What will be the result on solving of `(3/5 + 1/5 div 3/(10)) xx ((36)/(45) + (16)/(5))` ?

Options

A. `1/3`
B. `(76)/(15)`
C. `(32)/(25)`
D. `(64)/(125)`

Answer

To solve the expression `(3/5 + 1/5 div 3/(10)) xx ((36)/(45) + (16)/(5))`, we will follow the order of operations, often remembered by the acronym BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

### Step-by-Step Solution:

1. **Rewrite the expression**:
The expression can be rewritten for clarity:
\[ (3/5 + (1/5) \div (3/10)) \times ((36/45) + (16/5)) \]

2. **Solve the division inside the first bracket**:
To divide fractions, we multiply by the reciprocal:
\[ (1/5) \div (3/10) = (1/5) \times (10/3) = \frac{10}{15} = \frac{2}{3} \]
Now, substitute this back into the expression:
\[ (3/5 + 2/3) \times ((36/45) + (16/5)) \]

3. **Find a common denominator for the first bracket**:
The common denominator for 5 and 3 is 15:
\[ 3/5 = \frac{9}{15}, \quad 2/3 = \frac{10}{15} \]
Now add:
\[ \frac{9}{15} + \frac{10}{15} = \frac{19}{15} \]

4. **Simplify the second bracket**:
For the second bracket, find a common denominator for 45 and 5, which is 45:
\[ 16/5 = \frac{144}{45} \]
Now add:
\[ (36/45) + (144/45) = \frac{180}{45} \]

5. **Combine the two brackets**:
Now substitute back into the expression:
\[ \frac{19}{15} \times \frac{180}{45} \]

6. **Simplify the multiplication**:
Multiply the fractions:
\[ \frac{19 \times 180}{15 \times 45} \]
First, simplify \(180/45 = 4\):
\[ = \frac{19 \times 4}{15} = \frac{76}{15} \]

7. **Convert to mixed number**:
To convert \(\frac{76}{15}\) into a mixed number:
\[ 76 = 75 + 1 = 15 \times 5 + 1 \]
Thus:
\[ \frac{76}{15} = 5 \frac{1}{15} \]

### Final Result:
The final answer is:
\[ 5 \frac{1}{15} \]