To insert three rational numbers between \(\frac{3}{5}\) and \(\frac{2}{3}\), we'll follow these steps:
1. Convert the fractions to decimal form for easier manipulation:
[tex]\[
\frac{3}{5} = 0.6 \quad \text{and} \quad \frac{2}{3} \approx 0.6667
\][/tex]
2. Calculate the difference between the two bounds:
[tex]\[
0.6667 - 0.6 = 0.0667
\][/tex]
3. Divide the difference by 4 to determine the step size for our rational numbers:
[tex]\[
\frac{0.0667}{4} = 0.0167
\][/tex]
4. Add the step size progressively to the lower bound (\(0.6\)) to find the three numbers:
1. First rational number:
[tex]\[
0.6 + 0.0167 \approx 0.6167
\][/tex]
2. Second rational number:
[tex]\[
0.6 + 2 \times 0.0167 = 0.6 + 0.0334 \approx 0.6333
\][/tex]
3. Third rational number:
[tex]\[
0.6 + 3 \times 0.0167 = 0.6 + 0.0501 \approx 0.65
\][/tex]
Therefore, the three rational numbers inserted between \(\frac{3}{5}\) and \(\frac{2}{3}\) are approximately:
[tex]\[
0.6167, 0.6333, \text{ and } 0.65
\][/tex]
In fractions, these numbers would correspond to:
[tex]\[
\frac{37}{60}, \frac{19}{30}, \text{ and } \frac{13}{20}
\][/tex]
