Evaluate: a. `7 (17)/( 30) + 3 (3)/( 11) - 6 (6)/( 11)`

Answer

To evaluate the expression \( 7 \frac{17}{30} + 3 \frac{3}{11} - 6 \frac{6}{11} \), we will follow these steps:

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

1. For \( 7 \frac{17}{30} \):
\[ 7 \frac{17}{30} = \frac{7 \times 30 + 17}{30} = \frac{210 + 17}{30} = \frac{227}{30} \]

2. For \( 3 \frac{3}{11} \):
\[ 3 \frac{3}{11} = \frac{3 \times 11 + 3}{11} = \frac{33 + 3}{11} = \frac{36}{11} \]

3. For \( 6 \frac{6}{11} \):
\[ 6 \frac{6}{11} = \frac{6 \times 11 + 6}{11} = \frac{66 + 6}{11} = \frac{72}{11} \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using the improper fractions:
\[ \frac{227}{30} + \frac{36}{11} - \frac{72}{11} \]

### Step 3: Combine Like Terms
Next, we can combine the fractions with the same denominator:
\[ \frac{227}{30} + \left( \frac{36}{11} - \frac{72}{11} \right) = \frac{227}{30} + \frac{36 - 72}{11} = \frac{227}{30} - \frac{36}{11} \]

### Step 4: Find a Common Denominator
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 30 and 11 is 330.

1. Convert \( \frac{227}{30} \):
\[ \frac{227}{30} = \frac{227 \times 11}{30 \times 11} = \frac{2497}{330} \]

2. Convert \( \frac{36}{11} \):
\[ \frac{36}{11} = \frac{36 \times 30}{11 \times 30} = \frac{1080}{330} \]

### Step 5: Subtract the Fractions
Now we can perform the subtraction:
\[ \frac{2497}{330} - \frac{1080}{330} = \frac{2497 - 1080}{330} = \frac{1417}{330} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1417}{330} \]

---