Answer :
Sure, let's fill in the missing elements of the given table step-by-step. The table is initially provided as follows:
[tex]\[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & & \\ \hline \frac{2}{3} & & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
1. Calculate the missing element in the 2nd row, 3rd column:
[tex]\[ \left(\text{2nd row, 3rd column}\right) = \left(\text{2nd row, 1st column}\right) + \left(\text{1st row, 3rd column}\right) \][/tex]
[tex]\[ = \frac{1}{3} + \frac{2}{3} = \frac{1}{3} + \frac{2}{3} = 1.0 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & \\ \hline \frac{2}{3} & & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
2. Calculate the missing element in the 2nd row, 4th column:
[tex]\[ \left(\text{2nd row, 4th column}\right) = \left(\text{2nd row, 1st column}\right) + \left(\text{1st row, 4th column}\right) \][/tex]
[tex]\[ = \frac{1}{3} + \frac{3}{2} = \frac{1}{3} + 1.5 = 1.833 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
3. Calculate the missing element in the 3rd row, 2nd column:
[tex]\[ \left(\text{3rd row, 2nd column}\right) = \left(\text{3rd row, 1st column}\right) + \left(\text{1st row, 2nd column}\right) \][/tex]
[tex]\[ = \frac{2}{3} + \frac{1}{2} = \frac{2}{3} + 0.5 = 1.167 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
4. Calculate the missing element in the 3rd row, 4th column:
[tex]\[ \left(\text{3rd row, 4th column}\right) = \left(\text{3rd row, 1st column}\right) + \left(\text{1st row, 4th column}\right) \][/tex]
[tex]\[ = \frac{2}{3} + \frac{3}{2} = \frac{2}{3} + 1.5 = 2.167 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
5. Calculate the missing element in the 4th row, 2nd column:
[tex]\[ \left(\text{4th row, 2nd column}\right) = \left(\text{4th row, 1st column}\right) + \left(\text{1st row, 2nd column}\right) \][/tex]
[tex]\[ = \frac{1}{2} + \frac{1}{2} = 0.5 + 0.5 = 1.0 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & 1.0 & & \\ \hline \end{array} \right] \][/tex]
6. Calculate the missing element in the 4th row, 3rd column:
[tex]\[ \left(\text{4th row, 3rd column}\right) = \left(\text{4th row, 1st column}\right) + \left(\text{1st row, 3rd column}\right) \][/tex]
[tex]\[ = \frac{1}{2} + \frac{2}{3} = 0.5 + 0.667 = 1.167 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & 1.0 & 1.167 & \\ \hline \end{array} \right] \][/tex]
7. Calculate the missing element in the 4th row, 4th column:
[tex]\[ \left(\text{4th row, 4th column}\right) = \left(\text{4th row, 1st column}\right) + \left(\text{1st row, 4th column}\right) \][/tex]
[tex]\[ = \frac{1}{2} + \frac{3}{2} = 0.5 + 1.5 = 2.0 \][/tex]
Hence, the completed table is:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & 1.0 & 1.167 & 2.0 \\ \hline \end{array} \right] \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & & \\ \hline \frac{2}{3} & & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
1. Calculate the missing element in the 2nd row, 3rd column:
[tex]\[ \left(\text{2nd row, 3rd column}\right) = \left(\text{2nd row, 1st column}\right) + \left(\text{1st row, 3rd column}\right) \][/tex]
[tex]\[ = \frac{1}{3} + \frac{2}{3} = \frac{1}{3} + \frac{2}{3} = 1.0 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & \\ \hline \frac{2}{3} & & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
2. Calculate the missing element in the 2nd row, 4th column:
[tex]\[ \left(\text{2nd row, 4th column}\right) = \left(\text{2nd row, 1st column}\right) + \left(\text{1st row, 4th column}\right) \][/tex]
[tex]\[ = \frac{1}{3} + \frac{3}{2} = \frac{1}{3} + 1.5 = 1.833 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
3. Calculate the missing element in the 3rd row, 2nd column:
[tex]\[ \left(\text{3rd row, 2nd column}\right) = \left(\text{3rd row, 1st column}\right) + \left(\text{1st row, 2nd column}\right) \][/tex]
[tex]\[ = \frac{2}{3} + \frac{1}{2} = \frac{2}{3} + 0.5 = 1.167 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
4. Calculate the missing element in the 3rd row, 4th column:
[tex]\[ \left(\text{3rd row, 4th column}\right) = \left(\text{3rd row, 1st column}\right) + \left(\text{1st row, 4th column}\right) \][/tex]
[tex]\[ = \frac{2}{3} + \frac{3}{2} = \frac{2}{3} + 1.5 = 2.167 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & & & \\ \hline \end{array} \right] \][/tex]
5. Calculate the missing element in the 4th row, 2nd column:
[tex]\[ \left(\text{4th row, 2nd column}\right) = \left(\text{4th row, 1st column}\right) + \left(\text{1st row, 2nd column}\right) \][/tex]
[tex]\[ = \frac{1}{2} + \frac{1}{2} = 0.5 + 0.5 = 1.0 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & 1.0 & & \\ \hline \end{array} \right] \][/tex]
6. Calculate the missing element in the 4th row, 3rd column:
[tex]\[ \left(\text{4th row, 3rd column}\right) = \left(\text{4th row, 1st column}\right) + \left(\text{1st row, 3rd column}\right) \][/tex]
[tex]\[ = \frac{1}{2} + \frac{2}{3} = 0.5 + 0.667 = 1.167 \][/tex]
So, the table now looks like:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & 1.0 & 1.167 & \\ \hline \end{array} \right] \][/tex]
7. Calculate the missing element in the 4th row, 4th column:
[tex]\[ \left(\text{4th row, 4th column}\right) = \left(\text{4th row, 1st column}\right) + \left(\text{1st row, 4th column}\right) \][/tex]
[tex]\[ = \frac{1}{2} + \frac{3}{2} = 0.5 + 1.5 = 2.0 \][/tex]
Hence, the completed table is:
[tex]\[ \left[ \begin{array}{|c|c|c|c|} \hline + & \frac{1}{2} & \frac{2}{3} & \frac{3}{2} \\ \hline \frac{1}{3} & \frac{5}{6} & 1.0 & 1.833 \\ \hline \frac{2}{3} & 1.167 & \frac{4}{3} & 2.167 \\ \hline \frac{1}{2} & 1.0 & 1.167 & 2.0 \\ \hline \end{array} \right] \][/tex]
