To solve for [tex]\( f \)[/tex] in the given problem, we need to follow these steps:
1. Convert the mixed number to an improper fraction:
- The mixed number given is [tex]\( 3 \frac{5}{6} \)[/tex].
- To convert [tex]\( 3 \frac{5}{6} \)[/tex] into an improper fraction:
[tex]\[
3 \frac{5}{6} = \frac{(3 \times 6 + 5)}{6} = \frac{18 + 5}{6} = \frac{23}{6}
\][/tex]
2. Rewrite the equation:
- The equation given is [tex]\( \frac{2}{3} + f = 3 \frac{5}{6} \)[/tex].
- Substituting the improper fraction we converted, the equation becomes:
[tex]\[
\frac{2}{3} + f = \frac{23}{6}
\][/tex]
3. Find a common denominator to combine fractions:
- [tex]\( \frac{2}{3} \)[/tex] needs to be converted to a fraction with a denominator of 6:
[tex]\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\][/tex]
4. Isolate [tex]\( f \)[/tex]:
- Subtract [tex]\( \frac{4}{6} \)[/tex] from both sides to solve for [tex]\( f \)[/tex]:
[tex]\[
f = \frac{23}{6} - \frac{4}{6} = \frac{23 - 4}{6} = \frac{19}{6}
\][/tex]
5. Convert the improper fraction back to a mixed number:
- Now, [tex]\( f = \frac{19}{6} \)[/tex]:
- Divide 19 by 6:
[tex]\[
19 \div 6 = 3 \text{ with a remainder of } 1
\][/tex]
- So, [tex]\( \frac{19}{6} \)[/tex] can be expressed as:
[tex]\[
3 \frac{1}{6}
\][/tex]
Therefore, [tex]\( f \)[/tex] is [tex]\( \boxed{3 \frac{1}{6}} \)[/tex].
