# Intuition behind this algebraic question

1. Apr 5, 2009

### 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 Code (Text): 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? 2. Apr 5, 2009 ### qntty It looks like you've got it, what do you need help with? 3. Apr 5, 2009 ### 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? Last edited: Apr 5, 2009 4. Apr 6, 2009 ### 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 5. Apr 6, 2009 ### 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?

6. Apr 6, 2009

### Gear300

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

7. Apr 6, 2009

### lifelearner

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

8. Apr 6, 2009

### 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.

9. Apr 6, 2009

### lifelearner

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