Simplify the following expressions :
`2/3 " of " 21 4/9 - 9 4/11`

Answer

To simplify the expression \( \frac{2}{3} \text{ of } 21 \frac{4}{9} - 9 \frac{4}{11} \), we can follow these steps:

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

1. For \( 21 \frac{4}{9} \):
\[ 21 \frac{4}{9} = \frac{21 \times 9 + 4}{9} = \frac{189 + 4}{9} = \frac{193}{9} \]

2. For \( 9 \frac{4}{11} \):
\[ 9 \frac{4}{11} = \frac{9 \times 11 + 4}{11} = \frac{99 + 4}{11} = \frac{103}{11} \]

### Step 2: Multiply by \( \frac{2}{3} \)
Now, we will multiply \( \frac{2}{3} \) by \( \frac{193}{9} \):
\[ \frac{2}{3} \times \frac{193}{9} = \frac{2 \times 193}{3 \times 9} = \frac{386}{27} \]

### Step 3: Set Up the Expression
Now, we can rewrite the original expression:
\[ \frac{386}{27} - \frac{103}{11} \]

### Step 4: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \( 27 \) and \( 11 \):
\[ \text{LCM}(27, 11) = 27 \times 11 = 297 \]

### Step 5: Rewrite Each Fraction with the Common Denominator
Now, we will convert each fraction to have a denominator of \( 297 \):
1. For \( \frac{386}{27} \):
\[ \frac{386}{27} = \frac{386 \times 11}{27 \times 11} = \frac{4246}{297} \]

2. For \( \frac{103}{11} \):
\[ \frac{103}{11} = \frac{103 \times 27}{11 \times 27} = \frac{2781}{297} \]

### Step 6: Subtract the Fractions
Now that both fractions have the same denominator, we can subtract them:
\[ \frac{4246}{297} - \frac{2781}{297} = \frac{4246 - 2781}{297} = \frac{1465}{297} \]

### Step 7: Convert the Result to a Mixed Number
To convert \( \frac{1465}{297} \) to a mixed number:
1. Divide \( 1465 \) by \( 297 \):
\[ 1465 \div 297 \approx 4 \quad \text{(since } 297 \times 4 = 1188\text{)} \]
2. Find the remainder:
\[ 1465 - 1188 = 277 \]
Thus, we can express \( \frac{1465}{297} \) as:
\[ 4 \frac{277}{297} \]

### Final Answer
The simplified expression is:
\[ 4 \frac{277}{297} \]

---