To subtract the mixed numbers [tex]\( 9 \frac{1}{2} \)[/tex] and [tex]\( 11 \frac{6}{7} \)[/tex], follow these steps:
1. Convert the mixed numbers to improper fractions:
- For [tex]\( 9 \frac{1}{2} \)[/tex]:
[tex]\[
9 \frac{1}{2} = \frac{9 \times 2 + 1}{2} = \frac{18 + 1}{2} = \frac{19}{2}
\][/tex]
- For [tex]\( 11 \frac{6}{7} \)[/tex]:
[tex]\[
11 \frac{6}{7} = \frac{11 \times 7 + 6}{7} = \frac{77 + 6}{7} = \frac{83}{7}
\][/tex]
2. Find the common denominator:
The denominators are 2 and 7. The least common multiple (LCM) of these denominators is 14.
3. Adjust the numerators to this common denominator:
- For [tex]\( \frac{19}{2} \)[/tex]:
[tex]\[
\frac{19}{2} = \frac{19 \times (14 / 2)}{14} = \frac{19 \times 7}{14} = \frac{133}{14}
\][/tex]
- For [tex]\( \frac{83}{7} \)[/tex]:
[tex]\[
\frac{83}{7} = \frac{83 \times (14 / 7)}{14} = \frac{83 \times 2}{14} = \frac{166}{14}
\][/tex]
4. Subtract the numerators:
[tex]\[
\frac{133}{14} - \frac{166}{14} = \frac{133 - 166}{14} = \frac{-33}{14}
\][/tex]
5. Simplify the fraction if possible:
The greatest common divisor (GCD) of 33 and 14 is 1, which means the fraction [tex]\(\frac{-33}{14}\)[/tex] is already in its simplest form.
So, the simplest form of [tex]\( 9 \frac{1}{2} - 11 \frac{6}{7} \)[/tex] is:
[tex]\[
\boxed{\frac{-33}{14}}
\][/tex]
