Evaluate: a. `((2)/(5) xx (15)/(16)) - ((6)/(11) div (54)/(33))`

Answer

To evaluate the expression \(\left(\frac{2}{5} \times \frac{15}{16}\right) - \left(\frac{6}{11} \div \frac{54}{33}\right)\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \left(\frac{2}{5} \times \frac{15}{16}\right) - \left(\frac{6}{11} \div \frac{54}{33}\right) \]

### Step 2: Simplify the multiplication
Now, we calculate \(\frac{2}{5} \times \frac{15}{16}\):
\[ \frac{2 \times 15}{5 \times 16} \]
We can simplify this by cancelling common factors:
- \(2\) in the numerator and \(16\) in the denominator can be simplified to \(1\) and \(8\) respectively.
- \(5\) in the denominator and \(15\) in the numerator can be simplified to \(1\) and \(3\) respectively.

So we have:
\[ \frac{1 \times 3}{1 \times 8} = \frac{3}{8} \]

### Step 3: Simplify the division
Next, we simplify \(\frac{6}{11} \div \frac{54}{33}\). We can rewrite the division as multiplication by the reciprocal:
\[ \frac{6}{11} \times \frac{33}{54} \]
Now, simplify:
- \(6\) and \(54\) can be simplified to \(1\) and \(9\) respectively.
- \(33\) and \(11\) can be simplified to \(3\) and \(1\) respectively.

So we have:
\[ \frac{1 \times 3}{1 \times 9} = \frac{3}{9} = \frac{1}{3} \]

### Step 4: Rewrite the expression
Now we can rewrite the entire expression:
\[ \frac{3}{8} - \frac{1}{3} \]

### Step 5: Find a common denominator
The least common multiple (LCM) of \(8\) and \(3\) is \(24\). We need to convert both fractions to have this common denominator:
\[ \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \]
\[ \frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24} \]

### Step 6: Subtract the fractions
Now we can subtract:
\[ \frac{9}{24} - \frac{8}{24} = \frac{9 - 8}{24} = \frac{1}{24} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{24} \]

---