Find five rational numbers between `(3)/(5)` and `(2)/(3)`.

Answer

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

### Step 1: Convert the fractions to have a common denominator
First, we need to find a common denominator for \( \frac{3}{5} \) and \( \frac{2}{3} \).

- The denominators are 5 and 3. The least common multiple (LCM) of 5 and 3 is 15.

Now, we convert both fractions:

\[ \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \]

\[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \]

### Step 2: Identify the range between the two fractions
Now we have:

\[ \frac{9}{15} \quad \text{and} \quad \frac{10}{15} \]

We need to find five rational numbers between \( \frac{9}{15} \) and \( \frac{10}{15} \).

### Step 3: Increase the denominator to find more rational numbers
To find five rational numbers, we can multiply both the numerator and the denominator of \( \frac{9}{15} \) and \( \frac{10}{15} \) by a number. Here, we will multiply by 6 (since we need 5 numbers, we do \( 5 + 1 = 6 \)).

\[ \frac{9}{15} = \frac{9 \times 6}{15 \times 6} = \frac{54}{90} \]

\[ \frac{10}{15} = \frac{10 \times 6}{15 \times 6} = \frac{60}{90} \]

### Step 4: List the rational numbers between the two fractions
Now we can find rational numbers between \( \frac{54}{90} \) and \( \frac{60}{90} \):

The rational numbers are:
- \( \frac{55}{90} \)
- \( \frac{56}{90} \)
- \( \frac{57}{90} \)
- \( \frac{58}{90} \)
- \( \frac{59}{90} \)

### Step 5: Simplify the rational numbers (if needed)
Now we can simplify these fractions:

1. \( \frac{55}{90} \) can be simplified by dividing both by 5:
\[ \frac{55 \div 5}{90 \div 5} = \frac{11}{18} \]

2. \( \frac{56}{90} \) can be simplified by dividing both by 2:
\[ \frac{56 \div 2}{90 \div 2} = \frac{28}{45} \]

3. \( \frac{57}{90} \) cannot be simplified further.

4. \( \frac{58}{90} \) can be simplified by dividing both by 2:
\[ \frac{58 \div 2}{90 \div 2} = \frac{29}{45} \]

5. \( \frac{59}{90} \) cannot be simplified further.

### Final Answer
The five rational numbers between \( \frac{3}{5} \) and \( \frac{2}{3} \) are:
- \( \frac{11}{18} \)
- \( \frac{28}{45} \)
- \( \frac{57}{90} \)
- \( \frac{29}{45} \)
- \( \frac{59}{90} \)