Finding Rational Number Between $\frac{1}{4}$ and $\frac{1}{2}$

The goal is to identify the rational number from the given options that falls strictly between $\frac{1}{4}$ and $\frac{1}{2}$.

Comparing Fractions

To compare the fractions effectively, we first find a common denominator for the boundary fractions $\frac{1}{4}$ and $\frac{1}{2}$. The least common multiple of 4 and 2 is 4. We can express both fractions with a denominator of 8 for easier comparison with the options.

• $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
• $\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

Now, we need to find the option that is greater than $\frac{2}{8}$ and less than $\frac{4}{8}$.

Evaluating Options

Let's examine each option:

1. Option 1: $\frac{1}{8}$
   Compare $\frac{1}{8}$ with $\frac{2}{8}$. Since $1 < 2$, we have $\frac{1}{8} < \frac{2}{8}$. This number is not between $\frac{1}{4}$ and $\frac{1}{2}$.
2. Option 2: $\frac{3}{5}$
   Compare $\frac{3}{5}$ with $\frac{1}{2}$. We can cross-multiply: $3 \times 2 = 6$ and $5 \times 1 = 5$. Since $6 > 5$, we have $\frac{3}{5} > \frac{1}{2}$. This number is not between $\frac{1}{4}$ and $\frac{1}{2}$.
3. Option 3: $\frac{3}{8}$
   Compare $\frac{3}{8}$ with $\frac{2}{8}$ and $\frac{4}{8}$. Since $2 < 3 < 4$, we have $\frac{2}{8} < \frac{3}{8} < \frac{4}{8}$. Therefore, $\frac{3}{8}$ lies between $\frac{1}{4}$ and $\frac{1}{2}$.
4. Option 4: $\frac{1}{6}$
   Compare $\frac{1}{6}$ with $\frac{1}{4}$. Find a common denominator, like 12. $\frac{1}{6} = \frac{2}{12}$ and $\frac{1}{4} = \frac{3}{12}$. Since $2 < 3$, we have $\frac{1}{6} < \frac{1}{4}$. This number is not between $\frac{1}{4}$ and $\frac{1}{2}$.

Conclusion

Based on the comparison, the only rational number that lies between $\frac{1}{4}$ and $\frac{1}{2}$ is $\frac{3}{8}$.