Question

Some students planned for a get-together. The budget for food was $500. Five of the students failed to come because of the distance, and therefore the cost of food for each member increased by $5. How many students attended the get-together?

Answer 1

i think you are supposed to solve as
\[\frac{500}{x}+5=\frac{500}{x-5}\] and reason as follows
the pricer per person originally was
\[\frac{500}{x}\] when 5 people dropped out there were
\[x-5 \] people at a cost per person of
\[\frac{500}{x-5}\] but it was $5 more so you get the equation i wrote above

Answer 2

you still have to solve the quadratic equation. and there might be a simpler way to arrive at it without the stupid variable in the denominator, but i don't see it.
btw the answer is 25

Answer 3

I don't see a faster way than satellite's solution. Nicely done.

Answer 4

Let's see... if x is the number of students... the cost per student would be 500/x. The new cost per student would be (500/x) + 5. We could also say that 500/(x-5) is the new cost. If both those are ways of expressing the new cost, they should be equal:
(500/x) + 5 = 500/(x-5)
so you would multiply both sides by the two denominators (x)(x-5)

On the left, the x would cancel from the first term, so you would have 500(x-5) + 5(x)(x-5)

On the right, the x-5 would cancel, so you would have 500(x)

So, \[5x ^{2} + 500x -25x - 2500 = 500x\]
and yay, the 500x cancels out:
\[5x ^{2}- 25x - 2500 = 0\]
and you can divide out a 5, so:
\[x ^{2} - 5x - 500 = 0\]
and that factors to (x-25)(x+20) = 0
Since negative 20 doesn't make sense as the original number of students, it must be 25, but that isn't the answer, because 5 of them didn't show up. The number of students who attended was 20. That doesn't make either original denominator zero, so we're good!

Answer 5

Oops, I started out by defining x as the number of students who came, so 25 IS the answer.

Answer 6

(500/x)+5=500/(x-5)
(500+5x)/x=500/(x-5) cross multiply.
(x-5)(500+5x)=500x
500x-2500+5x^2-25x=500x
5x^2-25x-2500=0
5(x^2-5x-500)=0
5(x-25)(x=20=0
x-25=0
x=25
so 25-5=20.
20 is the actual number who attended

so does that look right