Simplify : `5(1)/(2) - [2 (1)/(3) div {(3)/(4) - (1)/(2) ((2)/(3) - bar((1)/(6)-(1)/(8)))}]`

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(6)`
D. `(2)/(3)`

Answer

To simplify the expression \( 5\left(\frac{1}{2}\right) - \left[2\left(\frac{1}{3}\right) \div \left\{ \left(\frac{3}{4}\right) - \left(\frac{1}{2}\right) \left(\left(\frac{2}{3}\right) - \overline{\left(\frac{1}{6} - \frac{1}{8}\right)}\right)\right\}\right] \), we will follow the B-BODMAS rule step by step.

### Step 1: Simplify the bar expression
First, we need to simplify the expression inside the bar: \( \frac{1}{6} - \frac{1}{8} \).

To do this, we find the least common multiple (LCM) of 6 and 8, which is 24.

\[ \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24} \]

Now, we subtract:

\[ \frac{1}{6} - \frac{1}{8} = \frac{4}{24} - \frac{3}{24} = \frac{1}{24} \]

### Step 2: Substitute back into the expression
Now, we substitute \( \frac{1}{24} \) back into the expression:

\[ 5\left(\frac{1}{2}\right) - \left[2\left(\frac{1}{3}\right) \div \left\{ \left(\frac{3}{4}\right) - \left(\frac{1}{2}\right) \left(\left(\frac{2}{3}\right) - \frac{1}{24}\right)\right\}\right] \]

### Step 3: Simplify the inner expression
Next, we simplify \( \left(\frac{2}{3}\right) - \frac{1}{24} \).

The LCM of 3 and 24 is 24.

\[ \frac{2}{3} = \frac{16}{24} \]

Now, we subtract:

\[ \frac{2}{3} - \frac{1}{24} = \frac{16}{24} - \frac{1}{24} = \frac{15}{24} = \frac{5}{8} \]

### Step 4: Substitute back into the expression
Now we substitute \( \frac{5}{8} \) back into the expression:

\[ 5\left(\frac{1}{2}\right) - \left[2\left(\frac{1}{3}\right) \div \left\{ \left(\frac{3}{4}\right) - \left(\frac{1}{2}\right) \left(\frac{5}{8}\right)\right\}\right] \]

### Step 5: Simplify \( \left(\frac{3}{4}\right) - \left(\frac{1}{2}\right) \left(\frac{5}{8}\right) \)
First, calculate \( \left(\frac{1}{2}\right) \left(\frac{5}{8}\right) \):

\[ \frac{1}{2} \cdot \frac{5}{8} = \frac{5}{16} \]

Now, we subtract:

\[ \frac{3}{4} - \frac{5}{16} \]

The LCM of 4 and 16 is 16.

\[ \frac{3}{4} = \frac{12}{16} \]

Now, we subtract:

\[ \frac{3}{4} - \frac{5}{16} = \frac{12}{16} - \frac{5}{16} = \frac{7}{16} \]

### Step 6: Substitute back into the expression
Now we substitute \( \frac{7}{16} \) back into the expression:

\[ 5\left(\frac{1}{2}\right) - \left[2\left(\frac{1}{3}\right) \div \frac{7}{16}\right] \]

### Step 7: Simplify \( 2\left(\frac{1}{3}\right) \div \frac{7}{16} \)
First, calculate \( 2\left(\frac{1}{3}\right) \):

\[ 2 \cdot \frac{1}{3} = \frac{2}{3} \]

Now, divide by \( \frac{7}{16} \):

\[ \frac{2}{3} \div \frac{7}{16} = \frac{2}{3} \cdot \frac{16}{7} = \frac{32}{21} \]

### Step 8: Substitute back into the expression
Now we have:

\[ 5\left(\frac{1}{2}\right) - \frac{32}{21} \]

### Step 9: Simplify \( 5\left(\frac{1}{2}\right) \)
Calculate \( 5\left(\frac{1}{2}\right) \):

\[ 5 \cdot \frac{1}{2} = \frac{5}{2} \]

### Step 10: Final subtraction
Now we need to subtract:

\[ \frac{5}{2} - \frac{32}{21} \]

The LCM of 2 and 21 is 42.

Convert both fractions:

\[ \frac{5}{2} = \frac{105}{42}, \quad \frac{32}{21} = \frac{64}{42} \]

Now, subtract:

\[ \frac{105}{42} - \frac{64}{42} = \frac{41}{42} \]

Thus, the final simplified expression is:

\[ \frac{41}{42} \]