A rational number between `(-2)/(3) and (1)/(2)` is

Options

A. `(-1)/(6)`
B. `(-1)/(12)`
C. `(-5)/(6)`
D. `(5)/(6)`

Answer

To find a rational number between \(-\frac{2}{3}\) and \(\frac{1}{2}\), we can use the concept of finding the average (or mean) of the two numbers. Here is the step-by-step solution:

### Step 1: Convert the fractions to a common denominator
To find the average, it is easier if both fractions have the same denominator. The denominators here are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6.

- Convert \(-\frac{2}{3}\) to a fraction with a denominator of 6:
\[ -\frac{2}{3} = -\frac{2 \times 2}{3 \times 2} = -\frac{4}{6} \]

- Convert \(\frac{1}{2}\) to a fraction with a denominator of 6:
\[ \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \]

### Step 2: Find the average of the two fractions
Now that we have both fractions with a common denominator, we can find the average:
\[ \text{Average} = \frac{\left(-\frac{4}{6} + \frac{3}{6}\right)}{2} \]

### Step 3: Simplify the sum
Now, we simplify the sum inside the parentheses:
\[ -\frac{4}{6} + \frac{3}{6} = \frac{-4 + 3}{6} = \frac{-1}{6} \]

### Step 4: Divide by 2 to find the average
Now we divide by 2:
\[ \text{Average} = \frac{\frac{-1}{6}}{2} = \frac{-1}{6} \times \frac{1}{2} = \frac{-1}{12} \]

### Conclusion
Thus, a rational number between \(-\frac{2}{3}\) and \(\frac{1}{2}\) is \(-\frac{1}{12}\).

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