A map has a scale bar. If the segment representing 10 m measures 15 mm on the print, what is the map distance for 10 m and 20 m? Also, what is the map distance for 42.4 km if the map scale is 1:50?

Based on the assumption, the length representing 10 m is 15 mm.
So, for 10 m, the map distance is $\boxed{15 \text{ mm}}$.

Since 20 m is twice 10 m, the map distance will be twice the map distance for 10 m.

So, for 20 m, the map distance is $\boxed{30 \text{ mm}}$.

The scale 1:50 means that 1 unit on the map represents 50 units on the actual ground.
This can be written as:

Given actual distance = 42.4 km.
We know that $1 \text{ km} = 1\,000 \text{ m}$ and $1 \text{ m} = 100 \text{ cm}$.
So, $1 \text{ km} = 1\,000 \times 100 \text{ cm} = 100\,000 \text{ cm}$.
Now, convert 42.4 km to cm:

Let the map distance be $x$ cm.
Using the ratio from the scale:

Solve for $x$:

The distance on the map is $\boxed{84\,800 \text{ cm}}$.