Find three rational numbers between `(1)/(10)and(2)/(15)`.

Explanation

To find three rational numbers between \( \frac{1}{10} \) and \( \frac{2}{15} \), we can follow these steps:

### Step 1: Convert the fractions to have a common denominator
We start with the fractions \( \frac{1}{10} \) and \( \frac{2}{15} \). To compare them easily, we need to convert them to have a common denominator.

The least common multiple (LCM) of 10 and 15 is 30.

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

\[ \frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30} \]

### Step 3: Convert \( \frac{2}{15} \) to a fraction with a denominator of 30
Next, we convert \( \frac{2}{15} \) to a fraction with a denominator of 30 by multiplying both the numerator and the denominator by 2:

\[ \frac{2}{15} = \frac{2 \times 2}{15 \times 2} = \frac{4}{30} \]

### Step 4: Identify the range between the two fractions
Now we have:

\[ \frac{1}{10} = \frac{3}{30} \quad \text{and} \quad \frac{2}{15} = \frac{4}{30} \]

We need to find three rational numbers between \( \frac{3}{30} \) and \( \frac{4}{30} \).

### Step 5: Find rational numbers between \( \frac{3}{30} \) and \( \frac{4}{30} \)
To find rational numbers between \( \frac{3}{30} \) and \( \frac{4}{30} \), we can express these fractions with a larger denominator. Let's multiply both fractions by 10 to make the denominators 300:

\[ \frac{3}{30} = \frac{3 \times 10}{30 \times 10} = \frac{30}{300} \]
\[ \frac{4}{30} = \frac{4 \times 10}{30 \times 10} = \frac{40}{300} \]

Now we have:

\[ \frac{30}{300} \quad \text{and} \quad \frac{40}{300} \]

### Step 6: Choose three rational numbers
We can now choose three rational numbers between \( \frac{30}{300} \) and \( \frac{40}{300} \). For example:

- \( \frac{31}{300} \)
- \( \frac{32}{300} \)
- \( \frac{33}{300} \)

### Conclusion
Thus, three rational numbers between \( \frac{1}{10} \) and \( \frac{2}{15} \) are:

\[ \frac{31}{300}, \quad \frac{32}{300}, \quad \frac{33}{300} \]