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.)
C (capacity) = r (rows) * s (seats per row)
r*s=C
(r+4)(s-2)=C
(r-7)(s+5)=C-1
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.
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.
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…
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.
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
You are.
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.
