Simplify`9 5/6 -4 3/8+2 7/12`

Answer

To simplify the expression \(9 \frac{5}{6} - 4 \frac{3}{8} + 2 \frac{7}{12}\), we will follow these steps:

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

1. For \(9 \frac{5}{6}\):
\[ 9 \frac{5}{6} = \frac{(9 \times 6) + 5}{6} = \frac{54 + 5}{6} = \frac{59}{6} \]

2. For \(4 \frac{3}{8}\):
\[ 4 \frac{3}{8} = \frac{(4 \times 8) + 3}{8} = \frac{32 + 3}{8} = \frac{35}{8} \]

3. For \(2 \frac{7}{12}\):
\[ 2 \frac{7}{12} = \frac{(2 \times 12) + 7}{12} = \frac{24 + 7}{12} = \frac{31}{12} \]

Now we have:
\[ \frac{59}{6} - \frac{35}{8} + \frac{31}{12} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \(6\), \(8\), and \(12\).

- The prime factorization of \(6\) is \(2 \times 3\).
- The prime factorization of \(8\) is \(2^3\).
- The prime factorization of \(12\) is \(2^2 \times 3\).

The LCM is found by taking the highest power of each prime:
- For \(2\), the highest power is \(2^3\).
- For \(3\), the highest power is \(3^1\).

Thus, the LCM is:
\[ 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Step 3: Convert Each Fraction to Have the Same Denominator
Now we convert each fraction to have a denominator of \(24\).

1. Convert \(\frac{59}{6}\):
\[ \frac{59}{6} = \frac{59 \times 4}{6 \times 4} = \frac{236}{24} \]

2. Convert \(\frac{35}{8}\):
\[ \frac{35}{8} = \frac{35 \times 3}{8 \times 3} = \frac{105}{24} \]

3. Convert \(\frac{31}{12}\):
\[ \frac{31}{12} = \frac{31 \times 2}{12 \times 2} = \frac{62}{24} \]

Now we have:
\[ \frac{236}{24} - \frac{105}{24} + \frac{62}{24} \]

### Step 4: Combine the Fractions
Now we can combine the fractions:
\[ \frac{236 - 105 + 62}{24} = \frac{236 - 105}{24} + \frac{62}{24} = \frac{131 + 62}{24} = \frac{193}{24} \]

### Step 5: Convert to Mixed Number
Finally, we convert \(\frac{193}{24}\) to a mixed number:
\[ 193 \div 24 = 8 \quad \text{(whole number part)} \]
The remainder is:
\[ 193 - (24 \times 8) = 193 - 192 = 1 \]
Thus, we have:
\[ \frac{193}{24} = 8 \frac{1}{24} \]

### Final Answer
The simplified form of \(9 \frac{5}{6} - 4 \frac{3}{8} + 2 \frac{7}{12}\) is:
\[ \boxed{8 \frac{1}{24}} \]