`9 (1)/(3) + 7 (1)/(2) = ? + 5 (1)/(6) + 6 (1)/(3)`

Options

A. 4
B. `4(1)/(2)`
C. 5
D. `5(1)/(3)`

Answer

To solve the equation \(9 \frac{1}{3} + 7 \frac{1}{2} = ? + 5 \frac{1}{6} + 6 \frac{1}{3}\), we will first convert all mixed numbers into improper fractions.

### Step 1: Convert mixed numbers to improper fractions

1. **Convert \(9 \frac{1}{3}\)**:
\[ 9 \frac{1}{3} = 9 \times 3 + 1 = 27 + 1 = \frac{28}{3} \]

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

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

4. **Convert \(6 \frac{1}{3}\)**:
\[ 6 \frac{1}{3} = 6 \times 3 + 1 = 18 + 1 = \frac{19}{3} \]

### Step 2: Rewrite the equation with improper fractions

Now, we can rewrite the equation:
\[ \frac{28}{3} + \frac{15}{2} = ? + \frac{31}{6} + \frac{19}{3} \]

### Step 3: Move all terms to one side

Rearranging gives:
\[ ? = \frac{28}{3} + \frac{15}{2} - \frac{31}{6} - \frac{19}{3} \]

### Step 4: Find a common denominator

The least common multiple (LCM) of the denominators \(3\), \(2\), and \(6\) is \(6\). We will convert each fraction to have a denominator of \(6\):

1. **Convert \(\frac{28}{3}\)**:
\[ \frac{28}{3} = \frac{28 \times 2}{3 \times 2} = \frac{56}{6} \]

2. **Convert \(\frac{15}{2}\)**:
\[ \frac{15}{2} = \frac{15 \times 3}{2 \times 3} = \frac{45}{6} \]

3. **Convert \(\frac{19}{3}\)**:
\[ \frac{19}{3} = \frac{19 \times 2}{3 \times 2} = \frac{38}{6} \]

### Step 5: Substitute back into the equation

Now substituting back, we have:
\[ ? = \frac{56}{6} + \frac{45}{6} - \frac{31}{6} - \frac{38}{6} \]

### Step 6: Combine the fractions

Combine the fractions:
\[ ? = \frac{56 + 45 - 31 - 38}{6} = \frac{32}{6} \]

### Step 7: Simplify the fraction

Now simplify \(\frac{32}{6}\):
\[ ? = \frac{16}{3} \]

### Step 8: Convert back to mixed number

Convert \(\frac{16}{3}\) back to a mixed number:
\[ \frac{16}{3} = 5 \frac{1}{3} \]

### Final Answer

Thus, the value of \(?\) is:
\[ ? = 5 \frac{1}{3} \]