Simplify the following expression :
`(2/3 xx 1/6) +(2/3 xx 7/2) -((13)/4 xx 4/3)=?`

Options

A. `(14)/3`
B. `-(17)/9`
C. `(29)/3`
D. `(23)/9`

Answer

To simplify the expression \((\frac{2}{3} \times \frac{1}{6}) + (\frac{2}{3} \times \frac{7}{2}) - \left(\frac{13}{4} \times \frac{4}{3}\right)\), we will follow these steps:

### Step 1: Calculate each term separately.

1. **First term**: \(\frac{2}{3} \times \frac{1}{6}\)
\[ = \frac{2 \times 1}{3 \times 6} = \frac{2}{18} = \frac{1}{9} \]

2. **Second term**: \(\frac{2}{3} \times \frac{7}{2}\)
\[ = \frac{2 \times 7}{3 \times 2} = \frac{14}{6} = \frac{7}{3} \]

3. **Third term**: \(\frac{13}{4} \times \frac{4}{3}\)
\[ = \frac{13 \times 4}{4 \times 3} = \frac{52}{12} = \frac{13}{3} \]

### Step 2: Combine the results from Step 1.

Now, we substitute these results back into the expression:
\[ \frac{1}{9} + \frac{7}{3} - \frac{13}{3} \]

### Step 3: Simplify the expression.

First, we can combine the fractions with a common denominator. The common denominator for \(9\) and \(3\) is \(9\).

1. Convert \(\frac{7}{3}\) and \(\frac{13}{3}\) to have a denominator of \(9\):
\[ \frac{7}{3} = \frac{7 \times 3}{3 \times 3} = \frac{21}{9} \]
\[ \frac{13}{3} = \frac{13 \times 3}{3 \times 3} = \frac{39}{9} \]

2. Now substitute back:
\[ \frac{1}{9} + \frac{21}{9} - \frac{39}{9} = \frac{1 + 21 - 39}{9} = \frac{22 - 39}{9} = \frac{-17}{9} \]

### Final Answer:
\[ \frac{-17}{9} \]