\( \left. \begin{array} { l } { \frac { 1 } { 3 } + \frac { 2 } { 5 } = \frac { ( L ) ( 1 ) } { ( ) ( 3 ) } + \frac { ( \llcorner ) ( 2 ) } { ( ) ( 5 ) } = \frac { 5 } { 15 } + \frac { 6 } { 15 } = \frac { - } { 15 } } \\ { \frac { 1 } { 2 } + \frac { 5 } { 10 } = \frac { ( ) ( 1 ) } { ( ) ( 2 ) } + \frac { 5 } { 10 } = \frac { 5 } { 10 } + \frac { 5 } { 10 } = - } \end{array} \right. \)

Let's analyze the equations step by step. 1. **First Equation:** \[ \frac{1}{3} + \frac{2}{5} \] To add these fractions, we need a common denominator. The least common multiple of 3 and 5 is 15. Converting each fraction: \[ \frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{2}{5} = \frac{6}{15} \] Now we can add them: \[ \frac{5}{15} + \frac{6}{15} = \frac{5 + 6}{15} = \frac{11}{15} \] So, the first equation simplifies to: \[ \frac{1}{3} + \frac{2}{5} = \frac{11}{15} \] 2. **Second Equation:** \[ \frac{1}{2} + \frac{5}{10} \] Here, we can see that \(\frac{5}{10}\) simplifies to \(\frac{1}{2}\). Therefore: \[ \frac{1}{2} + \frac{5}{10} = \frac{1}{2} + \frac{1}{2} = 1 \] Now, let's fill in the blanks in the original equations: - For the first equation: - \(L = 11\) - The denominator for both fractions is \(15\). - The numerator for the second fraction is \(6\). - The final result is \(11\). - For the second equation: - The first fraction has a numerator of \(5\) and a denominator of \(10\). - The final result is \(1\). Thus, we can summarize the filled-in values: \[ \left. \begin{array} { l } { \frac { 1 } { 3 } + \frac { 2 } { 5 } = \frac { ( 11 ) ( 1 ) } { ( 1 ) ( 3 ) } + \frac { ( 6 ) ( 2 ) } { ( 1 ) ( 5 ) } = \frac { 5 } { 15 } + \frac { 6 } { 15 } = \frac { 11 } { 15 } } \\ { \frac { 1 } { 2 } + \frac { 5 } { 10 } = \frac { ( 5 ) ( 1 ) } { ( 1 ) ( 2 ) } + \frac { 5 } { 10 } = \frac { 5 } { 10 } + \frac { 5 } { 10 } = 1 } \end{array} \right. \]