Find a rational number lying between `(1)/(3) and (1)/(2)`.

Options

A. `(5)/(12) `
B. `(4)/(12) `
C. `(6)/(12) `
D. `(3)/(12) `

Answer

To find a rational number lying between \( \frac{1}{3} \) and \( \frac{1}{2} \), we can follow these steps:

### Step 1: Identify the two rational numbers
Let \( x = \frac{1}{3} \) and \( y = \frac{1}{2} \).

### Step 2: Find the average of the two numbers
To find a rational number between \( x \) and \( y \), we can calculate the average using the formula:
\[ \text{Average} = \frac{x + y}{2} \]

### Step 3: Substitute the values of \( x \) and \( y \)
Substituting \( x \) and \( y \) into the formula:
\[ \text{Average} = \frac{\frac{1}{3} + \frac{1}{2}}{2} \]

### Step 4: Find a common denominator
To add \( \frac{1}{3} \) and \( \frac{1}{2} \), we need a common denominator. The least common multiple of 3 and 2 is 6. Thus, we convert both fractions:
\[ \frac{1}{3} = \frac{2}{6} \quad \text{and} \quad \frac{1}{2} = \frac{3}{6} \]

### Step 5: Add the fractions
Now we can add the fractions:
\[ \frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \]

### Step 6: Divide by 2
Now we need to divide the sum by 2:
\[ \text{Average} = \frac{\frac{5}{6}}{2} = \frac{5}{6} \times \frac{1}{2} = \frac{5}{12} \]

### Conclusion
Thus, a rational number lying between \( \frac{1}{3} \) and \( \frac{1}{2} \) is \( \frac{5}{12} \).

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