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?
To solve this problem, we can set up equations based on the information given.
Let's start by defining some variables:
Let A represent the number of adults initially present in the event hall.
Let C represent the number of children initially present in the event hall.
a) After 60 adults and 84 children entered the event hall, the number of adults became A + 60, and the number of children became C + 84. The ratio of adults to children is 5:7.
So, we can set up the equation: (A + 60)/(C + 84) = 5/7.
b) After 56 adults and 56 children left the hall, the number of adults became A + 60 - 56, and the number of children became C + 84 - 56. The ratio of adults to children became 7:13.
We can set up the equation: (A + 60 - 56)/(C + 84 - 56) = 7/13.
Now we have two equations. We can solve them simultaneously to find the initial number of children, C.
Step 1: Solve equation (1) for A in terms of C:
(A + 60) = (5/7)(C + 84)
A + 60 = (5/7)C + 60(5/7)
A = (5/7)C - 60(5/7) + 60
A = (5/7)C - 300/7
Step 2: Substitute equation (2) into equation (1):
((5/7)C - 300/7) + 60)/(C + 84) = 5/7
(5/7)C - 300/7 + 60 = (5/7)(C + 84)
(5/7)C - 300/7 + 420/7 = (5/7)C + (5/7)(84)
(5/7)C + 120/7 = (5/7)C + 420/7
Step 3: Cancel out (5/7)C:
120/7 = 420/7
120 = 420
The equation is inconsistent, which means there is no solution. This implies that there is no initial number of children in the event hall that satisfies both ratios.
Therefore, it is not possible to determine how many children were initially present in the event hall based on the given information.