Find five rational number between 3/5 and 2/3

To find five rational numbers between \( \frac{3}{5} \) and \( \frac{2}{3} \), we can follow these steps:

1. **Convert the fractions to have a common denominator:**

   The denominators of \( \frac{3}{5} \) and \( \frac{2}{3} \) are 5 and 3, respectively. The least common denominator (LCD) of 5 and 3 is 15.

   Convert \( \frac{3}{5} \) and \( \frac{2}{3} \) to fractions with a denominator of 15:
   \[
   \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
   \]
   \[
   \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
   \]

2. **Find five rational numbers between \( \frac{9}{15} \) and \( \frac{10}{15} \):**

   To find five rational numbers between \( \frac{9}{15} \) and \( \frac{10}{15} \), we can create fractions with a higher common denominator.

   Let's increase the denominator to 30:
   \[
   \frac{9}{15} = \frac{9 \times 2}{15 \times 2} = \frac{18}{30}
   \]
   \[
   \frac{10}{15} = \frac{10 \times 2}{15 \times 2} = \frac{20}{30}
   \]

   Now we can find rational numbers between \( \frac{18}{30} \) and \( \frac{20}{30} \):
   \[
   \frac{19}{30}
   \]

   To find more, let's use a higher denominator, say 45:
   \[
   \frac{9}{15} = \frac{9 \times 3}{15 \times 3} = \frac{27}{45}
   \]
   \[
   \frac{2}{3} = \frac{2 \times 15}{3 \times 15} = \frac{30}{45}
   \]

   Now we can find rational numbers between \( \frac{27}{45} \) and \( \frac{30}{45} \):
   \[
   \frac{28}{45}, \frac{29}{45}
   \]

   Using a higher denominator, 60:
   \[
   \frac{9}{15} = \frac{9 \times 4}{15 \times 4} = \frac{36}{60}
   \]
   \[
   \frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}
   \]

   Now we can find rational numbers between \( \frac{36}{60} \) and \( \frac{40}{60} \):
   \[
   \frac{37}{60}, \frac{38}{60}, \frac{39}{60}
   \]

Combining all the found rational numbers, we get:
- \( \frac{19}{30} \)
- \( \frac{28}{45} \)
- \( \frac{29}{45} \)
- \( \frac{37}{60} \)
- \( \frac{38}{60} \)

So, five rational numbers between \( \frac{3}{5} \) and \( \frac{2}{3} \) are:
1. \( \frac{19}{30} \)
2. \( \frac{28}{45} \)
3. \( \frac{29}{45} \)
4. \( \frac{37}{60} \)
5. \( \frac{38}{60} \)

Would you like more details on the process or have any questions?

Here are some related questions:
1. How do you find the least common denominator of two fractions?
2. What are rational numbers and how do they differ from irrational numbers?
3. How can you determine if a number is rational or irrational?
4. How can you convert a mixed number to an improper fraction?
5. What is the process to add or subtract fractions with different denominators?
6. How can you simplify a fraction?
7. What are some examples of rational numbers between 0 and 1?
8. How do you convert a repeating decimal to a fraction?

**Tip:** When finding numbers between two fractions, converting them to have a common denominator makes it easier to identify intermediate values.

Discover how to find five rational numbers between the fractions 3/5 and 2/3 using a step-by-step approach. This problem involves understanding fractions and rational numbers, suitable for Grades 6-8.

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