`1(1)/(8)+1(6)/(7)+3(3)/(5)=?`

Options

A. `8(121)/(140)`
B. `6(163)/(280)`
C. `9(197)/(280)`
D. `7(117)/(140)`

Explanation

To solve the expression \( \frac{11}{8} + \frac{16}{7} + \frac{33}{5} \), we will follow these steps:

### Step 1: Identify the fractions
The given expression consists of three fractions:
1. \( \frac{11}{8} \)
2. \( \frac{16}{7} \)
3. \( \frac{33}{5} \)

### Step 2: Find the Least Common Multiple (LCM)
To add these fractions, we need to find a common denominator. The denominators are 8, 7, and 5.

To find the LCM:
- The prime factorization of 8 is \( 2^3 \).
- The prime factorization of 7 is \( 7^1 \).
- The prime factorization of 5 is \( 5^1 \).

The LCM is obtained by taking the highest power of each prime:
\[ \text{LCM} = 2^3 \times 7^1 \times 5^1 = 8 \times 7 \times 5 = 280 \]

### Step 3: Convert each fraction to the common denominator
Now we convert each fraction to have the denominator of 280.

1. For \( \frac{11}{8} \):
\[ \frac{11}{8} = \frac{11 \times 35}{8 \times 35} = \frac{385}{280} \]

2. For \( \frac{16}{7} \):
\[ \frac{16}{7} = \frac{16 \times 40}{7 \times 40} = \frac{640}{280} \]

3. For \( \frac{33}{5} \):
\[ \frac{33}{5} = \frac{33 \times 56}{5 \times 56} = \frac{1848}{280} \]

### Step 4: Add the fractions
Now we can add the fractions:
\[ \frac{385}{280} + \frac{640}{280} + \frac{1848}{280} = \frac{385 + 640 + 1848}{280} \]

Calculating the numerator:
\[ 385 + 640 = 1025 \]
\[ 1025 + 1848 = 2873 \]

So, we have:
\[ \frac{2873}{280} \]

### Step 5: Simplify the fraction (if possible)
The fraction \( \frac{2873}{280} \) cannot be simplified further since 2873 is a prime number and does not share any common factors with 280.

### Final Answer
Thus, the final answer is:
\[ \frac{2873}{280} \]