Evaluate: e. `7 (1)/(6) + 8 (4)/(5) - 9 (2)/(3)`

Answer

To evaluate the expression \( 7 \frac{1}{6} + 8 \frac{4}{5} - 9 \frac{2}{3} \), we will follow these steps:

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

1. **Convert \( 7 \frac{1}{6} \)**:
\[ 7 \frac{1}{6} = \frac{7 \times 6 + 1}{6} = \frac{42 + 1}{6} = \frac{43}{6} \]

2. **Convert \( 8 \frac{4}{5} \)**:
\[ 8 \frac{4}{5} = \frac{8 \times 5 + 4}{5} = \frac{40 + 4}{5} = \frac{44}{5} \]

3. **Convert \( 9 \frac{2}{3} \)**:
\[ 9 \frac{2}{3} = \frac{9 \times 3 + 2}{3} = \frac{27 + 2}{3} = \frac{29}{3} \]

Now we can rewrite the expression:
\[ \frac{43}{6} + \frac{44}{5} - \frac{29}{3} \]

### Step 2: Find a Common Denominator
Next, we need to find a common denominator for the fractions. The denominators are 6, 5, and 3. The least common multiple (LCM) of these numbers is 30.

### Step 3: Convert Each Fraction to the Common Denominator
Now we convert each fraction to have a denominator of 30.

1. **Convert \( \frac{43}{6} \)**:
\[ \frac{43}{6} = \frac{43 \times 5}{6 \times 5} = \frac{215}{30} \]

2. **Convert \( \frac{44}{5} \)**:
\[ \frac{44}{5} = \frac{44 \times 6}{5 \times 6} = \frac{264}{30} \]

3. **Convert \( \frac{29}{3} \)**:
\[ \frac{29}{3} = \frac{29 \times 10}{3 \times 10} = \frac{290}{30} \]

Now we can rewrite the expression:
\[ \frac{215}{30} + \frac{264}{30} - \frac{290}{30} \]

### Step 4: Combine the Fractions
Now that all fractions have the same denominator, we can combine them:
\[ \frac{215 + 264 - 290}{30} = \frac{479 - 290}{30} = \frac{189}{30} \]

### Step 5: Simplify the Result
Finally, we simplify \( \frac{189}{30} \):
\[ \frac{189 \div 3}{30 \div 3} = \frac{63}{10} \]

### Final Answer
Thus, the final answer is:
\[ \frac{63}{10} \]

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