`3(1)/(4)+2(1)/(2)+6(1)/(6)=?`

Options

A. `13(11)/(12)`
B. `11(11)/(12)`
C. `12(11)/(12)`
D. `15(11)/(12)`

Explanation

To solve the expression \(3\frac{1}{4} + 2\frac{1}{2} + 6\frac{1}{6}\), we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
- For \(3\frac{1}{4}\):
\[ 3\frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4} \]

- For \(2\frac{1}{2}\):
\[ 2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \]

- For \(6\frac{1}{6}\):
\[ 6\frac{1}{6} = \frac{6 \times 6 + 1}{6} = \frac{36 + 1}{6} = \frac{37}{6} \]

### Step 2: Rewrite the expression
Now we can rewrite the expression as:
\[ \frac{13}{4} + \frac{5}{2} + \frac{37}{6} \]

### Step 3: Find the least common multiple (LCM)
The denominators are 4, 2, and 6. The LCM of these numbers is 12.

### Step 4: Convert each fraction to have a common denominator
- For \(\frac{13}{4}\):
\[ \frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12} \]

- For \(\frac{5}{2}\):
\[ \frac{5}{2} = \frac{5 \times 6}{2 \times 6} = \frac{30}{12} \]

- For \(\frac{37}{6}\):
\[ \frac{37}{6} = \frac{37 \times 2}{6 \times 2} = \frac{74}{12} \]

### Step 5: Add the fractions
Now we can add the fractions:
\[ \frac{39}{12} + \frac{30}{12} + \frac{74}{12} = \frac{39 + 30 + 74}{12} = \frac{143}{12} \]

### Final Answer
Thus, the final answer is:
\[ \frac{143}{12} \]