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?