To determine the sum of [tex]\(-2 \frac{3}{4}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex], 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]\[
-2 \frac{3}{4} = -2 - \frac{3}{4}.
\][/tex]
To do this, write [tex]\(-2\)[/tex] as [tex]\(\frac{-8}{4}\)[/tex]:
[tex]\[
-2 = \frac{-8}{4}.
\][/tex]
Now add these two fractions:
[tex]\[
-2 \frac{3}{4} = \frac{-8}{4} + \frac{-3}{4} = \frac{-8 - 3}{4} = \frac{-11}{4}.
\][/tex]
2. Add the Two Fractions:
- Now, we need to add [tex]\(\frac{-11}{4}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[
\frac{-11}{4} + \frac{5}{9}.
\][/tex]
To add these fractions, find a common denominator. The least common multiple of 4 and 9 is 36.
Convert each fraction to have the 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 add them:
[tex]\[
\frac{-99}{36} + \frac{20}{36} = \frac{-99 + 20}{36} = \frac{-79}{36}.
\][/tex]
3. Simplify the Result (if necessary):
- The fraction [tex]\(\frac{-79}{36}\)[/tex] is already in its simplest form because 79 and 36 share no common divisors other than 1.
The sum of [tex]\(-2 \frac{3}{4}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex] is [tex]\(\frac{-79}{36}\)[/tex]. This is a fraction, so the correct answer to the question is:
The sum is a fraction.
Additionally, the sum [tex]\(\frac{-79}{36}\)[/tex] when converted to decimal form is approximately [tex]\(-2.1944444444444446\)[/tex], which is a non-terminating, repeating decimal. The decimal form repeating part, however, reassures that [tex]\(\frac{-79}{36}\)[/tex] is a rational number.
The sum is rational.
Thus, the correct answer is also that the sum is a terminating and a repeating decimal.
In summary:
- The sum is a fraction.
- The sum is rational.
However, for the answer format given, selecting "The sum is a fraction" would be the simplest and most directly relevant choice.
