Simplify : e. `5 (1)/(3) xx (3)/(8) - (7)/(9) div 2 (1)/(10)`

Answer

Let's simplify the expression step by step.

**Given Expression:**
\[ 5 \frac{1}{3} \times \frac{3}{8} - \frac{7}{9} \div 2 \frac{1}{10} \]

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

For \( 5 \frac{1}{3} \):
\[ 5 \times 3 + 1 = 15 + 1 = 16 \]
So,
\[ 5 \frac{1}{3} = \frac{16}{3} \]

For \( 2 \frac{1}{10} \):
\[ 2 \times 10 + 1 = 20 + 1 = 21 \]
So,
\[ 2 \frac{1}{10} = \frac{21}{10} \]

Now we can rewrite the expression:
\[ \frac{16}{3} \times \frac{3}{8} - \frac{7}{9} \div \frac{21}{10} \]

### Step 2: Rewrite Division as Multiplication
Next, we rewrite the division as multiplication by the reciprocal:
\[ \frac{16}{3} \times \frac{3}{8} - \frac{7}{9} \times \frac{10}{21} \]

### Step 3: Simplify the Multiplications
Now we simplify each multiplication.

For \( \frac{16}{3} \times \frac{3}{8} \):
\[ \frac{16 \times 3}{3 \times 8} = \frac{16}{8} = 2 \]

For \( \frac{7}{9} \times \frac{10}{21} \):
\[ \frac{7 \times 10}{9 \times 21} = \frac{70}{189} \]
Now, we can simplify \( \frac{70}{189} \):
- The GCD of 70 and 189 is 7.
\[ \frac{70 \div 7}{189 \div 7} = \frac{10}{27} \]

### Step 4: Combine the Results
Now we substitute back into the expression:
\[ 2 - \frac{10}{27} \]

### Step 5: Find a Common Denominator
To subtract these fractions, we need a common denominator. The LCM of 1 and 27 is 27. We convert 2 into a fraction:
\[ 2 = \frac{2 \times 27}{1 \times 27} = \frac{54}{27} \]

### Step 6: Perform the Subtraction
Now we can subtract:
\[ \frac{54}{27} - \frac{10}{27} = \frac{54 - 10}{27} = \frac{44}{27} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{44}{27} \]

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