`((7)/(-26)+(16)/(39))=?`

Options

A. `(11)/(78)`
B. `(-11)/(78)`
C. `(11)/(39)`
D. `(-11)/(39)`

Answer

To solve the expression \(\frac{7}{-26} + \frac{16}{39}\), we will follow these steps:

### Step 1: Rewrite the expression
Since the denominator of the first fraction is negative, we can rewrite the expression as:
\[ -\frac{7}{26} + \frac{16}{39} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators 26 and 39.

- The prime factorization of 26 is \(2 \times 13\).
- The prime factorization of 39 is \(3 \times 13\).

To find the LCM, we take the highest power of each prime factor:
- From 26: \(2^1\) and \(13^1\)
- From 39: \(3^1\) and \(13^1\)

Thus, the LCM is:
\[ LCM = 2^1 \times 3^1 \times 13^1 = 2 \times 3 \times 13 = 78 \]

### Step 3: Convert each fraction to have the LCM as the denominator
Now we convert each fraction to have a denominator of 78.

For \(-\frac{7}{26}\):
\[ -\frac{7}{26} = -\frac{7 \times 3}{26 \times 3} = -\frac{21}{78} \]

For \(\frac{16}{39}\):
\[ \frac{16}{39} = \frac{16 \times 2}{39 \times 2} = \frac{32}{78} \]

### Step 4: Add the fractions
Now we can add the two fractions:
\[ -\frac{21}{78} + \frac{32}{78} = \frac{-21 + 32}{78} = \frac{11}{78} \]

### Final Answer
Thus, the final answer is:
\[ \frac{11}{78} \]

---