`3 5/6+6 1/7 - 2 1/3 - 1 1/2=`

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 to 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
Next, we need to find a common denominator for the fractions. The denominators are 6, 7, 3, and 2. The least common multiple (LCM) of these numbers is 42.

### Step 3: Convert Each Fraction to the Common Denominator
Now we convert each fraction to have a denominator of 42:

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. \( \frac{3}{2} = \frac{3 \times 21}{2 \times 21} = \frac{63}{42} \)

### Step 4: Rewrite the Expression
Now we can rewrite the expression with the common denominator:
\[ \frac{161}{42} + \frac{258}{42} - \frac{98}{42} - \frac{63}{42} \]

### Step 5: Combine the Fractions
Combine the fractions:
\[ \frac{161 + 258 - 98 - 63}{42} = \frac{258}{42} \]

### Step 6: Simplify the Result
Now simplify \( \frac{258}{42} \):
- The greatest common divisor (GCD) of 258 and 42 is 6.
- Divide both the numerator and the denominator by 6:
\[ \frac{258 \div 6}{42 \div 6} = \frac{43}{7} \]

### Step 7: Convert Back to Mixed Number
Finally, convert \( \frac{43}{7} \) back to a mixed number:
- \( 43 \div 7 = 6 \) remainder \( 1 \), so it becomes \( 6 \frac{1}{7} \).

Thus, the final answer is:
\[ \boxed{6 \frac{1}{7}} \]