The question asks for a rational number that lies between two given numbers in each part. Let's solve each one step-by-step:

a) Between $-5$ and $-7$

To find a rational number between $-5$ and $-7$, we look for a number greater than $-7$ but less than $-5$. A simple choice is the average:

$\text{Average} = \frac{-5 + (-7)}{2} = \frac{-12}{2} = -6$

So, a rational number between $-5$ and $-7$ is $-6$.

b) Between $\frac{1}{2}$ and $\frac{2}{3}$

To find a rational number between $\frac{1}{2}$ and $\frac{2}{3}$, we can also take their average:

$\text{Average} = \frac{\frac{1}{2} + \frac{2}{3}}{2} = \frac{\frac{3}{6} + \frac{4}{6}}{2} = \frac{\frac{7}{6}}{2} = \frac{7}{12}$

So, a rational number between $\frac{1}{2}$ and $\frac{2}{3}$ is $\frac{7}{12}$.

c) Between $-0.5$ and $-\frac{1}{4}$

Here, $-0.5$ is the same as $-\frac{1}{2}$. We need a number between $-\frac{1}{2}$ and $-\frac{1}{4}$. Again, we can use the average:

$\text{Average} = \frac{-\frac{1}{2} + (-\frac{1}{4})}{2} = \frac{-\frac{2}{4} + (-\frac{1}{4})}{2} = \frac{-\frac{3}{4}}{2} = -\frac{3}{8}$

So, a rational number between $-0.5$ and $-\frac{1}{4}$ is $-\frac{3}{8}$.

Summary of Answers:

a) $-6$
b) $\frac{7}{12}$
c) $-\frac{3}{8}$

Would you like more details on any of these solutions or have any further questions?

Related Questions:

1. How do you find the midpoint between two fractions?
2. What are other rational numbers between $-5$ and $-7$?
3. Can there be multiple rational numbers between two given numbers? If so, how can we find them?
4. How do we compare two rational numbers?
5. What are the methods to convert decimals to fractions?

Tip:

When finding a rational number between two others, averaging them is a quick and effective method!