To find a rational number between the two given rational numbers [tex]\(\frac{-7}{8}\)[/tex] and [tex]\(\frac{11}{9}\)[/tex] using the average method, we will use the following steps:
1. Express the Rational Numbers:
We have two rational numbers [tex]\(\frac{-7}{8}\)[/tex] and [tex]\(\frac{11}{9}\)[/tex].
2. Find a Common Denominator:
In order to add these two rational numbers, we first need to find a common denominator.
The denominators are 8 and 9. The common denominator will be the product of these two, which is [tex]\(8 \times 9 = 72\)[/tex].
3. Convert Each Fraction to Have the Common Denominator:
Convert [tex]\(\frac{-7}{8}\)[/tex] and [tex]\(\frac{11}{9}\)[/tex] so that both fractions have the denominator 72:
- For [tex]\(\frac{-7}{8}\)[/tex]: Multiply both numerator and denominator by 9:
[tex]\[
\frac{-7}{8} = \frac{-7 \times 9}{8 \times 9} = \frac{-63}{72}
\][/tex]
- For [tex]\(\frac{11}{9}\)[/tex]: Multiply both numerator and denominator by 8:
[tex]\[
\frac{11}{9} = \frac{11 \times 8}{9 \times 8} = \frac{88}{72}
\][/tex]
4. Add the Numerators:
Now that the denominators are the same, we can add the numerators:
[tex]\[
\frac{-63 + 88}{72} = \frac{25}{72}
\][/tex]
5. Simplify the Result:
The fraction [tex]\(\frac{25}{72}\)[/tex] cannot be simplified further because 25 and 72 have no common divisors other than 1.
Therefore, the average (and thus a rational number between [tex]\(\frac{-7}{8}\)[/tex] and [tex]\(\frac{11}{9}\)[/tex]) is:
[tex]\[
\boxed{\frac{25}{72}}
\][/tex]
