`9 (3)/(4) + [2(1)/(6) + {4 (1)/(3) - (2 (1)/(2) + (3)/(4))}]` is equal to

Options

A. 13
B. `(15)/(4)`
C. 14
D. `(17)/(4)`

Answer

To solve the expression \( 9 \frac{3}{4} + \left[ 2 \frac{1}{6} + \left\{ 4 \frac{1}{3} - \left( 2 \frac{1}{2} + \frac{3}{4} \right) \right\} \right] \), we will follow the order of operations (BODMAS/BIDMAS).

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

- \( 9 \frac{3}{4} = \frac{9 \times 4 + 3}{4} = \frac{36 + 3}{4} = \frac{39}{4} \)
- \( 2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6} \)
- \( 4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} \)
- \( 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \)

Now, the expression is:
\[ \frac{39}{4} + \left[ \frac{13}{6} + \left\{ \frac{13}{3} - \left( \frac{5}{2} + \frac{3}{4} \right) \right\} \right] \]

### Step 2: Simplify Inside the Parentheses
Next, we simplify \( \frac{5}{2} + \frac{3}{4} \).

To add these fractions, we need a common denominator. The LCM of 2 and 4 is 4.

- Convert \( \frac{5}{2} \) to have a denominator of 4:
\[ \frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4} \]
Now we can add:
\[ \frac{10}{4} + \frac{3}{4} = \frac{10 + 3}{4} = \frac{13}{4} \]

Now, substitute back into the expression:
\[ \frac{39}{4} + \left[ \frac{13}{6} + \left\{ \frac{13}{3} - \frac{13}{4} \right\} \right] \]

### Step 3: Simplify the Bracket
Now, simplify \( \frac{13}{3} - \frac{13}{4} \).

To subtract these fractions, we need a common denominator. The LCM of 3 and 4 is 12.

- Convert \( \frac{13}{3} \) to have a denominator of 12:
\[ \frac{13}{3} = \frac{13 \times 4}{3 \times 4} = \frac{52}{12} \]
- Convert \( \frac{13}{4} \) to have a denominator of 12:
\[ \frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12} \]

Now we can subtract:
\[ \frac{52}{12} - \frac{39}{12} = \frac{52 - 39}{12} = \frac{13}{12} \]

Substituting back, we have:
\[ \frac{39}{4} + \left[ \frac{13}{6} + \frac{13}{12} \right] \]

### Step 4: Simplify the Next Bracket
Now, we simplify \( \frac{13}{6} + \frac{13}{12} \).

The LCM of 6 and 12 is 12.

- Convert \( \frac{13}{6} \) to have a denominator of 12:
\[ \frac{13}{6} = \frac{13 \times 2}{6 \times 2} = \frac{26}{12} \]

Now we can add:
\[ \frac{26}{12} + \frac{13}{12} = \frac{26 + 13}{12} = \frac{39}{12} \]

Substituting back, we have:
\[ \frac{39}{4} + \frac{39}{12} \]

### Step 5: Add the Final Fractions
Now we need to add \( \frac{39}{4} + \frac{39}{12} \).

The LCM of 4 and 12 is 12.

- Convert \( \frac{39}{4} \) to have a denominator of 12:
\[ \frac{39}{4} = \frac{39 \times 3}{4 \times 3} = \frac{117}{12} \]

Now we can add:
\[ \frac{117}{12} + \frac{39}{12} = \frac{117 + 39}{12} = \frac{156}{12} \]

### Step 6: Simplify the Final Result
Now simplify \( \frac{156}{12} \):
\[ \frac{156}{12} = 13 \]

Thus, the final answer is:
\[ \boxed{13} \]