Question 1182018: Mia must buy some chocolates, roses, and ballons to sell. She has $3000 to spend on $25 cchocolates, $35 roses, and $25 ballons. She also wants to buy 100 items in total. She also wants the combined number of chocolates and balloons to be equal to that of roses. How many of each item should she buy?
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39623)   (Show Source):
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
In the response from the other tutor, they wrote down three equations in three variables directly from the information as given in the problem. Then they showed starting the process of solving the problem by playing with those equations to determine that the number of roses should be 50. (And they go no further with their response....)
That is a valid way to start on an algebraic solution to the problem. But there are much better ways!
If you read the problem and take a few seconds to analyze the given information, you will realize that "she also wants to buy 100 items in total" and "she also wants the combined number of chocolates and balloons to be equal to that of roses" together immediately tell you that the number of roses is 50 -- and also that the total number of chocolates and balloons is also 50.
The lesson there is this:
Read and understand the problem before you start writing equations!
Now the setup is easy, using a single variable:
50 = # of roses
c = # of chocolates
50-c = # of balloons
That gives us a single equation to try to solve:
50(35)+25(c)+25(50-c) = 3000
But trying to solve that equation shows that the problem is faulty:
1750+25c+1250-25c = 3000
3000 = 3000
This equation tells us nothing that we did not already know. And there is no information given that we haven't used, so we can't go any farther with the solution.
The fact that the prices of a chocolate and a balloon are the same makes it impossible to find a single answer to the problem.
In fact, from the given information, we can only tell that the number of roses must be 50 and the combined numbers of chocolates and balloons must be 50.
Answer by ikleyn(52835)  (Show Source):
You can put this solution on YOUR website! .
@josgarithmetic derived a system of 3 equation in 3 unknowns.
@greenestamps showed you that the problem does not admit a unique solution.
You may ask yourself --- how it may happen that the system of 3 equations in 3 unknowns has no a unique solution ?
The answer is: BECAUSE this system of equation is DEGENERATED : its determinant is equal to zero.
It is really rare (but still possible) case in word problems, when the system of three equations in three unknown
does not provide a unique solution ( ! )
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