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?
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.
Let's solve this step-by-step.
Let's assume the number of children at the birthday party is "x."
According to the first condition, if she gives each child 6 balloons, Alice will be short of 35 balloons. This can be represented as:
6x - 35 = Number of balloons Alice has
Similarly, according to the second condition, if she gives each child 4 balloons, Alice will be short of 5 balloons. This can be represented as:
4x - 5 = Number of balloons Alice has
Now, we have a system of two equations:
Equation 1: 6x - 35 = Number of balloons Alice has
Equation 2: 4x - 5 = Number of balloons Alice has
To find the number of children, we need to solve these equations simultaneously.
Equation 1:
6x - 35 = Number of balloons Alice has
Multiply both sides by -1:
-6x + 35 = -Number of balloons Alice has
Equation 2:
4x - 5 = Number of balloons Alice has
Now, we can set both equations equal to each other:
-6x + 35 = 4x - 5
Simplifying the equation:
-6x - 4x = -5 - 35
-10x = -40
Divide both sides by -10:
x = -40 / -10
x = 4
Therefore, there are 4 children at the birthday party.