Math CentralQuandaries & Queries


Question from Allen, a student:

An airplane whose capacity is 100 passengers is to be chartered for a flight to Europe. The fare is to be $\$150$ per person, if 60 people buy tickets. However, the airline agrees to reduce the fare for every passenger by $\$1$ for each additional ticket sold. How many tickets should be sold to maximize the ticket revenue for this flight?

Hi Allen,

If you are not an expert in algebra, the best way to solve this type of word problem is to first understand what’s going on by working examples.

Example 1: If 60 tickets are sold, then the revenue would be $\$150 \times 60.$
Example 2: If 61 … then $\$149 \times 61.$
Example 3: If 62 … then $\$148 \times 62.$

What do you notice about 150 + 60, 149 + 61, 148 + 62 … ?
What can you say about the largest possible value of the product $x \times y$ for two positive numbers whose sum is x + y = 210?

(Comment: It probably costs more than $\$150$ to fly to Europe if you leave from London. I wonder if there is any place from which you could fly to Europe for as little as $\$150.$)


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