Firstly, we can change the mixed numbers to improper fractions to make this easier.
4 [/tex]\frac{1}{8}[/tex] becomes [tex]\frac{33}{8}[/tex]
-13 [/tex]\frac{5}{9}[/tex] becomes [/tex]\{122}{9}[/tex]
Next, we can make the denominators the same by finding the LCD (lowest common multiple/denominator)
The LCD of 8 and 9 is 72
For [tex]\frac{33}{8}[/tex], we need to multiply both top and bottom by 9 to make the denominator 72.
It becomes [tex]\frac{297}{72}[/tex]
For [/tex]\{122}{9}[/tex], we need to multiply both top and bottom by 8 to make the denominator 72.
It becomes [tex]\frac{976}{72}[/tex]
Now, since adding a negative number is the same as subtracting that number, we do:
[tex]\frac{297}{72}[/tex] - [tex]\frac{976}{72}[/tex], which is [tex]\frac{-679}{72}[/tex]
Now we just simplify the fraction.
I personally divide the fraction to find its whole number first.
[tex]\frac{-679}{72}[/tex] is -9 and a remainder.
To find the remainder, multiply the denominator (72) by -9, which is -648.
Now, subtract -648 from -679 (which is basically -679+648)
Now our remainder is -31, which is our numerator.
[tex]\frac{-31}{72}[/tex] is now the remainder in fraction form.
[tex]\frac{-31}{72}[/tex] is not exactly [tex]\frac{1}{2}[/tex], but it's close enough. Since the other answer choices don't make sense, -9 [tex]\frac{-1}{2}[/tex] is the answer.
