Final answer:
Antoine can invite a minimum of 14 guests and a maximum of 18 guests to his banquet, based on his budget ranging from $350 to $450 and the cost of $25 per guest.
Step-by-step explanation:
To determine the minimum and maximum number of guests Antoine can invite, we first need to establish the range of his budget. Antoine's budget is $400 with a possible variance of $50. This means the minimum amount he could spend is $350 (which is $400 minus $50), and the maximum is $450 (which is $400 plus $50). Since each guest costs $25, we divide the total minimum and maximum budget by the cost per guest to find the range of guests he can invite.
The minimum number of guests is calculated by dividing the minimum budget ($350) by the cost per guest ($25), which is 14 guests.
The maximum number of guests is calculated by dividing the maximum budget ($450) by the cost per guest ($25), which is 18 guests.
Therefore, Antoine can invite between 14 and 18 guests to his banquet, depending on his final budget.