Let's denote the total area of the yard as \( Y \).
Travis has mowed \( \frac{1}{2}Y \) of the yard by lunch time.
His sister has raked \( \frac{1}{3} \) of what he has mowed. Since he mowed \( \frac{1}{2}Y \), the amount that his sister has raked is:
\[ \frac{1}{3} \times \frac{1}{2}Y = \frac{1}{6}Y \]
Now, to find the total amount of the yard that has been mowed and raked together, we add the area mowed by Travis and the area raked by his sister:
\[ \text{Total mowed and raked} = \frac{1}{2}Y + \frac{1}{6}Y \]
To add these fractions, we need a common denominator. The least common denominator of 2 and 6 is 6.
Rewriting \( \frac{1}{2}Y \) in terms of sixths:
\[ \frac{1}{2}Y = \frac{3}{6}Y \]
Now we can add the two fractions:
\[ \frac{3}{6}Y + \frac{1}{6}Y = \frac{4}{6}Y \]
This simplifies to:
\[ \frac{4}{6}Y = \frac{2}{3}Y \]
Therefore, the total area of the yard that has been mowed and raked is
\[ \frac{2}{3} \text{ of the yard.} \]