3/4 of seats in a concert hall were occupied. The ratio of adults to children was 7:3. If 90 more adults attended the concert, the number of adults would be 3 times of number of children. How many seats were in the concert hall?
we have
a/c = 7/3
a+90 = 3c
a = 315
c = 135
but, that's only 3/4 of the seats, so the total is
4/3 (315+135) = 600
600
Well, it seems like this concert hall was a pretty popular spot! Let's break down the problem step by step.
Let's say the total number of seats in the concert hall is represented by "x".
According to the information provided, 3/4 of these seats were occupied. So, the number of occupied seats would be 3/4 of x.
Now, let's focus on the ratio of adults to children, which is 7:3. Since we know that 3/4 of the seats were occupied, we can say that 7/10 of the occupied seats were adults, and 3/10 of the occupied seats were children.
So, the number of adults attending the concert initially would be (7/10) * (3/4) * x = (21/40) * x.
According to the problem, if 90 more adults attended the concert, the number of adults would be 3 times the number of children. Let's represent the number of children as "y".
So, the new number of adults attending the concert would be (21/40) * x + 90, and the number of children would be (3/10) * (3/4) * x = (9/40) * x.
Putting the information together, we can create an equation:
(21/40) * x + 90 = 3 * (9/40) * x
Now we can solve this equation to find the value of "x", which represents the total number of seats in the concert hall.
To solve this problem, we can set up a system of equations.
Let's assume the total number of seats in the concert hall is S.
Given that 3/4 of the seats were occupied, we can express this as (3/4)S.
Next, we can determine the number of adults and children attending the concert. Let's say there are 7A adults and 3A children, where A is a constant.
According to the question, if 90 more adults attend the concert, then the number of adults will be three times the number of children, which can be expressed as:
7A + 90 = 3(3A)
Simplifying this equation, we get:
7A + 90 = 9A
90 = 2A
A = 45
Now, we can substitute the value of A back into our equations to find the total number of seats:
Seats occupied by adults: 7A = 7 * 45 = 315
Seats occupied by children: 3A = 3 * 45 = 135
Total occupied seats: 315 + 135 = 450
Since the total occupied seats was given as 3/4 of the total seats in the concert hall, we can set up the equation:
(3/4)S = 450
Now, we can solve for S:
(3/4)S = 450
Multiplying both sides by 4/3 to cancel out the fraction:
S = (450 * 4) / 3
S = 600
Therefore, the concert hall has a total of 600 seats.