A rational number between `(-2)/(3) and (1)/(4)` is

Options

A. `(5)/(12)`
B. `(-5)/(12)`
C. `(5)/(24)`
D. `(-5)/(24)`

Answer

To find a rational number between \(-\frac{2}{3}\) and \(\frac{1}{4}\), we can use the method of finding the average of the two given rational numbers. Here are the steps to find the rational number:

### Step-by-Step Solution:

1. **Identify the given rational numbers:**
\[ x = -\frac{2}{3}, \quad y = \frac{1}{4} \]

2. **Calculate the average of \(x\) and \(y\):**
\[ \text{Average} = \frac{x + y}{2} \]

3. **Add the two rational numbers:**
\[ x + y = -\frac{2}{3} + \frac{1}{4} \]

4. **Find a common denominator to add the fractions:**
The least common multiple (LCM) of 3 and 4 is 12. So, convert the fractions to have a common denominator of 12:
\[ -\frac{2}{3} = -\frac{2 \times 4}{3 \times 4} = -\frac{8}{12} \]
\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \]

5. **Add the fractions with the common denominator:**
\[ -\frac{8}{12} + \frac{3}{12} = \frac{-8 + 3}{12} = \frac{-5}{12} \]

6. **Divide the sum by 2 to find the average:**
\[ \text{Average} = \frac{\frac{-5}{12}}{2} = \frac{-5}{12} \times \frac{1}{2} = \frac{-5}{24} \]

So, a rational number between \(-\frac{2}{3}\) and \(\frac{1}{4}\) is \(\frac{-5}{24}\).