To add [tex]\(\frac{1}{4}\)[/tex], [tex]\(\frac{5}{14}\)[/tex], and [tex]\(\frac{6}{7}\)[/tex], we need to find a common denominator. Let's break down the steps to solve this problem:
1. First, identify the denominators of the fractions: 4, 14, and 7. The least common multiple (LCM) of these numbers will be our common denominator.
2. Determine the LCM of 4, 14, and 7:
- The prime factorization of 4 is [tex]\(2^2\)[/tex].
- The prime factorization of 14 is [tex]\(2 \times 7\)[/tex].
- The prime factorization of 7 is [tex]\(7^1\)[/tex].
The LCM is found by taking the highest power of each prime number present in the factorizations:
- For 2: [tex]\(2^2\)[/tex]
- For 7: [tex]\(7^1\)[/tex]
Thus, the LCM is [tex]\(2^2 \times 7 = 4 \times 7 = 28\)[/tex].
3. Rewrite each fraction with the common denominator of 28:
[tex]\[
\frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28}
\][/tex]
[tex]\[
\frac{5}{14} = \frac{5 \times 2}{14 \times 2} = \frac{10}{28}
\][/tex]
[tex]\[
\frac{6}{7} = \frac{6 \times 4}{7 \times 4} = \frac{24}{28}
\][/tex]
4. Add the fractions:
[tex]\[
\frac{7}{28} + \frac{10}{28} + \frac{24}{28} = \frac{7 + 10 + 24}{28} = \frac{41}{28}
\][/tex]
5. Simplify the fraction [tex]\(\frac{41}{28}\)[/tex]. Since 41 is a prime number and does not divide evenly by 28, the fraction is already in its simplest form.
6. Convert the improper fraction [tex]\(\frac{41}{28}\)[/tex] to a mixed number:
- Divide 41 by 28. The quotient is 1, which is the whole number part.
- The remainder is [tex]\(41 - 28 = 13\)[/tex].
So, [tex]\(\frac{41}{28}\)[/tex] can be written as the mixed number [tex]\(1 \frac{13}{28}\)[/tex].
Therefore, the final answer is:
[tex]\[
\frac{1}{4} + \frac{5}{14} + \frac{6}{7} = \frac{41}{28} = 1 \frac{13}{28}
\][/tex]
The answer choices match as follows:
- [tex]\(\frac{12}{25}\)[/tex] is incorrect.
- [tex]\(\frac{41}{28}\)[/tex] is correct but not in mixed number form.
- [tex]\(1 \frac{13}{28}\)[/tex] is correct
- [tex]\(\frac{3}{7}\)[/tex] is incorrect.
So, the simplified answer in mixed number form is:
[tex]\[
\boxed{1 \frac{13}{28}}
\][/tex]
