Evaluate: c. `5 (1)/(4) xx 3 (5)/(6) + 1 (3)/(5) div 5 (1)/(3)`

Answer

To evaluate the expression \( 5 \frac{1}{4} \times 3 \frac{5}{6} + 1 \frac{3}{5} \div 5 \frac{1}{3} \), we will follow these steps:

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

1. **Convert \( 5 \frac{1}{4} \)**:
\[ 5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4} \]

2. **Convert \( 3 \frac{5}{6} \)**:
\[ 3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6} \]

3. **Convert \( 1 \frac{3}{5} \)**:
\[ 1 \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{5 + 3}{5} = \frac{8}{5} \]

4. **Convert \( 5 \frac{1}{3} \)**:
\[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \]

Now, we can rewrite the expression:
\[ \frac{21}{4} \times \frac{23}{6} + \frac{8}{5} \div \frac{16}{3} \]

### Step 2: Perform Division
Next, we handle the division part first:
\[ \frac{8}{5} \div \frac{16}{3} = \frac{8}{5} \times \frac{3}{16} \]
Now, multiply:
\[ \frac{8 \times 3}{5 \times 16} = \frac{24}{80} \]
We can simplify \( \frac{24}{80} \):
\[ \frac{24 \div 8}{80 \div 8} = \frac{3}{10} \]

### Step 3: Perform Multiplication
Now we have:
\[ \frac{21}{4} \times \frac{23}{6} + \frac{3}{10} \]
Let's calculate \( \frac{21}{4} \times \frac{23}{6} \):
\[ \frac{21 \times 23}{4 \times 6} = \frac{483}{24} \]

### Step 4: Add the Two Fractions
Now we need to add \( \frac{483}{24} + \frac{3}{10} \). To do this, we need a common denominator. The LCM of 24 and 10 is 120.

Convert both fractions:
1. **Convert \( \frac{483}{24} \)**:
\[ \frac{483 \times 5}{24 \times 5} = \frac{2415}{120} \]

2. **Convert \( \frac{3}{10} \)**:
\[ \frac{3 \times 12}{10 \times 12} = \frac{36}{120} \]

Now we can add:
\[ \frac{2415}{120} + \frac{36}{120} = \frac{2415 + 36}{120} = \frac{2451}{120} \]

### Step 5: Simplify the Result
Now, we simplify \( \frac{2451}{120} \):
- The GCD of 2451 and 120 is 3.
\[ \frac{2451 \div 3}{120 \div 3} = \frac{817}{40} \]

Thus, the final answer is:
\[ \frac{817}{40} \]