One way is to notice that 360 is actually three times of 120.

Currently the 3 : 5 and 3 : 4 ratios seem to suggest that the number of red beads had not changed at all while the number of blue beads had decreased by 1 unit. This does not make sense at all.

So, we need to reflect the ratios in such a way which reflects the corresponding decrease in values, that is, the decrease in blue beads is three times that of the red beads. This method is the fastest to get to the solution, but might require some guessing on your end.

3 : 5 ==> 6 : 10
3 : 4 ==> 3 : 4

The above does not work because the 6 units decrease in blue beads is not three times the 3 units decrease in red beads.

3 : 5 ==> 9 : 15
3 : 4 ==> 6 : 8

The above does not work because the 7 units decrease in blue beads is not three times the 3 units decrease in red beads.

3 : 5 ==> 12 : 20
3 : 4 ==> 9 : 12

The above does not work because the 8 units decrease in blue beads is not three times the 3 units decrease in red beads.

3 : 5 ==> 15 : 25
3 : 4 ==> 12 : 16

The above works because the 9 units decrease in blue beads is indeed three times the 3 units decrease in red beads.

We have found it. We will use 15 : 25 starting and 12 : 16 final in the units below.

3 units must correspond to the 120 beads.

3 units ==> 120 beads
1 unit ==> 40 beads
28 units ==> 1120 beads
40 units ==> 1600 beads

So, there were 1120 beads in the end and 1600 beads at first.

There is also the unit-part method to do this, but it's much more confusing and difficult so I won't cover it here.