Find three rational numbers between `(3)/(5)` and `(7)/(8)`. How many rational numbers can be determined between these two numbers ?

Answer

To find three rational numbers between \( \frac{3}{5} \) and \( \frac{7}{8} \), we can follow these steps:

### Step 1: Find a common denominator
First, we need to convert both fractions to have a common denominator. The denominators are 5 and 8. We can find the least common multiple (LCM) of these two numbers.

**LCM of 5 and 8**:
Since 5 and 8 have no common factors, the LCM is simply \( 5 \times 8 = 40 \).

### Step 2: Convert the fractions
Now we convert both fractions to have the denominator of 40.

- For \( \frac{3}{5} \):
\[ \frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40} \]

- For \( \frac{7}{8} \):
\[ \frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40} \]

### Step 3: Identify rational numbers between the two fractions
Now that we have both fractions with a common denominator, we can find rational numbers between \( \frac{24}{40} \) and \( \frac{35}{40} \).

The rational numbers between these two fractions can be found by selecting numerators that lie between 24 and 35.

### Step 4: Choose three rational numbers
We can select any three numbers between 24 and 35. For example:
- \( \frac{25}{40} \)
- \( \frac{26}{40} \)
- \( \frac{27}{40} \)

### Conclusion
Thus, three rational numbers between \( \frac{3}{5} \) and \( \frac{7}{8} \) are:
1. \( \frac{25}{40} \)
2. \( \frac{26}{40} \)
3. \( \frac{27}{40} \)

### How many rational numbers can be determined between these two numbers?
There are infinitely many rational numbers between any two distinct rational numbers. This is because you can always find another rational number by taking the average of any two rational numbers or by finding fractions with numerators that lie between the two given numerators.