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To understand how the rations work when more people join the detachment, let’s break this down. Originally, the detachment has rations to feed 15 people for 8 days. Therefore, the total amount of food can be calculated as follows:

Total food = 15 people × 8 days = 120 "person-days" of food.

This means the rations can sustain one person for 120 days, or 120 people for one day, or any combination thereof that equals 120.

Now, when 5 more people join, the total number of people becomes 20. To find out how many days the rations would last for these 20 people, we divide the total food available (120 person-days) by the number of people:

Days = Total food / Number of people

Days = 120 person-days / 20 people = 6 days.

Now, to find out how many fewer days the rations last compared to the original 8 days, we subtract the new duration from the original duration:

Fewer days = Original days - New days

Fewer days = 8 days - 6 days = 2 days.

Therefore, the rations will last for 2 fewer days