Some Territorial Army battalions were split to form cadres for new units. In some regiments these were simply sequentially numbered (typically 6th and 7th) but in others they were given a number which showed their ancestry i.e. if the 5th King's Own was split the two resulting battalions were numbered 1/5th and 2/5th.
2nd/5th Bn King's Own was a second line Territorial Force battalion. During WW1 the Territorial Force was expanded each infantry battalion forming two additonal battalions. The 5th Bn King's Own became 1st/5th Bn King's Own and the two new battalions were 2nd/5th and 3rd/5th. 2nd/5th was a front line battalion and the number of Territorial Force divisions was doubled to provide the command structure for these additional battalions. 3rd/5th was a training battalion providing trained drafts for the two front line battalions. The following link outlines the WW1 history of these and other King's Own battalions.
As the TF battalions mobilised and marched off to war they then recruited their own replacement battalions thus the 4th Battalion marched off to war as the 1/4th while its replacement became the 2/4th Battalion. This process was repeated in a number of cases thus giving rise to a 3/4th Battalion.
Other battalions raised for the war were known as service battalions. These took their numbers immediately after the original TF battalions. So using The Hampshire Regiment as an example again it ended up looking like this:
3/4th Battalion (TF) - Later 4th (Reserve) Battalion
3/5th Battalion (TF) - Later 5th (Reserve) Battalion
No 3/6th battalion was raised
3/7th Battalion - Later 7th (Reserve) Battalion
No 3/8th battalion was raised
3/9th (Cyclist) Battalion (TF)
17th Battalion (TF) - This was made up of TF men from the other TF battalions who were unable to serve overseas
10th (Service) Battalion
11th (Service) Battalion (Pioneers)
12th (Service) Battalion
13th (Reserve) Battalion - Later became 34th Training Reserve Battalion
14th (Service) Battalion (1st Portsmouth)
15th (Service) Battalion (2nd Portsmouth) - Later became 15th (Hampshire Yeomanry) Battalion*
16th (Reserve) Battalion - Later became 18th (Home Service) Battalion
1st (Garrison) Battalion - Later became 19th (Garrison) Battalion
Late in the war the 1/1st Hampshire Carabiner Yeomany (the TF cavalry) was absorbed into the infantry with most of them going to the 15th Battalion, hence the name change.
Hope this makes things clear for you.
Now just to make things interesting. During the Second World War the 4th and 5th Battalions did the same thing. However, during this war and later most double numbering was the result of amalgamation of battalions, for example 6th/7th QUEENS was the result of two TA battalions merging.
Finally, of course there are also numbered regiments which have more then one battalion. In this example we have: 1st/2nd and 2nd/2nd Battalions being the 1st and 2nd Battalions of the 2nd King Edward VII's Own Gurkha Rifles.
Hope this makes sense. If you need any more PM me.
Many thanks for the replies, as an aside, my interest was sparked by a King's Own CWGC grave in Rivington Churchyard, oddly (to me) without King's Crown on capbadge. I am also humbled by the sheer number of Battalions raised given the population of the UK at the time.