Explanation:

- Let's define the two numbers as \( x \) and \( y \).

- Their sum is given by \( x + y = 40 \).

- The arithmetic mean (AM) is \( \frac{x + y}{2} = 20 \).

- The geometric mean (GM) is \( \sqrt{xy} \).

- We know GM is 20% less than AM:

$$

\sqrt{xy} = 0.8 \times 20 = 16

$$

- So, \( xy = 16^2 = 256 \).

- These lead us to the quadratic equation:

$$

t^2 - 40t + 256 = 0

$$

- Solving this, \( t = \frac{40 \pm \sqrt{40^2 - 4 \times 256}}{2} \).

- This solves to \( t = 32 \) or \( t = 8 \) leading to numbers 32 and 8.

- Difference: \( 32 - 8 = 24 \).

- ? Option 3 - 24 is correct.

.