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.