`2(1)/(7) + 3(6)/(7) + 5(1)/(6) = ?`

Options

A. `11(6)/(7)`
B. `11(1)/(6)`
C. `11(5)/(6)`
D. `10(5)/(6)`

Answer

To solve the expression \(2\frac{1}{7} + 3\frac{6}{7} + 5\frac{1}{6}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers into improper fractions.

1. For \(2\frac{1}{7}\):
\[ 2\frac{1}{7} = \frac{2 \times 7 + 1}{7} = \frac{14 + 1}{7} = \frac{15}{7} \]

2. For \(3\frac{6}{7}\):
\[ 3\frac{6}{7} = \frac{3 \times 7 + 6}{7} = \frac{21 + 6}{7} = \frac{27}{7} \]

3. For \(5\frac{1}{6}\):
\[ 5\frac{1}{6} = \frac{5 \times 6 + 1}{6} = \frac{30 + 1}{6} = \frac{31}{6} \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using the improper fractions:
\[ \frac{15}{7} + \frac{27}{7} + \frac{31}{6} \]

### Step 3: Combine Like Terms
Next, we can combine the fractions with the same denominator:
\[ \frac{15}{7} + \frac{27}{7} = \frac{15 + 27}{7} = \frac{42}{7} = 6 \]

Now, we have:
\[ 6 + \frac{31}{6} \]

### Step 4: Convert Whole Number to Fraction
To add \(6\) and \(\frac{31}{6}\), we convert \(6\) into a fraction with a denominator of \(6\):
\[ 6 = \frac{6 \times 6}{6} = \frac{36}{6} \]

### Step 5: Add the Fractions
Now we can add the two fractions:
\[ \frac{36}{6} + \frac{31}{6} = \frac{36 + 31}{6} = \frac{67}{6} \]

### Final Answer
Thus, the final answer is:
\[ \frac{67}{6} \]

### Step 6: Convert to Mixed Number (if necessary)
If we want to convert \(\frac{67}{6}\) back to a mixed number:
\[ 67 \div 6 = 11 \quad \text{(remainder 1)} \]
So, \(\frac{67}{6} = 11\frac{1}{6}\).

### Summary of the Answer
The final result is:
\[ 11\frac{1}{6} \]