Between two rational numbers

Options

A. there is no rational number
B. there is exactly one rational number
C. there are infinitely many rational number
D. there are only rational numbers and no irrational numbers

Answer

To find rational numbers between two given rational numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Two Rational Numbers**:
Let's take two rational numbers as examples: \( \frac{3}{5} \) and \( \frac{4}{5} \).

2. **Convert to a Common Denominator**:
To find rational numbers between them, it's often easier to convert both fractions to have a common denominator. The least common denominator of \( 5 \) is \( 10 \).
- Convert \( \frac{3}{5} \) to \( \frac{6}{10} \) (by multiplying both the numerator and denominator by \( 2 \)).
- Convert \( \frac{4}{5} \) to \( \frac{8}{10} \) (by multiplying both the numerator and denominator by \( 2 \)).

3. **Identify a Rational Number Between Them**:
Now that we have \( \frac{6}{10} \) and \( \frac{8}{10} \), we can find a rational number between them. The simplest way is to take the average of the two:
\[ \text{Average} = \frac{6 + 8}{2} = \frac{14}{2} = \frac{7}{10} \]
Thus, \( \frac{7}{10} \) is one rational number between \( \frac{3}{5} \) and \( \frac{4}{5} \).

4. **Finding More Rational Numbers**:
To find more rational numbers, we can multiply the original fractions by different factors. For instance, if we multiply both \( \frac{3}{5} \) and \( \frac{4}{5} \) by \( 10 \):
- \( \frac{3}{5} \) becomes \( \frac{30}{50} \)
- \( \frac{4}{5} \) becomes \( \frac{40}{50} \)

Now, we can find rational numbers between \( \frac{30}{50} \) and \( \frac{40}{50} \):
- \( \frac{31}{50}, \frac{32}{50}, \frac{33}{50}, \frac{34}{50}, \frac{35}{50}, \frac{36}{50}, \frac{37}{50}, \frac{38}{50}, \frac{39}{50} \)

5. **Conclusion**:
There are infinitely many rational numbers between any two rational numbers. In this case, we found \( \frac{7}{10} \) as one example and listed several others between \( \frac{30}{50} \) and \( \frac{40}{50} \).