To add the fractions [tex]\(\frac{1}{6}\)[/tex], [tex]\(\frac{1}{4}\)[/tex], and [tex]\(\frac{2}{3}\)[/tex], we first need to find a common denominator.
The denominators are 6, 4, and 3. The least common multiple (LCM) of these numbers will be our common denominator.
- The prime factorization of 6 is [tex]\(2 \times 3\)[/tex].
- The prime factorization of 4 is [tex]\(2^2\)[/tex].
- The prime factorization of 3 is [tex]\(3\)[/tex].
The LCM needs to include each prime number to the highest power that appears in these factorizations:
- The highest power of 2 is [tex]\(2^2\)[/tex] (which is 4).
- The highest power of 3 is [tex]\(3\)[/tex].
Therefore, the LCM of 6, 4, and 3 is [tex]\(4 \times 3 = 12\)[/tex].
Now, we convert each fraction to have this common denominator of 12:
1. [tex]\(\frac{1}{6}\)[/tex] becomes [tex]\(\frac{1 \times 2}{6 \times 2} = \frac{2}{12}\)[/tex].
2. [tex]\(\frac{1}{4}\)[/tex] becomes [tex]\(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)[/tex].
3. [tex]\(\frac{2}{3}\)[/tex] becomes [tex]\(\frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)[/tex].
Now we add the fractions:
[tex]\[
\frac{2}{12} + \frac{3}{12} + \frac{8}{12} = \frac{2 + 3 + 8}{12} = \frac{13}{12}
\][/tex]
The resulting fraction [tex]\(\frac{13}{12}\)[/tex] is an improper fraction. It can be converted into a mixed number.
To convert [tex]\(\frac{13}{12}\)[/tex] to a mixed number, divide the numerator by the denominator:
[tex]\[
13 \div 12 = 1 \text{ (remainder } 1)
\][/tex]
So, [tex]\(\frac{13}{12}\)[/tex] can be written as:
[tex]\[
1 \frac{1}{12}
\][/tex]
Therefore, the simplified answer in mixed number form is [tex]\(1 \frac{1}{12}\)[/tex].
So, the answer to [tex]\(\frac{1}{6} + \frac{1}{4} + \frac{2}{3}\)[/tex] is [tex]\(1 \frac{1}{12}\)[/tex].
