Add the following fractions. Be sure to simplify your answer.

[tex]\[
\begin{array}{r}
6 \frac{1}{6} = 6 \frac{1}{6} \\
+ 7 \frac{1}{2} = 7 \frac{1}{2} \\
\hline
\end{array}
\][/tex]

[tex]\[ 13 \frac{2}{3} = \][/tex]



Answer :

Let's add the mixed fractions [tex]\(6 \frac{1}{6}\)[/tex] and [tex]\(7 \frac{1}{2}\)[/tex].

1. Convert the mixed fractions to improper fractions:

- [tex]\(6 \frac{1}{6}\)[/tex] can be expressed as:
[tex]\[ 6 + \frac{1}{6} = \frac{36}{6} + \frac{1}{6} = \frac{37}{6} \][/tex]

- [tex]\(7 \frac{1}{2}\)[/tex] can be expressed as:
[tex]\[ 7 + \frac{1}{2} = \frac{14}{2} + \frac{1}{2} = \frac{15}{2} \][/tex]

2. Find a common denominator for the two fractions:

- The common denominator for [tex]\(6\)[/tex] and [tex]\(2\)[/tex] is [tex]\(6\)[/tex].

- Convert [tex]\(\frac{15}{2}\)[/tex] to a fraction with a denominator of [tex]\(6\)[/tex]:
[tex]\[ \frac{15}{2} = \frac{15 \times 3}{2 \times 3} = \frac{45}{6} \][/tex]

3. Add the two fractions:

- Now, we have [tex]\(\frac{37}{6}\)[/tex] and [tex]\(\frac{45}{6}\)[/tex]:

[tex]\[ \frac{37}{6} + \frac{45}{6} = \frac{37 + 45}{6} = \frac{82}{6} \][/tex]

4. Convert the improper fraction back to a mixed number:

- Divide [tex]\(82\)[/tex] by [tex]\(6\)[/tex] to find the whole number part:
[tex]\[ 82 \div 6 = 13 \quad \text{(quotient is the whole number part)} \][/tex]

- Find the remainder:
[tex]\[ 82 \mod 6 = 4 \quad \text{(remainder is the numerator of the fractional part)} \][/tex]

- The fractional part with the remainder as the numerator is:
[tex]\[ \frac{4}{6} \][/tex]

- Simplify the fractional part:
[tex]\[ \frac{4}{6} = \frac{2}{3} \][/tex]

5. Combine the whole number part and the simplified fractional part:

- The final answer is:
[tex]\[ 13 \frac{2}{3} \][/tex]

So, when you add [tex]\(6 \frac{1}{6}\)[/tex] and [tex]\(7 \frac{1}{2}\)[/tex], the result is [tex]\(13 \frac{2}{3}\)[/tex].