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Question 1204971: Andre is organizing a charity event where he plans to sell two types of items: T-shirts and hats. The selling price for a T-shirt is $15, and the selling price for a hat is $10. Andre wants to sell at least 50 items in total and make a minimum of $600 in sales. If he knows that he can sell at most 40 T-shirts, and he wants to sell at least twice as many T-shirts as hats, how many of each item should he sell to meet his goals?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
The problem as presented does not have a single solution.
40 t-shirts alone makes the required $600, so if the number of t-shirts is 40, the number of hats can be anywhere from 10 (minimum number of total items is 50) to 20 (number of t-shirts is at least twice the number of hats).
And there are other solutions with 39 t-shirts, or 38, or....
It appears something is missing in the statement of the problem.
Answer by ikleyn(52824)  (Show Source):
You can put this solution on YOUR website! .
Andre is organizing a charity event where he plans to sell two types of items: T-shirts and hats.
The selling price for a T-shirt is $15, and the selling price for a hat is $10.
Andre wants to sell at least 50 items in total and make a minimum of $600 in sales.
If he knows that he can sell at most 40 T-shirts, and he wants to sell at least twice as many
T-shirts as hats, how many of each item should he sell to meet his goals?
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This problem is on solving a system of inequalities in integer non-negative numbers.
Let x be the number of T-shirt;
y be the number of hats.
The inequalities are
x + y >= 50 (1) (Andre wants to sell at least 50 items in total)
15x + 10y >= 600 (2) (Andre wants to make a minimum of $600 in sales)
x <= 40 (3) (he can sell at most 40 T-shirts_
x >= 2y (4) (he wants to sell at least twice as many T-shirts as hats)
x >= 0, y >= 0 (5) (standard non-negativity inequalities)
Next you take a graph paper and draw the corresponding lines and mark corresponding areas.
Your solution are integer points of the graph paper in the intersection of all areas (if such points do exist).
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It is "WHAT to do" and "HOW to do" to solve the problem on your own.
The problem may have more than one solution, but it is not a catastrophic event - it is very normal, if it is so.
It is a standard procedure to solve a school problem on inequalities in integer numbers in 2D plane
(= for two variables) graphically.
To facilitate your job, you can use free of charge plotting tool at web-site
www.desmos.com/calculator
In this case, you should print each and every inequality (1) - (5) in their specialized window.
The intersection of all relevant regions is the solution set, and you need integer points of the grid in the solution set.
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