How many fractions are between 5/7 and 6/7

To determine how many fractions are between \(\frac{5}{7}\) and \(\frac{6}{7}\), we need to consider the fractions with the same denominator, i.e., fractions of the form \(\frac{k}{7}\) where \(k\) is an integer.

Given the fractions \(\frac{5}{7}\) and \(\frac{6}{7}\), any fraction \(\frac{k}{7}\) with \(5 < k < 6\) would be between them. However, since \(k\) must be an integer, there is no integer value between 5 and 6. Therefore, there are **no fractions** between \(\frac{5}{7}\) and \(\frac{6}{7}\) with a denominator of 7.

However, if we consider fractions with different denominators that are equivalent to \(\frac{5}{7}\) and \(\frac{6}{7}\) but have a common denominator greater than 7, then there will be fractions between them. For example, multiplying both fractions by 2:

\[
\frac{5 \times 2}{7 \times 2} = \frac{10}{14} \quad \text{and} \quad \frac{6 \times 2}{7 \times 2} = \frac{12}{14}
\]

Here, \(\frac{11}{14}\) is a fraction that lies between \(\frac{10}{14}\) and \(\frac{12}{14}\), which corresponds to \(\frac{5.5}{7}\).

Generally, to find fractions between \(\frac{5}{7}\) and \(\frac{6}{7}\), you can create fractions with larger denominators and then find those that lie between the two given fractions.

Would you like more details or have any questions?

Here are some questions to consider:
1. How can we determine the number of fractions between two fractions with different denominators?
2. What is the process of finding fractions between two given fractions on a number line?
3. How does increasing the denominator affect the number of fractions between two given fractions?
4. Can we find a general formula to count the fractions between any two given fractions?
5. What is the significance of the mediant fraction in identifying fractions between two others?

**Tip:** When comparing fractions, converting them to have the same denominator can make it easier to see which fractions lie between them.

Discover how to determine the number of fractions between 5/7 and 6/7 on a number line. Explore the concept of equivalent fractions and learn how increasing denominators affects the number of fractions between two given fractions. Suitable for students in Grades 5-6.

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