Simplify the following expression :
`((3)/(2)xx(1)/(6))+((5)/(3)xx(7)/(2))-((13)/(4)xx(4)/(3))=?`

Options

A. `(12)/(21)`
B. `(21)/(12)`
C. `(36)/(21)`
D. `(41)/(12)`

Answer

To simplify the expression \(\frac{3}{2} \times \frac{1}{6} + \frac{5}{3} \times \frac{7}{2} - \frac{13}{4} \times \frac{4}{3}\), we can follow these steps:

### Step 1: Calculate each multiplication separately.
1. **First term:**
\[ \frac{3}{2} \times \frac{1}{6} = \frac{3 \times 1}{2 \times 6} = \frac{3}{12} = \frac{1}{4} \]

2. **Second term:**
\[ \frac{5}{3} \times \frac{7}{2} = \frac{5 \times 7}{3 \times 2} = \frac{35}{6} \]

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.
Now we will combine the results of the multiplications:
\[ \frac{1}{4} + \frac{35}{6} - \frac{13}{3} \]

### Step 3: Find a common denominator.
The least common multiple (LCM) of the denominators \(4\), \(6\), and \(3\) is \(12\).

- Convert \(\frac{1}{4}\) to twelfths:
\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \]

- Convert \(\frac{35}{6}\) to twelfths:
\[ \frac{35}{6} = \frac{35 \times 2}{6 \times 2} = \frac{70}{12} \]

- Convert \(\frac{13}{3}\) to twelfths:
\[ \frac{13}{3} = \frac{13 \times 4}{3 \times 4} = \frac{52}{12} \]

### Step 4: Substitute back into the expression.
Now substitute back into the expression:
\[ \frac{3}{12} + \frac{70}{12} - \frac{52}{12} \]

### Step 5: Combine the fractions.
Combine the numerators:
\[ \frac{3 + 70 - 52}{12} = \frac{21}{12} \]

### Step 6: Simplify the fraction.
The fraction \(\frac{21}{12}\) can be simplified:
\[ \frac{21}{12} = \frac{7}{4} \]

### Final Answer:
Thus, the simplified expression is:
\[ \frac{7}{4} \]

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