Evaluate: b. `((6)/(7) xx (35)/(48)) + ((5)/(6) div (15)/(16))`

Answer

To evaluate the expression \(\left(\frac{6}{7} \times \frac{35}{48}\right) + \left(\frac{5}{6} \div \frac{15}{16}\right)\), we can follow these steps:

### Step 1: Evaluate the multiplication
First, we will calculate \(\frac{6}{7} \times \frac{35}{48}\).

\[ \frac{6 \times 35}{7 \times 48} \]

### Step 2: Simplify the multiplication
Now, we can simplify before multiplying. We can cancel \(7\) from \(35\) and \(6\) from \(48\):

\[ \frac{6}{7} \times \frac{35}{48} = \frac{6 \div 6}{7 \div 7} \times \frac{35 \div 7}{48 \div 6} = \frac{1}{1} \times \frac{5}{8} = \frac{5}{8} \]

### Step 3: Evaluate the division
Next, we will calculate \(\frac{5}{6} \div \frac{15}{16}\).

Using the rule for division of fractions, we can rewrite this as:

\[ \frac{5}{6} \times \frac{16}{15} \]

### Step 4: Simplify the division
Now, we can simplify before multiplying:

\[ \frac{5 \times 16}{6 \times 15} \]

We can cancel \(5\) from \(15\):

\[ = \frac{1 \times 16}{6 \div 3} = \frac{16}{18} = \frac{8}{9} \]

### Step 5: Add the results
Now we need to add \(\frac{5}{8}\) and \(\frac{8}{9}\).

To do this, we need a common denominator. The least common multiple of \(8\) and \(9\) is \(72\).

### Step 6: Convert to common denominator
Convert \(\frac{5}{8}\) to have a denominator of \(72\):

\[ \frac{5}{8} = \frac{5 \times 9}{8 \times 9} = \frac{45}{72} \]

Convert \(\frac{8}{9}\) to have a denominator of \(72\):

\[ \frac{8}{9} = \frac{8 \times 8}{9 \times 8} = \frac{64}{72} \]

### Step 7: Add the fractions
Now we can add the two fractions:

\[ \frac{45}{72} + \frac{64}{72} = \frac{45 + 64}{72} = \frac{109}{72} \]

### Final Answer
Thus, the final answer is:

\[ \frac{109}{72} \]

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