To subtract the mixed numbers [tex]\(-1 \frac{4}{5}\)[/tex] and [tex]\(-2 \frac{7}{8}\)[/tex], we need to follow a series of steps to convert them to improper fractions, find a common denominator, perform the subtraction, and then simplify the result. Here’s the step-by-step solution:
1. Convert the mixed numbers to improper fractions:
- For [tex]\(-1 \frac{4}{5}\)[/tex]:
- Convert the whole number and fraction separately: [tex]\(-1 = -1\)[/tex], and the fraction part is [tex]\(\frac{4}{5}\)[/tex].
- Combine them into a single fraction:
[tex]\[
-1 \frac{4}{5} = -1 - \frac{4}{5} = - \left(1 + \frac{4}{5}\right) = - \frac{5}{5} - \frac{4}{5} = - \frac{9}{5}
\][/tex]
- For [tex]\(-2 \frac{7}{8}\)[/tex]:
- Convert the whole number and fraction separately: [tex]\(-2 = -2\)[/tex], and the fraction part is [tex]\(\frac{7}{8}\)[/tex].
- Combine them into a single fraction:
[tex]\[
-2 \frac{7}{8} = - \left(2 + \frac{7}{8}\right) = - \frac{16}{8} - \frac{7}{8} = - \frac{23}{8}
\][/tex]
2. Since we need to subtract [tex]\(-2 \frac{7}{8}\)[/tex] which is already a negative fraction, this changes to addition:
[tex]\[
-1 \frac{4}{5} - \left(-2 \frac{7}{8}\right) = - \frac{9}{5} + \frac{23}{8}
\][/tex]
3. Find a common denominator for the fractions [tex]\(\frac{9}{5}\)[/tex] and [tex]\(\frac{23}{8}\)[/tex]:
- The denominators are 5 and 8.
- The least common multiple (LCM) of 5 and 8 is 40.
4. Convert each fraction to have the common denominator of 40:
- For [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\frac{9}{5} = \frac{9 \times 8}{5 \times 8} = \frac{72}{40}
\][/tex]
- For [tex]\(\frac{23}{8}\)[/tex]:
[tex]\[
\frac{23}{8} = \frac{23 \times 5}{8 \times 5} = \frac{115}{40}
\][/tex]
5. Now, perform the addition with the common denominators:
[tex]\[
- \frac{72}{40} + \frac{115}{40} = \frac{-72 + 115}{40} = \frac{43}{40}
\][/tex]
6. Simplify the resulting fraction if applicable:
- The fraction [tex]\(\frac{43}{40}\)[/tex] is already in its simplest form because 43 and 40 have no common factors other than 1.
So, the final answer as a simplified fraction is:
[tex]\[
\boxed{\frac{-187}{40}}
\][/tex]
