What is the value of `28 div (11 - 4) + 384 div (15 + 16 div 4 + 4)` ?

Options

A. `(472)/(23)`
B. `(476)/(23)`
C. `(512)/(23)`
D. `(524)/(23)`

Answer

To solve the expression \( 28 \div (11 - 4) + 384 \div (15 + 16 \div 4 + 4) \), we will follow the order of operations, often remembered by the acronym BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

### Step 1: Solve the brackets
First, we will evaluate the expressions inside the brackets.

1. Calculate \( 11 - 4 \):
\[ 11 - 4 = 7 \]

2. For the second bracket \( 15 + 16 \div 4 + 4 \):
- First, calculate \( 16 \div 4 \):
\[ 16 \div 4 = 4 \]
- Now substitute back into the expression:
\[ 15 + 4 + 4 \]
- Now, calculate \( 15 + 4 + 4 \):
\[ 15 + 4 + 4 = 23 \]

### Step 2: Substitute back into the expression
Now, substitute the results from the brackets back into the original expression:
\[ 28 \div 7 + 384 \div 23 \]

### Step 3: Perform the divisions
1. Calculate \( 28 \div 7 \):
\[ 28 \div 7 = 4 \]

2. Calculate \( 384 \div 23 \):
- Perform the division:
\[ 384 \div 23 \approx 16.65 \quad (\text{exactly } 16 \text{ remainder } 16) \]

### Step 4: Combine the results
Now, we combine the results from the divisions:
\[ 4 + \frac{384}{23} \]
To express \( 4 \) in terms of a fraction with a denominator of \( 23 \):
\[ 4 = \frac{92}{23} \]

Now, add the two fractions:
\[ \frac{92}{23} + \frac{384}{23} = \frac{92 + 384}{23} = \frac{476}{23} \]

### Final Answer
Thus, the value of the expression \( 28 \div (11 - 4) + 384 \div (15 + 16 \div 4 + 4) \) is:
\[ \frac{476}{23} \]