# Theater seating math, where to start **URL:** \ **Category:** Factual Questions\ **Created:** [December 19, 2008, 1:37pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898 "2008-12-19T13:37:00Z")\ **Posts on this page:** 10\ **Page:** 1
**Author:** ![dauerbach](https://avatars.discourse-cdn.com/v4/letter/d/c2a13f/32.png) [@dauerbach](https://boards.straightdope.com/u/dauerbach)\ **Post date:** [December 19, 2008, 1:37pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/1 "2008-12-19T13:37:00Z")
Another case where I can’t even figure out where to start. Any helpful hints? The owners of a theater plan to reconfigure theseating. Adding 4 rows, they can reduce the number of seats per row by 2 without changing capacity. If, instead, they add 5 seats per row, they can reduce the number of rows by 7, thus decreasing capacity by 1 seat. What is the theater’s current capacity? (Assume that each row has the same number of seats.) ---
**Author:** ![muttrox](https://avatars.discourse-cdn.com/v4/letter/m/a8b319/32.png) [@muttrox](https://boards.straightdope.com/u/muttrox)\ **Post date:** [December 19, 2008, 1:54pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/2 "2008-12-19T13:54:40Z")
C (capacity) = r (rows) \* s (seats per row) r\*s=C (r+4)(s-2)=C (r-7)(s+5)=C-1 ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [December 19, 2008, 1:56pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/3 "2008-12-19T13:56:40Z")
Introduce two variables for the current number of rows and the current number of seats per row. The two given scenarios (adding rows so as to give people more elbow-room, and squishing seats together so as to give people more leg room) provide a system of equations that can be solved for the two unknowns. Once the two variables are determined, their product is the current theater capacity. ---
**Author:** ![dauerbach](https://avatars.discourse-cdn.com/v4/letter/d/c2a13f/32.png) [@dauerbach](https://boards.straightdope.com/u/dauerbach)\ **Post date:** [December 19, 2008, 2:38pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/4 "2008-12-19T14:38:33Z")
Ok, well I tried to solve the two equations using the subtraction method and ended up with s=7r-26/11 for the substitution value. That is not the correct answer. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [December 19, 2008, 3:13pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/5 "2008-12-19T15:13:26Z")
I come up with 7r - 11s = 26 Solving that gives me r = 21, s = 11. Unfortunately that doesn’t work when you plug the numbers back in: Original capacity = 21 rows x 11 seats = 231 Add four rows, remove 2 seats from each = 25 rows x 9 seats = 225 Remove 7 rows, add 5 seats to each = 14 rows x 16 seats = 224 Not quite sure where I’ve gone wrong… ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 19, 2008, 4:02pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/6 "2008-12-19T16:02:28Z")
> [@muttrox](#): > > C (capacity) = r (rows) \* s (seats per row) > r\*s=C > (r+4)(s-2)=C > (r-7)(s+5)=C-1 If you re-write the second two equations as: (r+4)(s-2)=rs (r-7)(s+5)=rs-1 you can reduce these to: -2r+4s-8=0 5r-7s-35=-1 which quickly yields the answer ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [December 19, 2008, 6:45pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/7 "2008-12-19T18:45:00Z")
> [@Xema](#): > > If you re-write the second two equations as: > (r+4)(s-2)=rs > (r-7)(s+5)=rs-1 > > you can reduce these to: > -2r+4s-8=0 > 5r-7s-35=-1 > > which quickly yields the answer Yep, that’s the way I did it. ---
**Author:** ![RedSwinglineOne](https://avatars.discourse-cdn.com/v4/letter/r/a9adbd/32.png) [@RedSwinglineOne](https://boards.straightdope.com/u/RedSwinglineOne)\ **Post date:** [December 19, 2008, 7:51pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/8 "2008-12-19T19:51:36Z")
> [@Xema](#): > > If you re-write the second two equations as: > (r+4)(s-2)=rs > (r-7)(s+5)=rs-1 > > you can reduce these to: > -2r+4s-8=0 > 5r-7s-35=-1 > > which quickly yields the answer Less quickly for me, its been a few years… So am I correct..? -r+2s=4 5r-7s=34 multiply the first by 5… -5r+10s=20 added to 5r-7s=34 3s=54 s=18 so r=32. **C=576** 32x18=576 36x16=576 25x23=575 ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 19, 2008, 7:55pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/9 "2008-12-19T19:55:06Z")
> [@RedSwinglineOne](#): > > So am I correct..? You are. ---
**Author:** ![RedSwinglineOne](https://avatars.discourse-cdn.com/v4/letter/r/a9adbd/32.png) [@RedSwinglineOne](https://boards.straightdope.com/u/RedSwinglineOne)\ **Post date:** [December 19, 2008, 8:40pm UTC](https://boards.straightdope.com/t/theater-seating-math-where-to-start/477898/10 "2008-12-19T20:40:43Z")
I realized only after I posted that the OP was only asking for clues so I’m sorry if I gave anything away. I should have figured that out when others purposely stopped short of the answer. I was just happy (and slightly surprised) to have cleared 20+ years of brain cobwebs to remember my high school math. 🙂