To solve the problem of finding the sum of the fractions [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex], follow these detailed steps:
1. Identify the fractions to be added:
[tex]\[
\frac{4}{5} \quad \text{and} \quad \frac{9}{11}
\][/tex]
2. Find a common denominator:
The common denominator of the fractions [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex] is the least common multiple (LCM) of the denominators [tex]\(5\)[/tex] and [tex]\(11\)[/tex]. Since [tex]\(5\)[/tex] and [tex]\(11\)[/tex] are both prime and do not share any common factors other than [tex]\(1\)[/tex], their LCM is simply their product:
[tex]\[
\text{LCM}(5, 11) = 5 \times 11 = 55
\][/tex]
3. Convert each fraction to an equivalent fraction with the common denominator:
- Convert [tex]\(\frac{4}{5}\)[/tex] to a fraction with a denominator of [tex]\(55\)[/tex]:
[tex]\[
\frac{4}{5} = \frac{4 \times 11}{5 \times 11} = \frac{44}{55}
\][/tex]
- Convert [tex]\(\frac{9}{11}\)[/tex] to a fraction with a denominator of [tex]\(55\)[/tex]:
[tex]\[
\frac{9}{11} = \frac{9 \times 5}{11 \times 5} = \frac{45}{55}
\][/tex]
4. Add the fractions:
With the fractions now having a common denominator, we can simply add the numerators:
[tex]\[
\frac{44}{55} + \frac{45}{55} = \frac{44 + 45}{55} = \frac{89}{55}
\][/tex]
5. Simplify the fraction if necessary:
In this case, [tex]\(\frac{89}{55}\)[/tex] is already in its simplest form because [tex]\(89\)[/tex] is a prime number and does not share any common factors with [tex]\(55\)[/tex] other than [tex]\(1\)[/tex].
6. Final result:
Therefore, the sum of the fractions [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex] is:
[tex]\[
\frac{4}{5} + \frac{9}{11} = \frac{89}{55}
\][/tex]
The resulting fraction is [tex]\(\frac{89}{55}\)[/tex], with a numerator of [tex]\(89\)[/tex] and a denominator of [tex]\(55\)[/tex].
