`((31)/(-4)+(-5)/(8))=?`

Options

A. `(67)/(8)`
B. `(57)/(8)`
C. `(-57)/(8)`
D. `(-67)/(8)`

Answer

To solve the expression \(\frac{31}{-4} + \frac{-5}{8}\), we will follow these steps:

### Step 1: Rewrite the expression
We can rewrite the expression by recognizing that \(\frac{31}{-4}\) can be expressed as \(-\frac{31}{4}\) and \(\frac{-5}{8}\) is already negative. Thus, we have:
\[ -\frac{31}{4} - \frac{5}{8} \]

### Step 2: Find a common denominator
To add the two fractions, we need a common denominator. The denominators are 4 and 8. The least common multiple (LCM) of 4 and 8 is 8.

### Step 3: Convert the fractions
Now we convert \(-\frac{31}{4}\) to have a denominator of 8:
\[ -\frac{31}{4} = -\frac{31 \times 2}{4 \times 2} = -\frac{62}{8} \]
So, we can rewrite the expression as:
\[ -\frac{62}{8} - \frac{5}{8} \]

### Step 4: Combine the fractions
Now that both fractions have the same denominator, we can combine them:
\[ -\frac{62}{8} - \frac{5}{8} = -\frac{62 + 5}{8} = -\frac{67}{8} \]

### Step 5: Final answer
Thus, the final answer is:
\[ -\frac{67}{8} \]

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