Intuition behind this algebraic question

[lifelearner]

Hi PF members!

I'd like some guidance with the intuition behind the following problem. I have provided my intuition as far as I can below but if anyone can help clarify it I'd be very thankful.

Question: Tickets to a concert cost $9.00 for adults and $6.50 for students. A total of 950 people paid $7550 to attend. How many students attended the concert?

I. The Algebra

The answer basically involves solving two equations with two unknowns:

s = students, a = adults

i) a + s = 950 and 9a + 6.5s = 7550

ii) a = 950 - s

iii) sub (ii) into 9a in (i)

iv) 8550 - 7550 = 2.5s

v) s = 400 (?)

II. My intuition (logic) behind the algebra


i) total # of people who came (students and adults) equals 950. Revenue from adults is ($9 per adult)*(total number of adults = a). Revenue from students is ($6.50 per student)*(total number of students = s)

ii) if in total 950 people came and assuming we know how many students came then those that are not students must be adults. In other words, a = 950 - s

iii) 9(950-s) + 6.5(s) = 8550 - 9s + 6.5s = 7550

That is, assume that we charge all 950 people $9. This revenue comes to $8550. Then we take out the students to whom we charged $9, whom are -9s. Then we will be left on one hand with the adults to whom we charged $9 (8550 - 9s). Plus we add the students to whom we charge $6.5 (6.5s). Sum will equal the actual revenue of $7550.


iv) 8550-7550 = 1000 = 2.5s

The revenue that comes from charging students AND adults $9 minus the actual revenue of $7550 from charging ONLY adults $9 gives $1000, which equals the extra $2.5 that all students WOULD'VE paid HAD we charged them $9. How many such students are there? 1000/2.5 = 400.


So, I need your help with iv in part II. How do I interpret equation iv? And that, too, logically?

 

[qntty]

Hi PF members!
iv) 8550-7550 = 1000 = 2.5s

The revenue that comes from charging students AND adults $9 minus the actual revenue of $7550 from charging ONLY adults $9 gives $1000, which equals the extra $2.5 that all students WOULD'VE paid HAD we charged them $9. How many such students are there? 1000/2.5 = 400.

It looks like you've got it, what do you need help with?

 

[lifelearner]

How do you interpret in your words 2.5s = 1000?

That is, within the context of this problem, how do you interpret $2.5 per student times some number of students equals $1000?

 

[Kaimyn]

lifelearner,

Am I correct in assuming that you did not find the algebra, but are rather looking at the logic behing the algebra? If so:

Part iv has skipped a couple of steps to get where it is. Basic algebra. Looking at step iii we get:

a = 950 - s
This can be altered to get,
9a = 8550 - 9s
Then substituting into equation i,
8550 - 9s + 6.5s = 7550
Simplify,
8550 - 2.5s = 7550
8550 = 7550 + 2.5s
8550 - 7550 = 2.5s
2.5s = 1000
s = 400

To find a,
a = 950 - s
a = 950 - 400
a = 550

And checking,
9a + 6.5s = 7550
9*550 + 6.5*400 = 7550
4950 + 2600 = 7550


So there you go, the 2.5s merely arises through algebra.

Now, if you did find this already, and you worked out the algebra... I don't know what the problem is

 

[lifelearner]

I understand completely the algebra. However, the physical meaning of 2.5s is what I don't understand. Somehow I feel Venn diagrams might be useful; i.e: 8550 is revenue you get if you charge all persons $9. When you subtract 9s then you effectively "filter" out the students whom you charge $9, which will leave you with revenue from adults, each of whom paid $9.

I understand 2.5s arises through algebra but can we put some meaning behind it?

 

[Gear300]

I understand completely the algebra. However, the physical meaning of 2.5s is what I don't understand. Somehow I feel Venn diagrams might be useful; i.e: 8550 is revenue you get if you charge all persons $9. When you subtract 9s then you effectively "filter" out the students whom you charge $9, which will leave you with revenue from adults, each of whom paid $9.

I understand 2.5s arises through algebra but can we put some meaning behind it?

I suppose the meaning might be 2.5 times the number of students is 1000?

 

[lifelearner]

What is the 1000 mean in physical meaning? And of what significance is 2.5 times # of students?

 

[arildno]

Why should it mean anything at all, other than being provably logically equivalent to a previous statement?

You could multiply that equation with, say, 3.72 and you'd still have a logically equivalent equation, but it wouldn't have much "meaning" in the sense you are seeking.

 

[lifelearner]

Why should it mean anything at all, other than being provably logically equivalent to a previous statement?

You could multiply that equation with, say, 3.72 and you'd still have a logically equivalent equation, but it wouldn't have much "meaning" in the sense you are seeking.

Should it not mean something as we are dealing with a practical question?