To find 9 rational numbers between \( \frac{1}{2} \) and \( \frac{3}{5} \), we can follow these steps:
### Step 1: Convert the fractions to have a common denominator
To compare the two fractions easily, we need to convert them to have the same denominator.
1. The denominators are 2 and 5. The least common multiple (LCM) of 2 and 5 is 10.
2. Convert \( \frac{1}{2} \) to a fraction with a denominator of 10:
\[
\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}
\]
3. Convert \( \frac{3}{5} \) to a fraction with a denominator of 10:
\[
\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}
\]
### Step 2: Identify the range of rational numbers
Now we have:
\[
\frac{5}{10} \quad \text{and} \quad \frac{6}{10}
\]
This means we need to find rational numbers between \( \frac{5}{10} \) and \( \frac{6}{10} \).
### Step 3: Generate rational numbers
To find 9 rational numbers between \( \frac{5}{10} \) and \( \frac{6}{10} \), we can divide the interval into equal parts.
1. The difference between \( \frac{5}{10} \) and \( \frac{6}{10} \) is:
\[
\frac{6}{10} - \frac{5}{10} = \frac{1}{10}
\]
2. To find 9 rational numbers, we can divide this interval into 10 equal parts (which gives us 9 segments). Each segment will have a width of:
\[
\frac{1}{10} \div 10 = \frac{1}{100}
\]
### Step 4: List the rational numbers
Starting from \( \frac{5}{10} \) (or \( \frac{50}{100} \)), we can add \( \frac{1}{100} \) repeatedly to find the rational numbers:
1. \( \frac{50}{100} \)
2. \( \frac{51}{100} \)
3. \( \frac{52}{100} \)
4. \( \frac{53}{100} \)
5. \( \frac{54}{100} \)
6. \( \frac{55}{100} \)
7. \( \frac{56}{100} \)
8. \( \frac{57}{100} \)
9. \( \frac{58}{100} \)
Thus, the 9 rational numbers between \( \frac{1}{2} \) and \( \frac{3}{5} \) are:
\[
\frac{51}{100}, \frac{52}{100}, \frac{53}{100}, \frac{54}{100}, \frac{55}{100}, \frac{56}{100}, \frac{57}{100}, \frac{58}{100}
\]
### Final Answer
The 9 rational numbers between \( \frac{1}{2} \) and \( \frac{3}{5} \) are:
\[
\frac{51}{100}, \frac{52}{100}, \frac{53}{100}, \frac{54}{100}, \frac{55}{100}, \frac{56}{100}, \frac{57}{100}, \frac{58}{100}
\]
