**Three Rational Numbers Between -5/6 and 5/8**

To find three rational numbers between -5/6 and 5/8, we can employ the method of averaging. By taking the average of two given rational numbers, we can obtain a rational number that lies between them. We will use this method to find the three rational numbers.

**1. Finding the First Rational Number**

To find the first rational number between -5/6 and 5/8, we can take the average of these two numbers:

Average = (-5/6 + 5/8) / 2

To simplify the expression, we need to find a common denominator:

-5/6 = -20/24
5/8 = 15/24

Now, we can calculate the average:

Average = (-20/24 + 15/24) / 2 = -5/24

Therefore, the first rational number between -5/6 and 5/8 is -5/24.

**2. Finding the Second Rational Number**

To find the second rational number, we will take the average of -5/6 and -5/24. Let's calculate:

Average = (-5/6 + -5/24) / 2

Again, we need to find a common denominator:

-5/6 = -20/24
-5/24 = -5/24

Now, we can calculate the average:

Average = (-20/24 + -5/24) / 2 = -25/48

Hence, the second rational number between -5/6 and 5/8 is -25/48.

**3. Finding the Third Rational Number**

To find the third rational number, we will take the average of -5/24 and 5/8:

Average = (-5/24 + 5/8) / 2

Using a common denominator:

-5/24 = -5/24
5/8 = 15/24

Now, we can calculate the average:

Average = (-5/24 + 15/24) / 2 = 5/24

Therefore, the third rational number between -5/6 and 5/8 is 5/24.

In conclusion, the three rational numbers between -5/6 and 5/8 are -5/24, -25/48, and 5/24. These numbers are obtained by averaging the given rational numbers.