The sum of two rational numbers is `(-1)/(2)`. If one of the numbers is `(5)/(6)`, find the other.

Options

A. `(-4)/(3) `
B. `(4)/(7)`
C. `(-4)/(7)`
D. `(4)/(3)`

Answer

To find the other rational number when the sum of two rational numbers is \(-\frac{1}{2}\) and one of the numbers is \(\frac{5}{6}\), we can follow these steps:

### Step 1: Set up the equation
Let the other rational number be \(x\). According to the problem, we have:
\[ \frac{5}{6} + x = -\frac{1}{2} \]

### Step 2: Isolate \(x\)
To find \(x\), we need to isolate it on one side of the equation. We can do this by subtracting \(\frac{5}{6}\) from both sides:
\[ x = -\frac{1}{2} - \frac{5}{6} \]

### Step 3: Find a common denominator
To subtract the fractions, we need a common denominator. The denominators are 2 and 6. The least common multiple (LCM) of 2 and 6 is 6. We will convert \(-\frac{1}{2}\) to a fraction with a denominator of 6:
\[ -\frac{1}{2} = -\frac{3}{6} \]

### Step 4: Substitute and simplify
Now we can substitute back into the equation:
\[ x = -\frac{3}{6} - \frac{5}{6} \]
Now, we can combine the fractions:
\[ x = \frac{-3 - 5}{6} = \frac{-8}{6} \]

### Step 5: Simplify the fraction
We can simplify \(\frac{-8}{6}\) by dividing both the numerator and the denominator by 2:
\[ x = \frac{-4}{3} \]

### Final Answer
The other rational number is:
\[ x = -\frac{4}{3} \]

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