Number of seats in each row

Question

A rectangular auditorium seats 2244 people. The number of seats in each row exceeds the number of rows by 7. Find the number of seats in each row. 

Answer 1

There are 2244 seats = R (rows) * S (seats per row)

   and R = S - 7    (written this way because we want to know S)

 

  (S-7)*S = 2244

  S² - 7S - 2244 = 0

 

Using the quadratic formula,

   S = (7 + SQRT(49+8976))/2   or   (7 - SQRT(49+8976))/2

   S =      51                              or     (-44)                 can't be negative

 

Checking:

    Is  (51-7)*51 = 2244      yes

 

note: we actually never need to find R, so we save that step.

Answer 2

We have a quadratic equation....

Let r= # rows

let s = # seats per row

r(s) = 2244

 

We know that s= r + 7 so substitute that in place of s

 

r(r+7)=2244

r²+7r=2244

r² + 7r - 2244=0

 

The factors of -2244 that add to 7 are 51(-44) so we factor this to...

(r+51)(r-44)=0

so r = -51 or r = 44

 

since we can't jave a negative number of rows we discard the extraneous answer ...  so r = 44

 

We have 44 rows each with s= r+7 or  44+7 = 51 seats per row

Answer 3

Let's say there are x rows.

 

This means, no. of seats in each row = x + 7

 

Therefore,

total seats = no. of rows * no. of seats in each row

2244 = x(x+7)

x2 + 7x -2244 = 0

x2 + 51x -44x -2244 = 0

x(x + 51) -44(x + 51) = 0

(x+51) (x-44) = 0

x = -51 or x = 44

 

Since no. of rows can not be negative, we disregard x = -51

x = no. of rows = 44

No. of seats in each row = x + 7 = 51

Answer 4

We can think of  this as the rectangle's dimensions according to seats as opposed to a formal defined length like feet.

Here's what we know:

A = lw

A = 2244

l = w + 7

 

2244 = (w + 7)w

2244 = w² + 7w

0 = w² + 7w - 2244

 

When we factor this out, we're going to need to have the outer values be a difference of 7 (to get that middle term)

2244 = 3 * 4 * 11 * 17

3 * 17 = 51

4 * 11 = 44

 

51 * 44 = 2244 (confirming what we already knew from factoring it out earlier.)

51 - 44 = 7

 

Since we have a positive 7w, the larger value will be positive in our factorization:

0 = (w + 51)(w - 44)

w = -51, 44

We can't have a negative number of seats, so:

w = 44

 

The number of seats per row is w + 7.

44 + 7 = 51

51 seat per row. 

Answer 5

Solve for x:

 

x = 2244/(x+7);  guess x; try x = 50 to start:

 

 

                                                                                          |check: ans.                    
2244 div.by (      guess      plus 7 )   = possible ans.   |must be …
2244 div.by (         50        plus 7 )    = 39.36842105   |more than 39 and less than 50
2244 div.by ( 39.36842105 plus 7 ) = 48.39500568  |> 39.36842105 < 48.39500568
2244 div.by ( 48.39500568 plus 7 ) = 40.50906707  |> 40.50906707 < 48.39500568
2244 div.by ( 40.50906707 plus 7 ) = 47.23308915  |> 40.50906707 < 47.23308915
2244 div.by ( 47.23308915 plus 7 ) = 41.37695336  |> 41.37695336 < 47.23308915
2244 div.by ( 41.37695336 plus 7 ) = 46.38572387  |> 41.37695336 < 46.38572387
2244 div.by ( 46.38572387 plus 7 ) = 42.03370934  |> 42.03370934 < 46.38572387
2244 div.by ( 42.03370934 plus 7 ) = 45.76443492  |> 42.03370934 < 45.76443492
2244 div.by ( 45.76443492 plus 7 ) = 42.52864649  |> 42.52864649 < 45.76443492
2244 div.by ( 42.52864649 plus 7 ) = 45.30711334  |> 42.52864649 < 45.30711334
2244 div.by ( 45.30711334 plus 7 ) = 42.90047485  |> 42.90047485 < 45.30711334
2244 div.by ( 42.90047485 plus 7 ) = 44.96951195  |> 42.90047485 < 44.96951195
2244 div.by ( 44.96951195 plus 7 ) = 43.17916247  |> 43.17916247 < 44.96951195
2244 div.by ( 43.17916247 plus 7 ) = 44.71975796  |> 43.17916247 < 44.71975796

 

Obviously x = 44 and x + 7 = 51.  Easy.  And I only had to come back for about 10 updates to line things up.