A lecture hall has 108 chairs arranged in rows with the same number of chairs in each row. You eliminate three rows by adding six chairs to each of the other rows. How many rows are there now ? How many chairs are in each row ?
A small theater had
6
rows of
25
chairs each. Workers just removed
9
of these chairs. How many chairs are left?
To solve this problem, let's break it down step by step.
1. Let's assume there were initially "x" rows in the lecture hall. Each row had the same number of chairs.
2. Since there were 108 chairs and "x" rows, the number of chairs in each row would be 108 divided by "x".
So, the original number of chairs in each row is 108 / x.
Now, we can move on to the next part of the problem.
3. If three rows are removed from the lecture hall, there are now (x - 3) rows remaining.
4. Additionally, six chairs are added to each of the remaining rows. Therefore, the new number of chairs in each row is the original number of chairs plus the added chairs:
(108 / x) + 6.
To find the new number of rows and the new number of chairs in each row, we need to set up an equation.
According to the given information:
(108 / x) + 6 = (108 / (x - 3))
Now, we can solve the equation:
1. Multiply both sides of the equation by x(x - 3) to eliminate the denominators:
x(x - 3) * [(108 / x) + 6] = x(x - 3) * (108 / (x - 3))
2. Simplify the equation:
108(x - 3) + 6x(x - 3) = 108x
3. Distribute and combine like terms:
108x - 324 + 6x^2 - 18x = 108x
4. Subtract 108x from both sides of the equation and rearrange the terms:
6x^2 - 18x - 108x - 324 = 0
5. Simplify the equation:
6x^2 - 126x - 324 = 0
Now we can solve this quadratic equation either by factoring or using the quadratic formula.
By factoring:
6x^2 - 126x - 324 = 0
6(x^2 - 21x - 54) = 0
6(x - 27)(x + 2) = 0
From here, we can see that the possible solutions for x are x = 27 or x = -2. However, since we are considering the number of rows, the solution can only be a positive integer. Therefore, we conclude that x = 27.
So, there are now 27 rows remaining in the lecture hall, and the new number of chairs in each row can be calculated as:
(108 / 27) + 6 = 4 + 6 = 10.
Therefore, there are now 27 rows in the lecture hall, and each row contains 10 chairs.