To determine which of the given real numbers does NOT lie between \( \sqrt{100} \) and \( \sqrt{120} \), we first need to calculate the square roots of 100 and 120:
\[ \sqrt{100} = 10 \]
\[ \sqrt{120} \approx 10.95 \quad \text{(since } \sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30} \text{ and } \sqrt{30} \approx 5.477 \text{, so } 2\sqrt{30} \approx 10.95\text{)} \]
The interval we are considering on the number line is \( (10, 10.95) \).
Now we will evaluate each option to see if it lies within this interval:
F. \( \frac{21}{2} = 10.5 \)
- This lies within the interval \( (10, 10.95) \).
G. \( 9.2 \)
- This does NOT lie within the interval \( (10, 10.95) \) (it is less than 10).
H. \( \sqrt{115} \approx 10.73 \)
- This lies within the interval \( (10, 10.95) \).
J. \( 10 \frac{2}{3} = 10.6667 \)
- This lies within the interval \( (10, 10.95) \).
So, the only option that does NOT lie between \( \sqrt{100} \) and \( \sqrt{120} \) is:
\[ \boxed{G} \quad (9.2) \]