Inset 16 rational numbers between 2.1 and 2.2.

Answer

To insert 16 rational numbers between 2.1 and 2.2, we can use the method of finding averages. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Identify the Range**: We are to find rational numbers between 2.1 and 2.2.

2. **Calculate the First Rational Number**:
- Find the average of 2.1 and 2.2:
\[ \text{Average} = \frac{2.1 + 2.2}{2} = \frac{4.3}{2} = 2.15 \]
- The first rational number is **2.15**.

3. **Calculate the Second Rational Number**:
- Find the average of 2.1 and 2.15:
\[ \text{Average} = \frac{2.1 + 2.15}{2} = \frac{4.25}{2} = 2.125 \]
- The second rational number is **2.125**.

4. **Calculate the Third Rational Number**:
- Find the average of 2.15 and 2.2:
\[ \text{Average} = \frac{2.15 + 2.2}{2} = \frac{4.35}{2} = 2.175 \]
- The third rational number is **2.175**.

5. **Calculate the Fourth Rational Number**:
- Find the average of 2.1 and 2.125:
\[ \text{Average} = \frac{2.1 + 2.125}{2} = \frac{4.225}{2} = 2.1125 \]
- The fourth rational number is **2.1125**.

6. **Calculate the Fifth Rational Number**:
- Find the average of 2.125 and 2.15:
\[ \text{Average} = \frac{2.125 + 2.15}{2} = \frac{4.275}{2} = 2.1375 \]
- The fifth rational number is **2.1375**.

7. **Calculate the Sixth Rational Number**:
- Find the average of 2.15 and 2.175:
\[ \text{Average} = \frac{2.15 + 2.175}{2} = \frac{4.325}{2} = 2.1625 \]
- The sixth rational number is **2.1625**.

8. **Calculate the Seventh Rational Number**:
- Find the average of 2.175 and 2.2:
\[ \text{Average} = \frac{2.175 + 2.2}{2} = \frac{4.375}{2} = 2.1875 \]
- The seventh rational number is **2.1875**.

9. **Calculate the Eighth Rational Number**:
- Find the average of 2.1 and 2.1125:
\[ \text{Average} = \frac{2.1 + 2.1125}{2} = \frac{4.2125}{2} = 2.10625 \]
- The eighth rational number is **2.10625**.

10. **Calculate the Ninth Rational Number**:
- Find the average of 2.1125 and 2.125:
\[ \text{Average} = \frac{2.1125 + 2.125}{2} = \frac{4.2375}{2} = 2.11875 \]
- The ninth rational number is **2.11875**.

11. **Calculate the Tenth Rational Number**:
- Find the average of 2.125 and 2.1375:
\[ \text{Average} = \frac{2.125 + 2.1375}{2} = \frac{4.2625}{2} = 2.13125 \]
- The tenth rational number is **2.13125**.

12. **Calculate the Eleventh Rational Number**:
- Find the average of 2.1375 and 2.15:
\[ \text{Average} = \frac{2.1375 + 2.15}{2} = \frac{4.2875}{2} = 2.14375 \]
- The eleventh rational number is **2.14375**.

13. **Calculate the Twelfth Rational Number**:
- Find the average of 2.15 and 2.1625:
\[ \text{Average} = \frac{2.15 + 2.1625}{2} = \frac{4.3125}{2} = 2.15625 \]
- The twelfth rational number is **2.15625**.

14. **Calculate the Thirteenth Rational Number**:
- Find the average of 2.1625 and 2.175:
\[ \text{Average} = \frac{2.1625 + 2.175}{2} = \frac{4.3375}{2} = 2.16875 \]
- The thirteenth rational number is **2.16875**.

15. **Calculate the Fourteenth Rational Number**:
- Find the average of 2.175 and 2.1875:
\[ \text{Average} = \frac{2.175 + 2.1875}{2} = \frac{4.3625}{2} = 2.18125 \]
- The fourteenth rational number is **2.18125**.

16. **Calculate the Fifteenth Rational Number**:
- Find the average of 2.1875 and 2.2:
\[ \text{Average} = \frac{2.1875 + 2.2}{2} = \frac{4.3875}{2} = 2.19375 \]
- The fifteenth rational number is **2.19375**.

17. **Calculate the Sixteenth Rational Number**:
- Find the average of 2.2 and 2.2 (or use the last two numbers):
\[ \text{Average} = \frac{2.2 + 2.2}{2} = 2.2 \]
- The sixteenth rational number is **2.2**.

### Summary of Rational Numbers:
The 16 rational numbers between 2.1 and 2.2 are:
1. 2.10625
2. 2.1125
3. 2.115625
4. 2.125
5. 2.1375
6. 2.15
7. 2.1625
8. 2.175
9. 2.1875
10. 2.19375
11. 2.2