Given a tempo marking in bpm (beats-per-minute), how can I work out the length of a crotchet (quarter-note) beat?
To find the length in seconds of each beat for any given metronome marking in beats-per-minute (bpm), you would divide 60 (the number of seconds in a minute) by the bpm marking. For instance, if a piece has a metronome marking of crotchet (quarter-note) = 120, each crotchet beat is 0.5 seconds long (60/120).
You can follow this simple rule to find the length of a crotchet whenever the tempo marking shows how many crotchet beats per minute there are. This will usually be for time-signatures with crotchet beats (i.e. with a 4 at the bottom, eg. 4/4, 3/4, 2/4 etc.)
However, if you wanted to find the length of a crotchet beat where the tempo marking shows the bpm of a different note value, you need to do an extra calculation. For instance, a time-signature of 3/8, tells you that there are 3 quaver (eighth) beats per bar. So, this would probably have a tempo marking showing the number of eighth beats per minute. You could follow the calculation above to find the length of each quaver beat, but would then need to double it to find the length of a crotchet beat, as a crotchet is twice the length of a quaver.
Of course, other types of beat can be used for a tempo marking. To continue the 3/8 example above, this could be shown with how many dotted crotchet beats there are per minute. (In other words, how many whole-bar, but not whole-note, beats there are per minute.) In this case, a crotchet is two-thirds the length of a dotted-crotchet, so each crotchet beat is two-thirds the length in seconds you would arrive at following the initial calculation.