<h2>Explanation:</h2><h3>Part A.</h3>
You are asked to define variables that "model the situation" without being told what you care about in the situation. For that purpose, the variables you have chosen, ...
- d = number of coins
- q = their total value
are completely appropriate. Your choice will give rise to two equations, already solved:
- d = 90 . . . . . . . . the equation for the amount of coins
- q = 17.55 . . . . . . the equation for the value of the coins
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However, that is not going to be helpful for Part B.
In Part B, you're asked for the <em>number of quarters</em>. You might expect that choosing a variable to represent this quantity would be helpful, and you'd be right. For a variety of reasons, choosing another variable to represent the <em>number of dimes</em> is also helpful. Now, we have the definitions
- let q = the number of quarters in Dylan's bank
- let d = the number of dimes in Dylan's bank
The corresponding system of equations is ...
- q + d = 90 . . . . . the number of coins Dylan has
- 0.25q +0.10d = 17.55 . . . . the value of the coins Dylan has
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<h3>Part B.</h3>
Since you're only interested in the number of quarters, it can be convenient to choose a solution method that focuses only on the q variable. Here, we'll multiply the first equation by -0.10 and add the result to the second equation. That eliminates d and gives an equation only in q.
-0.10(q +d) +(0.25q +0.10d) = -0.10(90) +(17.55)
0.15q = 8.55 . . . . . simplify
8.55/0.15 = q = 57 . . . . . . divide by the coefficient of q to find the solution
Dylan has 57 quarters in his bank.
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<em>Comment on "this situation" problems</em>
As you have discovered, modeling a particular "situation" needs to be informed by the aspect(s) of the situation that you care about. Many modern math texts ask for a model of "this situation" without bothering to tell you that crucial piece of information.
I find it unfortunate, and not particularly instructive in good critical thinking practice or problem solving technique.
Here, we found it helpful to read ahead for clues as to what the problem is really about. That will often be a helpful approach.
