Question 1125350: A theater can seat 270 people. The number of rows is 3 less than the number of seats in each row. How many rows of seats are there? Found 3 solutions by solver91311, ikleyn, greenestamps:Answer by solver91311(24713) (Show Source):
Let represent the number of rows. Then must be the number of seats in a row. The total number of seats is the product of the number of rows and the number of seats in a row:
Solve for the positive root of the quadratic.
John
My calculator said it, I believe it, that settles it
We need to find a decomposition of the number 270 into the product of two positive integer numbers that differ by 3.
In addition, these factors should be close enough to , which is about 17.
Having these two hints, it is the matter of seconds to find the decomposition
270 = 15*18.
Answer. The number of rows is 15.
Note from the other two responses you have received that formal algebra is not always the way to solve a problem.
In the algebraic solution outlined by the first tutor, the algebra leads you to where you need to finish solving the problem by finding two numbers whose product is 270 and whose difference is 3.
But that is what you need to do in the original problem! So the algebra didn't get you any closer to the solution.
The response from the other tutor shows a solution method that is much better because it uses logical reasoning -- which is excellent brain exercise.
The two numbers you are looking for differ by 3, and their product is 270. So the two numbers will be close to the square root of 270, which is between 16 and 17.
And so the most likely numbers are 15 and 18 -- and they work.
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note to tutor ikleyn
in your discussion of the problem you at one point say the product is 330 instead of 270; you might want to fix that
