A firefighter climbed up to the middle of a ladder to see if there was any fire on the second floor of a building and there was none; he then climbed up 10 more steps and saw some smoke so he came back down 14 steps and waited for 5 seconds to check his equipment. He then climbed 20 steps all the way to the top of the ladder and got off on the roof. How many steps are there on the ladder?

First, the firefighter climbed up to the middle of a ladder. so to reach the middle of the ladder he must have climbed [tex]\frac n 2[/tex] steps.

Then, after climbed 10 more steps, the total number of stairs from the ground he climbed [tex]= \frac n 2 +10[/tex].

Then, he came back down 14 steps, so the total number of stairs from the ground he climbed [tex]= \frac n 2 +10 -14=\frac n 2 -4[/tex].

Remaining steps [tex]= n - \left(\frac n 2 -4 \right)=\frac n 2 +4[/tex]

Given that he climbed 20 steps all the way to the top of the ladder which is equal to the remaining steps. So,

[tex]\frac n 2 +4 = 20[/tex]

[tex]\Rightarrow \frac n 2 = 20-4=16[/tex]

[tex]\Rightarrow n = 16 \times 2 = 32.[/tex]

Hence, there are 32 steps on the ladder.