To add the fractions [tex]\(\frac{1}{4}\)[/tex], [tex]\(\frac{9}{14}\)[/tex], and [tex]\(\frac{6}{7}\)[/tex], we need to follow these steps:
1. Find a common denominator: The denominators are 4, 14, and 7. The least common multiple (LCM) of these numbers is 28. Therefore, we will convert each fraction to have a denominator of 28.
2. Convert each fraction to the common denominator:
- For [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28}
\][/tex]
- For [tex]\(\frac{9}{14}\)[/tex]:
[tex]\[
\frac{9}{14} = \frac{9 \times 2}{14 \times 2} = \frac{18}{28}
\][/tex]
- For [tex]\(\frac{6}{7}\)[/tex]:
[tex]\[
\frac{6}{7} = \frac{6 \times 4}{7 \times 4} = \frac{24}{28}
\][/tex]
3. Add the fractions by summing their numerators (keeping the common denominator):
[tex]\[
\frac{7}{28} + \frac{18}{28} + \frac{24}{28} = \frac{7 + 18 + 24}{28} = \frac{49}{28}
\][/tex]
4. Simplify the resulting fraction if possible:
The fraction [tex]\(\frac{49}{28}\)[/tex] can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator.
- The GCD of 49 and 28 is 7.
- Thus, simplify [tex]\(\frac{49}{28}\)[/tex] by dividing both the numerator and the denominator by 7:
[tex]\[
\frac{49}{28} = \frac{49 \div 7}{28 \div 7} = \frac{7}{4}
\][/tex]
5. Convert the improper fraction to a mixed number:
[tex]\(\frac{7}{4}\)[/tex] can be written as a mixed number. Divide 7 by 4:
- 7 divided by 4 is 1 with a remainder of 3.
- This gives us:
[tex]\[
\frac{7}{4} = 1 \frac{3}{4}
\][/tex]
Therefore, the simplified mixed number is [tex]\(1 \frac{3}{4}\)[/tex]. The correct answer is:
[tex]\[
\boxed{1 \frac{3}{4}}
\][/tex]
