To add the mixed numbers [tex]\(-3 \frac{1}{6}\)[/tex] and [tex]\(5 \frac{3}{4}\)[/tex], we need to follow a series of steps to convert these mixed numbers to improper fractions, find a common denominator, add the fractions, and then convert back to a mixed number. Here is a step-by-step solution:
1. Convert the mixed numbers to improper fractions:
- [tex]\(-3 \frac{1}{6}\)[/tex]:
[tex]\[
-3 \frac{1}{6} = \left(-3\right) + \frac{1}{6} = -\frac{18}{6} + \frac{1}{6} = -\frac{19}{6}
\][/tex]
- [tex]\(5 \frac{3}{4}\)[/tex]:
[tex]\[
5 \frac{3}{4} = 5 + \frac{3}{4} = \frac{20}{4} + \frac{3}{4} = \frac{23}{4}
\][/tex]
2. Find the least common denominator (LCD) of the fractions [tex]\(\frac{-19}{6}\)[/tex] and [tex]\(\frac{23}{4}\)[/tex]:
- The denominators are 6 and 4. The least common denominator is 12.
3. Convert the fractions to have the same denominator:
- For [tex]\(\frac{-19}{6}\)[/tex]:
[tex]\[
\frac{-19}{6} = \frac{-19 \cdot 2}{6 \cdot 2} = \frac{-38}{12}
\][/tex]
- For [tex]\(\frac{23}{4}\)[/tex]:
[tex]\[
\frac{23}{4} = \frac{23 \cdot 3}{4 \cdot 3} = \frac{69}{12}
\][/tex]
4. Add the fractions:
- [tex]\(\frac{-38}{12} + \frac{69}{12}\)[/tex]:
[tex]\[
\frac{-38 + 69}{12} = \frac{31}{12}
\][/tex]
5. Convert the improper fraction back to a mixed number:
- [tex]\(\frac{31}{12}\)[/tex]:
- Divide 31 by 12:
[tex]\[
31 \div 12 = 2 \text{ remainder } 7
\][/tex]
- So, [tex]\(\frac{31}{12} = 2 \frac{7}{12}\)[/tex]
Therefore, the answer to [tex]\(-3 \frac{1}{6} + 5 \frac{3}{4}\)[/tex] is:
[tex]\[
-3 \frac{1}{6} + 5 \frac{3}{4} = 2 \frac{7}{12}
\][/tex]
