The ratio of the number of adults to the number of children in an event hall was 5:7.

(a)After 60 adults and 84 children entered the event hall and no one left the event hall, what was the ratio of the number of adults to the number of children in the event hall?
(b)A while later ,56 adults and 56 children left the hall and the ratio of the number of adults to the number of children became 7:13.
How many children were there in the event hall at first?

13

(a) Before anyone entered the event hall, the ratio of the number of adults to the number of children was 5:7. After 60 adults and 84 children entered the hall, the new number of adults and children can be calculated as follows:

Number of adults = 5 + 60 = 65
Number of children = 7 + 84 = 91

Therefore, the new ratio of adults to children in the event hall is 65:91, which can be simplified to 5:7.

(b) After some time, 56 adults and 56 children left the hall. The new ratio of adults to children became 7:13. Let's assume the original number of adults and children in the hall were A and C, respectively.

After 56 adults and 56 children left, the new number of adults and children becomes:

Number of adults = A - 56
Number of children = C - 56

According to the new ratio:

(A - 56) / (C - 56) = 7 / 13

Cross-multiplying, we get:

13(A - 56) = 7(C - 56)
13A - 728 = 7C - 392

Rearranging the equation:

13A - 7C = 336

From the information given in part (a), we know that the number of adults and children in the hall satisfy the equation:

5A - 7C = 0

We can solve this system of equations by substitution or elimination to find the values of A and C.

To solve this problem, we need to use a system of equations. Let's assign variables to the unknown quantities.

Let's say the initial number of adults in the event hall is "5x", and the initial number of children is "7x".

(a) After 60 adults and 84 children entered the hall, we add 60 to the initial number of adults and 84 to the initial number of children.

The new number of adults is (5x + 60), and the new number of children is (7x + 84).

Therefore, the new ratio of adults to children is:
(5x + 60) : (7x + 84)

(b) After 56 adults and 56 children left the hall, we subtract 56 from the initial number of adults and 56 from the initial number of children.

The final number of adults is (5x - 56), and the final number of children is (7x - 56).

Given that the new ratio of adults to children is 7:13, we can set up the following equation:

(5x - 56) : (7x - 56) = 7 : 13

Now, we can solve the system of equations to find the initial number of children (7x):

Equation from part (a): (5x + 60) : (7x + 84)
Equation from part (b): (5x - 56) : (7x - 56) = 7 : 13

Solving for x will give us the initial number of children.