To solve the problem of adding the mixed fractions and the proper fraction, follow these detailed, step-by-step instructions:
1. Convert the mixed fractions to improper fractions:
- For [tex]\(2 \frac{2}{3}\)[/tex]:
- The whole number part is 2.
- The fractional part is [tex]\( \frac{2}{3} \)[/tex].
- Convert it into an improper fraction:
[tex]\[ 2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \][/tex]
- For [tex]\(1 \frac{1}{6}\)[/tex]:
- The whole number part is 1.
- The fractional part is [tex]\( \frac{1}{6} \)[/tex].
- Convert it into an improper fraction:
[tex]\[ 1 \frac{1}{6} = 1 + \frac{1}{6} = \frac{6}{6} + \frac{1}{6} = \frac{7}{6} \][/tex]
2. Given fraction:
- The given proper fraction is [tex]\( \frac{7}{18} \)[/tex].
3. Find the common denominator:
- The denominators of the fractions are 3, 6, and 18. The least common denominator (LCM) of these numbers is 18.
4. Convert each fraction to have the common denominator of 18:
- For [tex]\( \frac{8}{3} \)[/tex]:
[tex]\[ \frac{8}{3} = \frac{8 \times 6}{3 \times 6} = \frac{48}{18} \][/tex]
- For [tex]\( \frac{7}{6} \)[/tex]:
[tex]\[ \frac{7}{6} = \frac{7 \times 3}{6 \times 3} = \frac{21}{18} \][/tex]
- The fraction [tex]\( \frac{7}{18} \)[/tex] is already in the correct form.
5. Add the fractions:
- Add the numerators of the fractions while keeping the common denominator:
[tex]\[ \frac{48}{18} + \frac{21}{18} + \frac{7}{18} = \frac{48 + 21 + 7}{18} = \frac{76}{18} \][/tex]
6. Simplify the resulting improper fraction:
- To simplify [tex]\(\frac{76}{18}\)[/tex], find the greatest common divisor (GCD) of 76 and 18, which is 2:
[tex]\[
\frac{76 \div 2}{18 \div 2} = \frac{38}{9}
\][/tex]
Thus, the sum of [tex]\( 2 \frac{2}{3} \)[/tex], [tex]\( 1 \frac{1}{6} \)[/tex], and [tex]\( \frac{7}{18} \)[/tex] is:
[tex]\[
\frac{38}{9}
\][/tex]
Or as a mixed number, it can also be represented as:
[tex]\[
4 \frac{2}{9}
\][/tex]
