An ice cream shop makes $1.25 on each small cone and $2.25 on each large cone. The ice cream shop sells between 60 and 80 small cones and between 120 and 150 large cones in one day. If it costs the owners $350 per day to run the shop, how many of each cone do they need to sell to make a profit each day?

Let x and y represent the numbers of small and large cones sold daily, respectively 60 ≤ x ≤ 80 120 ≤ y ≤ 150 1.25x + 2.25y > 350

Step-by-step explanation:

To answer the question, we need to be able to find the number of each type of cone that needs to be sold. For the purpose, it is convenient to define variables that represent those numbers. We have chosen to use the variables x and y to represent the number of small cones and the number of large cones, respectively ("Let" statement in above answer). One could use "s" and "l" as being more mnemonic, but "l" can be confused with a variety of other symbols, so we like not to use it.

The problem statement puts limits on the numbers of small and large cones sold, so our system of inequalities needs to reflect those limits. (We suspect these are not hard limits, but represent historical data. We doubt the shop would decline to sell more, and we can conceive of conditions under which they might sell fewer.)

The heart of the matter is that the profit from each type of cone must total more than $350 (per day). The problem statement requires the shop make a profit, not just break even, so we use the > symbol for profit, instead of ≥.

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See above for the "Let" statement and the system of inequalities. (This problem does not require we solve them.)