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To determine how many fewer days the rations will last when additional people join the naval detachment, it’s important to calculate the total amount of rations available and how it changes with the new number of people.

Initially, the rations are sufficient to feed 16 people for 10 days. Therefore, the total amount of rations can be represented in person-days, calculated as follows:

Total person-days = Number of people × Number of days = 16 people × 10 days = 160 person-days.

With the addition of 4 people, the new total number of people becomes 20. To find out how many days the rations will last for these 20 people, we can use the total rations available (160 person-days):

New duration = Total person-days / New number of people = 160 person-days / 20 people = 8 days.

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

Original duration - New duration = 10 days - 8 days = 2 days.

Since the rations now last 8 days instead of 10, the total fewer days the rations will last due to the increase in