Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

moneybird

An auditorium has a rectangular array of chairs. There are exactly 14 boys seated in each row and exactly 10 girls seated in each column. If exactly 3 chairs empty, prove that there are at least 567 chairs in the auditorium.

  • 2 years ago
  • 2 years ago

  • This Question is Closed
  1. Stom
    Best Response
    You've already chosen the best response.
    Medals 0

    lets say , there are 3 empty chairs and all in one row, that will imply that we have 14 colums and 13 rows . that would mean that we have 182 chairs, which is lesser that 567

    • 2 years ago
  2. moneybird
    Best Response
    You've already chosen the best response.
    Medals 0

    That is just an example though. The question is asking for a solid proof

    • 2 years ago
  3. joemath314159
    Best Response
    You've already chosen the best response.
    Medals 16

    I believe I have a solution. Let the auditorium have m rows and n columns. Then there will be 14m boys, and 10n girls in the auditorium. There are 3 empty seats, and m*n total seats. This gives us the equation:\[mn=14m+10n+3\Longrightarrow mn-14m-10n=3\]\[(m-10)(n-14)-140=3\Longrightarrow (m-10)(n-14)=143\]Let:\[x=m-10,y=n-14\]The only integer solutions to the equation:\[xy=143\] with x and y positive are (143,1), (1,143), (11, 13), (13,11). This will generate pairs in m and n of: (m,n) = (153, 15), (11, 157), (21, 27), (23,25). Hence the total number off seats in the auditorium is at least 567 (which is the product of 21 and 27, the smallest product of the bunch).

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.