`3 (5)/(6) + 6 (1)/(7) - 2 (1)/(3) - 1 (1)/(2)` is equal to

Options

A. `6 (1)/(7)`
B. `6 (1)/(21)`
C. `6 (5)/(28)`
D. `6 (1)/(14)`

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert each mixed number into an improper fraction.

1. \(3 \frac{5}{6} = \frac{(3 \times 6) + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}\)
2. \(6 \frac{1}{7} = \frac{(6 \times 7) + 1}{7} = \frac{42 + 1}{7} = \frac{43}{7}\)
3. \(2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}\)
4. \(1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}\)

Now, we rewrite the expression:
\[ \frac{23}{6} + \frac{43}{7} - \frac{7}{3} - \frac{3}{2} \]

### Step 2: Find a Common Denominator
The denominators are 6, 7, 3, and 2. The least common multiple (LCM) of these numbers is 42.

### Step 3: Convert Each Fraction to Have the Common Denominator
Now we convert each fraction:

1. \(\frac{23}{6} = \frac{23 \times 7}{6 \times 7} = \frac{161}{42}\)
2. \(\frac{43}{7} = \frac{43 \times 6}{7 \times 6} = \frac{258}{42}\)
3. \(\frac{7}{3} = \frac{7 \times 14}{3 \times 14} = \frac{98}{42}\)
4. \(\frac{3}{2} = \frac{3 \times 21}{2 \times 21} = \frac{63}{42}\)

Now, we can rewrite the expression:
\[ \frac{161}{42} + \frac{258}{42} - \frac{98}{42} - \frac{63}{42} \]

### Step 4: Combine the Fractions
Now that all fractions have the same denominator, we can combine them:
\[ \frac{161 + 258 - 98 - 63}{42} = \frac{258}{42} \]

### Step 5: Simplify the Result
Now we simplify \(\frac{258}{42}\):
1. Find the GCD of 258 and 42, which is 6.
2. Divide both the numerator and the denominator by 6:
\[ \frac{258 \div 6}{42 \div 6} = \frac{43}{7} \]

### Step 6: Convert Back to a Mixed Number
Finally, we convert \(\frac{43}{7}\) back to a mixed number:
\[ 43 \div 7 = 6 \quad \text{(remainder 1)} \]
So, \(\frac{43}{7} = 6 \frac{1}{7}\).

### Final Answer
The final answer is:
\[ 6 \frac{1}{7} \]