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How do you know that the sum of [tex]\left(-2 \frac{3}{4}\right)[/tex] and [tex]\frac{5}{9}[/tex] is rational?

A. The sum is a terminating and a repeating decimal.
B. The sum is a non-terminating and a non-repeating decimal.
C. The sum is a fraction.
D. The sum is an integer.

Sagot :

GinnyAnswer
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.
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