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: Find the LCM of the Denominators
The denominators of the two fractions are 10 and 15. We need to find the least common multiple (LCM) of these two numbers.

- The prime factorization of 10 is \( 2 \times 5 \).
- The prime factorization of 15 is \( 3 \times 5 \).
- The LCM is found by taking the highest power of each prime factor:
- From 10: \( 2^1 \) and \( 5^1 \)
- From 15: \( 3^1 \) and \( 5^1 \)

Thus, the LCM is \( 2^1 \times 3^1 \times 5^1 = 30 \).

### Step 2: Convert the Fractions to Have the Same Denominator
Now, we will convert both fractions to have the denominator of 30.

- For \( \frac{1}{10} \):
\[ \frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30} \]

- For \( \frac{2}{15} \):
\[ \frac{2}{15} = \frac{2 \times 2}{15 \times 2} = \frac{4}{30} \]

### Step 3: Identify Rational Numbers Between the Two Converted Fractions
Now we have:
\[ \frac{3}{30} \quad \text{and} \quad \frac{4}{30} \]
We need to find three rational numbers between \( \frac{3}{30} \) and \( \frac{4}{30} \).

### Step 4: Choose Rational Numbers
We can choose rational numbers by simply taking fractions with the same denominator. The numbers between 3 and 4 are 3.1, 3.2, and 3.3.

Thus, the rational numbers are:
- \( \frac{31}{300} \) (which is \( \frac{3.1 \times 30}{30} \))
- \( \frac{32}{300} \) (which is \( \frac{3.2 \times 30}{30} \))
- \( \frac{33}{300} \) (which is \( \frac{3.3 \times 30}{30} \))

### Step 5: Final Rational Numbers
The three rational numbers we found between \( \frac{1}{10} \) and \( \frac{2}{15} \) are:
- \( \frac{31}{300} \)
- \( \frac{32}{300} \)
- \( \frac{33}{300} \)

### Summary of the Solution
The three rational numbers between \( \frac{1}{10} \) and \( \frac{2}{15} \) are:
1. \( \frac{31}{300} \)
2. \( \frac{32}{300} \)
3. \( \frac{33}{300} \)