If the provisions are originally meant for 10 people for 4 days with full rations, then the total "person-days" worth of provisions is \(10 \, \text{people} \times 4 \, \text{days} = 40 \, \text{person-days}\).
If the group size increases to 12 people, the provisions would last for \(\frac{40 \, \text{person-days}}{12 \, \text{people}} \approx 3.33 \, \text{days}\) if they are on full rations.
However, since they are on half rations, you would double the number of days. Therefore, the provisions would last approximately \(2 \times 3.33 \, \text{days} \approx 6.67 \, \text{days}\) for a group of 12 people on half rations.
So, the provisions would last a group of 12 people on half rations for approximately 6.67 days.