Alice has some balloons to distribute to some children at a birthday party. If she gives each child 6 balloons, she will be short of 35 balloons. If she gives each child 4 balloons she will be short of 5 balloons. How many balloons does Alice have?

Answer 1

Let's assume Alice has x balloons.
If she gives each child 6 balloons, she will be short of 35 balloons, so she gives balloons to (x-35)/6 children.
And if she gives 4 balloons to each child, she will be short of 5 balloons, so she gives balloons to (x-5)/4 children.
Since she is giving the same number of balloons, we can set these two equations as equal to each other, so (x-35)/6 = (x-5)/4.
Multiplying both sides of the equation by 6*4=24 we get 24(x-35)=6(x-5).
Opening the brackets we get 24x-840=6x-30.
Subtracting 6x from both sides we get 18x-840=-30.
Adding 840 to both sides we get 18x=810.
Dividing both sides by 18 we get x = 45. Answer: \boxed{45}.