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The problem statement:

The expenses of a tuition class are partly fixed and partly variable with the number of students.The charge is $40\$$ per head when there are $25$ students and $60\$$ per head when there are $50$ students.Find the charge per head when there are $100$ students.

I am looking for some hints/ideas about approaching this problem.

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It may be worthwhile to check whether any numbers have been transposed. – André Nicolas Aug 21 '11 at 7:04
@André Nicolas:Sorry,I don't understand. – Quixotic Aug 21 '11 at 7:18
@FoolForMath as written the problem seems unsolvable, so he is asking if you have written it incorrectly, or perhaps if we had more context we could make an assumption to allow it to be solved (as in nerdbert's answer) – Deven Ware Aug 21 '11 at 7:26
If there are fixed costs (rent?) and costs per student (food?) one would expect the charge per head to fall as the number of heads increases. The obvious model, $C=F+ an$, where $C$ is total cost, $F$ is fixed costs, $a$ additional cost per head, $n$ number of students yields, with your numbers, $F=-1000$, so negative fixed costs! – André Nicolas Aug 21 '11 at 8:05
I am pretty sure that I have copied it correctly. – Quixotic Aug 21 '11 at 8:06
up vote 2 down vote accepted

If $x$ is the number of students, and $y$ is the tuition costs, you have

$$y = ax + b$$

in which $a$ controls the variable part and $b$ controls the fixed part of the total... if you knew $a$ and $b$ you could calculate the costs for any given number of students. To find them, just use the two known values for $x$ and $y$, which gives you two equations (for your two unknowns $a$ and $b$).

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but how do you know this is a linear function with respect to the part that is "varying with the number of students"; I would read it rather as $y = f(x) + b$ where $f(x)$ is some function of $x$. – Deven Ware Aug 21 '11 at 7:13
You are right... without further information the problem as stated is unsolvable! But it looks like a question from an introductory algebra textbook, so linearity is the most likely assumption. – nerdbert Aug 21 '11 at 7:24
It's not much of a stretch to interpret "varying with the number of students" as "varying directly with the number of students," which is another way to say "proportional to the number of students." And if you don't make some stretch, there's no way to solve the problem, so presumably this is what was intended. – Gerry Myerson Aug 21 '11 at 8:07

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