`(2)7/9+(9)11/12div(12)17/18=?`

Options

A. 2
B. 3
C. 4
D. none of these

Answer

To solve the expression \( (2) \frac{7}{9} + (9) \frac{11}{12} \div (12) \frac{17}{18} \), we can break it down into manageable steps.

### Step 1: Convert mixed numbers to improper fractions
First, we need to convert the mixed numbers into improper fractions.

1. **Convert \( (2) \frac{7}{9} \)**:
\[ 2 \frac{7}{9} = \frac{2 \times 9 + 7}{9} = \frac{18 + 7}{9} = \frac{25}{9} \]

2. **Convert \( (9) \frac{11}{12} \)**:
\[ 9 \frac{11}{12} = \frac{9 \times 12 + 11}{12} = \frac{108 + 11}{12} = \frac{119}{12} \]

3. **Convert \( (12) \frac{17}{18} \)**:
\[ 12 \frac{17}{18} = \frac{12 \times 18 + 17}{18} = \frac{216 + 17}{18} = \frac{233}{18} \]

### Step 2: Substitute the improper fractions back into the expression
Now we can rewrite the original expression:
\[ \frac{25}{9} + \frac{119}{12} \div \frac{233}{18} \]

### Step 3: Perform the division
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{119}{12} \div \frac{233}{18} = \frac{119}{12} \times \frac{18}{233} = \frac{119 \times 18}{12 \times 233} \]

Calculating the numerator and denominator:
- Numerator: \( 119 \times 18 = 2142 \)
- Denominator: \( 12 \times 233 = 2796 \)

So we have:
\[ \frac{119}{12} \div \frac{233}{18} = \frac{2142}{2796} \]

### Step 4: Simplify the fraction \( \frac{2142}{2796} \)
To simplify \( \frac{2142}{2796} \), we find the GCD of 2142 and 2796, which is 6:
\[ \frac{2142 \div 6}{2796 \div 6} = \frac{357}{466} \]

### Step 5: Add the fractions
Now we need to add \( \frac{25}{9} + \frac{357}{466} \). To do this, we need a common denominator. The least common multiple (LCM) of 9 and 466 is 4194.

1. Convert \( \frac{25}{9} \):
\[ \frac{25}{9} = \frac{25 \times 466}{9 \times 466} = \frac{11650}{4194} \]

2. Convert \( \frac{357}{466} \):
\[ \frac{357}{466} = \frac{357 \times 9}{466 \times 9} = \frac{3213}{4194} \]

Now we can add the two fractions:
\[ \frac{11650}{4194} + \frac{3213}{4194} = \frac{11650 + 3213}{4194} = \frac{14863}{4194} \]

### Step 6: Final simplification
Now, we check if \( \frac{14863}{4194} \) can be simplified. The GCD of 14863 and 4194 is 1, so it is already in its simplest form.

### Final Answer
Thus, the final answer is:
\[ \frac{14863}{4194} \]