Simplify : `5(1)/(2)` of `((2)/(3)-(3)/(5))+(1)/(2)+(5)/(11)`

Answer

To simplify the expression \( 5 \left( \frac{1}{2} \right) \left( \left( \frac{2}{3} - \frac{3}{5} \right) + \frac{1}{2} + \frac{5}{11} \right) \), we will follow the order of operations (BODMAS/BIDMAS). Here’s the step-by-step solution:

### Step 1: Simplify the expression inside the parentheses
First, we need to simplify \( \left( \frac{2}{3} - \frac{3}{5} \right) \).

To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 5 is 15.

\[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \]
\[ \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \]

Now we can subtract:
\[ \frac{2}{3} - \frac{3}{5} = \frac{10}{15} - \frac{9}{15} = \frac{1}{15} \]

### Step 2: Substitute back into the expression
Now substitute back into the original expression:
\[ 5 \left( \frac{1}{2} \right) \left( \frac{1}{15} + \frac{1}{2} + \frac{5}{11} \right) \]

### Step 3: Simplify the expression inside the parentheses
Next, we need to simplify \( \frac{1}{15} + \frac{1}{2} + \frac{5}{11} \).

To do this, we need a common denominator. The LCM of 15, 2, and 11 is 330.

Convert each fraction:
\[ \frac{1}{15} = \frac{1 \times 22}{15 \times 22} = \frac{22}{330} \]
\[ \frac{1}{2} = \frac{1 \times 165}{2 \times 165} = \frac{165}{330} \]
\[ \frac{5}{11} = \frac{5 \times 30}{11 \times 30} = \frac{150}{330} \]

Now add them together:
\[ \frac{22}{330} + \frac{165}{330} + \frac{150}{330} = \frac{22 + 165 + 150}{330} = \frac{337}{330} \]

### Step 4: Substitute back into the expression
Now substitute back into the expression:
\[ 5 \left( \frac{1}{2} \right) \left( \frac{337}{330} \right) \]

### Step 5: Multiply the fractions
Now we multiply:
\[ 5 \times \frac{1}{2} \times \frac{337}{330} = \frac{5 \times 337}{2 \times 330} = \frac{1685}{660} \]

### Step 6: Simplify the fraction
Now we simplify \( \frac{1685}{660} \). The greatest common divisor (GCD) of 1685 and 660 is 5.

Dividing both the numerator and the denominator by 5:
\[ \frac{1685 \div 5}{660 \div 5} = \frac{337}{132} \]

### Final Answer
Thus, the simplified form of the expression is:
\[ \frac{337}{132} \]

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