More than 40, fewer than 60 means the possible total number of pens is between 41 to 59 inclusive.

Subtract 22 gel pens first to make the number of each pen type the same.

41 - 22 = 19
59 - 22 = 37

Now the remaining number of pens is between 19 and 37 inclusive.

But the gel and ballpoint pens are at the same amount now.

The current number of pens must be even, such that it is divisible by 2 so that we can get a whole number of each type.

We can only consider even numbers from 20 to 36.

Method : listing

20,22,24,26,28,30,32,34,36

Dividing by 2, we get :

10,11,12,13,14,15,16,17,18

These are the possible numbers of ballpoint pens.

But since ballpoint pens are only sold in boxes of 6, we only consider the multiples of 6.

We are left with: 12,18

Add back the 22, the possible number of gel pens are :

34,40

Since gel pens are only sold in boxes of 4, then only 40 gel pens is possible since 34 is not a multiple of 4.


So,
number of gel pens = 40
number of ballpoint pens = 18
Total = 40 + 18 = 58

Alternative method :

Since gel pens are only sold in boxes of 4 and ballpoint pens in boxes of 6,

①number of gel pens has to be a multiple of 4.
②number of ballpoint pens has to be a multiple of 6.


We can list a few multiples of 6 first for the ballpoint pens.

6,12,18,24,30

Add 22 to get the corresponding possible number of gel pens.

28,34,40,46,52

The only multiple of 4 here is 40.

So ballpoint pens = 18, gel pens = 40.
Total = 58

(Note that we don't even need to consider 24 ballpoint pens and beyond, since the gel pens would be 46 and beyond The total pens would already be 60 and above, which is too many.)