Rational Number Between 5/-13 and 7/-9

Step 1: Find the Average of the two fractions
To find the rational number exactly halfway between 5/-13 and 7/-9, we need to calculate the average of these two fractions.

Step 2: Calculate the Average
The formula to find the average of two fractions is (a/b + c/d) / 2 = (ad + bc) / 2bd.
In this case, the fractions are 5/-13 and 7/-9.
So, the average would be ((5*-9) + (7*-13)) / (2*-13*-9).

Step 3: Simplify the Average
Calculating the numerator gives us (-45 - 91) = -136.
Calculating the denominator gives us -234.
Therefore, the average of 5/-13 and 7/-9 is -136 / -234.

Step 4: Simplify the Average to a Single Fraction
To simplify the average fraction -136 / -234, we need to divide both the numerator and denominator by their greatest common divisor.
In this case, the greatest common divisor is 2.
Dividing both -136 and -234 by 2, we get the simplified average fraction as 68 / 117.

Step 5: Rational Number Between the Fractions
Therefore, the rational number exactly halfway between 5/-13 and 7/-9 is 68 / 117. This fraction falls exactly between the given fractions on the number line.
By following these steps, we have successfully found the rational number exactly halfway between 5/-13 and 7/-9, which is 68 / 117.