SOLUTION: A rectangular auditorium seats 1505 people. The number of seats in each row exceeds the number of rows by 8. Find the number of seats in each row.

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Question 818108: A rectangular auditorium seats 1505 people. The number of seats in each row exceeds the number of rows by 8. Find the number of seats in each row.
Found 4 solutions by mananth, amalm06, ikleyn, greenestamps:
Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!
seats in each row =x
number of rows = y
x=y+8
x-y=8....................(1)
xy=1505
x= 1505/y..................(2)
1505/y -y = 8
1505-y^2=8y
y^2+8y-1505=0
y^2+43y-35y-1505=0
y(y+43)-35(y+43)=0
(y+43)(y-35)=0
y=-43 OR 35
There are 35 seats in each row and 43 rows



Answer by amalm06(224)   (Show Source): You can put this solution on YOUR website!






x=35 rows
x+8=43 seats/row (Answer)
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
There is much more elegant solution which allows to avoid boring calculations and solve the problem MENTALLY. 

    Let "m" be an unknown number which is 4 more than the number of rows: m = rows +4.


    Then "m" is 4 less then the number of seats in a row: m = seats - 4.


    The product seats*rows is 1505, or   (m+4)*(m-4) = 1505.


    Hence,   = 1505 + 16 = 1521,  and  m =  = 39.


    Then the number of rows is  m-4 = 39-4 = 35  and the number of seats in each row is  m+4 = 39+4 = 43.


Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


I definitely concur with tutor ikleyn that this kind of problem is much better solved using a little logical reasoning, instead of formal mathematics. Indeed, when you try to solve it using the traditional algebraic approach, you end up having to factor a quadratic by finding two numbers whose product is 1505 and whose difference is 8. But THAT IS WHAT THE ORIGINAL PROBLEM asks you to do.

So the formal algebra is a waste of time....

And the "trick" tutor ikleyn uses is a very useful one. If you know that

then you know that

and so the answers are 39-4=35 and 39+4=43.

It did not occur to me to use that trick on this problem. I just started playing with numbers to find a pair with a product of 1505 and a difference of 8. There weren't many possibilities....

I of course first saw


Then I had to find a factorization of 301; when I found it, I had my answer.

, so .

Answer: 35 rows of 43 seats each.
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