Find a rational number between `(1)/(10)and(1)/(30)`.

Explanation

To find a rational number between \( \frac{1}{10} \) and \( \frac{1}{30} \), we can follow these steps:

### Step 1: Find a common denominator
To compare the two fractions, we need to express them with a common denominator. The denominators are 10 and 30. The least common multiple (LCM) of 10 and 30 is 30.

### Step 2: Convert \( \frac{1}{10} \) to have a denominator of 30
To convert \( \frac{1}{10} \) to a fraction with a denominator of 30, we can multiply both the numerator and the denominator by 3:
\[ \frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30} \]

### Step 3: Write the fractions with the common denominator
Now we have:
- \( \frac{1}{10} = \frac{3}{30} \)
- \( \frac{1}{30} = \frac{1}{30} \)

### Step 4: Identify a rational number between \( \frac{3}{30} \) and \( \frac{1}{30} \)
Now that both fractions have the same denominator, we can easily find a rational number between them. A simple choice is to take the average of the two fractions:
\[ \text{Average} = \frac{\frac{3}{30} + \frac{1}{30}}{2} = \frac{3 + 1}{2 \times 30} = \frac{4}{60} = \frac{2}{30} \]

### Step 5: Simplify the fraction
We can simplify \( \frac{2}{30} \) by dividing both the numerator and the denominator by 2:
\[ \frac{2}{30} = \frac{1}{15} \]

### Conclusion
Thus, a rational number between \( \frac{1}{10} \) and \( \frac{1}{30} \) is \( \frac{1}{15} \).

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