The sum of two rational numbers is `(-5)/(8)`. If one of these numbers is `(-7)/(12)`, find the other.

Options

A. `(-5)/(24)`
B. `(-7)/(24)`
C. `(-1)/(22)`
D. `(-1)/(24)`

Answer

To find the other rational number, we can follow these steps:

### Step 1: Set Up the Equation
We know the sum of two rational numbers is \(-\frac{5}{8}\), and one of the numbers is \(-\frac{7}{12}\). Let the other number be \(x\).

\[ x + \left(-\frac{7}{12}\right) = -\frac{5}{8} \]

### Step 2: Rearrange the Equation
To find \(x\), we can rearrange the equation:

\[ x = -\frac{5}{8} + \frac{7}{12} \]

### Step 3: Find the Least Common Multiple (LCM)
We need to find the LCM of the denominators 8 and 12 to add the fractions. The LCM of 8 and 12 is 24.

### Step 4: Convert the Fractions
Now, we convert each fraction to have a denominator of 24.

For \(-\frac{5}{8}\):
\[ -\frac{5}{8} = -\frac{5 \times 3}{8 \times 3} = -\frac{15}{24} \]

For \(\frac{7}{12}\):
\[ \frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24} \]

### Step 5: Add the Fractions
Now we can add the two fractions:

\[ x = -\frac{15}{24} + \frac{14}{24} = \frac{-15 + 14}{24} = \frac{-1}{24} \]

### Step 6: Conclusion
Thus, the other rational number is:

\[ x = -\frac{1}{24} \]

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