). * Evaluate the distinct fractions present across all the options: * $\frac{4}{25} = \frac{16}{100} = 0.16$ * $\frac{9}{50} = \frac{18}{100} = 0.18$ * $\frac{17}{100} = 0.17$ * $\frac{167}{1000} = 0.167$ * Check the conditions for each value: * $0.16$ is less than $0.1666...$, so it is out of bounds. * $0.18, 0.17,$ and $0.167$ all correctly fall between $0.1666...$ and $0.20$. * Since options (a), (b), and (c) contain $\frac{4}{25}$, they are incorrect. Option (d) contains only valid numbers. ### Exam Strategy & Shortcut **Option Elimination via Decimals:** Instead of finding a common denominator for 5, 6, 25, 50, 100, and 1000 (which would be 3000), use decimals immediately. The moment you calculate $\frac{4}{25} = 0.16$, you can instantly eliminate options (a), (b), and (c) because $0.16 < 0.166...$. This leaves only (d) as the correct answer in seconds. ### Common Pitfall Students often waste immense amounts of time trying to find the LCM for all denominators to compare numerators. Another common mistake is assuming $0.16$ is greater than or equal to $0.1666...$ by misinterpreting the repeating digits. ### Final Answer **Therefore, the correct answer is $\frac{9}{50}, \frac{17}{100}, \frac{167}{1000}$.**