To determine the nature of the sum of [tex]\(\left(-2 \frac{3}{4}\right)\)[/tex] and [tex]\(\frac{5}{9}\)[/tex], we'll follow these steps:
1. Convert the Mixed Number to an Improper Fraction:
- The mixed number [tex]\(-2 \frac{3}{4}\)[/tex] can be converted to an improper fraction.
- [tex]\(-2\)[/tex] can be written as [tex]\(-2\)[/tex], and [tex]\(\frac{3}{4}\)[/tex] is a fraction. Together, they form:
[tex]\[
-2 \frac{3}{4} = -2 - \frac{3}{4}
\][/tex]
- Converting [tex]\(-2\)[/tex] into a fraction with the denominator 4:
[tex]\[
-2 = -\frac{8}{4}
\][/tex]
- Now, combine the fractions:
[tex]\[
-\frac{8}{4} - \frac{3}{4} = -\frac{11}{4}
\][/tex]
2. Express the Given Fraction
- The given fraction is [tex]\(\frac{5}{9}\)[/tex].
3. Sum the Two Fractions:
- To add the fractions [tex]\(-\frac{11}{4}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex], first find a common denominator.
- The least common multiple (LCM) of 4 and 9 is 36.
- Convert each fraction to have this common denominator:
[tex]\[
-\frac{11}{4} = -\frac{11 \times 9}{4 \times 9} = -\frac{99}{36}
\][/tex]
[tex]\[
\frac{5}{9} = \frac{5 \times 4}{9 \times 4} = \frac{20}{36}
\][/tex]
- Now that both fractions have the same denominator, add them:
[tex]\[
-\frac{99}{36} + \frac{20}{36} = \frac{-99 + 20}{36} = \frac{-79}{36}
\][/tex]
4. Understanding the Sum:
- The sum [tex]\(\frac{-79}{36}\)[/tex] is a fraction.
5. Conclusion:
- A rational number is defined as any number that can be expressed as the quotient [tex]\(\frac{p}{q}\)[/tex] of two integers [tex]\(p\)[/tex] and [tex]\(q\)[/tex], where [tex]\(q \neq 0\)[/tex].
- Since [tex]\(\frac{-79}{36}\)[/tex] is expressed as the quotient of two integers, it is indeed a fraction (and therefore a rational number).
Hence, the sum is a fraction.
