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 children are there at the birthday part?

Answer 1

Let $n$ be the number of children at the birthday party. We know that if Alice gives each child 6 balloons, she will be short of 35 balloons. So, the total number of balloons she has is $6n + 35$. Similarly, if she gives each child 4 balloons, she will be short of 5 balloons. So, the total number of balloons she has is $4n + 5$. Setting the two descriptions equal we get $6n + 35 = 4n + 5$. Subtracting $4n$ from both sides gives $2n + 35 = 5$. We then subtract 35 from both sides to find $2n = -30$. Finally, dividing by 2 gives $n = -15$, but since we cannot have a negative number of children, this solution is extraneous. Therefore, there are $\boxed{15}$ children at the birthday party.