In general optimization problems, you are asked to maximize or minimize some quantity, given constraints on other quantities. These quantities are all related through some equation.
Consider the following problem:
The guests at a football banquet consumed a total of 401 pounds of food. If no individual guest consumed more than 2.5 pounds of food, what is the minimum number of guests that could have attended the banquet?
You can visualize the underlying equation in the following table:
| Pounds of food per guest | × | Guests | = | Total pounds of food |
| At MOST | At LEAST | EXACTLY | ||
| 2.5 | × | ??? | = | 401 |
| maximize | minimize | constant |
Notice that finding the minimum value of the number of guests involves using the maximum pounds of food per guest, because the two quantities multiply to a constant. This sort of inversion (i.e., maximizing one thing to minimize another) is typical.
Begin by considering the extreme case in which each guest eats as much food as possible, or 2.5 pounds apiece. The corresponding number of guests at the banquet works out to 401/2.5 = 160.4 people.
However, you obviously cannot have a fractional number of guests at the banquet. Thus, the answer must be rounded. To determine whether to round up or down, consider the explicit constraint: the amount of food per guest is a maximum of 2.5 pounds per guest. Therefore, the minimum number of guests is 160.4 (if guests could be fractional), and you must round up to make the number of guests an integer: 161.
Note the careful reasoning required! Although the phrase “minimum number of guests” may tempt you to round down, you will get an incorrect answer if you do so. In general, as you solve this sort of problem, put the extreme case into the underlying equation, and solve. Then round appropriately.
If no one in a group of friends has more than $75, what is the smallest number of people who could be in the group if the group purchases a flat-screen TV that costs $1,100?