math problem confused

Question

A group of children traveled with a group of parents to Silverwood Theme Park in northern Idaho. Kids admission to Silverwood costs $28 per child, adult admission is $49 per adult. There were seven more children than adults, and the total cost of admission for the whole group was $504. How many children were in the group?



Answer 1

The goal in a problem like this is to break down the words and language into math.

So if I wanted to know how much the kids cost (total), what would I need to do?

I would just multiply the cost/kid by the number of kids to get cost. Same for the parents. As the money spent on parents is no different from the money spent on kids, their units match and they can be added.

kid cost = 28*c (child)

parent cost = 49*a (adult)

Total cost = kid cost and parent cost = 504

And is equivalent to +. because of the first two expressions, we can combine things as such:

28c + 49a = 504.

But in algebra, we need the degrees of freedom (DOF) to match. In other words, we need the same number of equations as we have unknowns. We have one equation, but two unknowns: c and a.

So we need another equation or relationship. So let's keep reading. We don't know the total, but we do know there are 7 more children than adults. Sometimes in algebra, I like to use simple, but real numbers to translate.

If I have 2 adults, how many children would there be?

9, because 2+7 = 9

Well, I am not sure how many adults I have, but I have a variable that represents adults (a). So I do have a second expression/equation:

a+7 = c

Now I have two equations and two unknowns. I can substitute similar to before:

28 (a+7) +49a = 504

Now I just need to multiply out, and combine as needed:

28a + 28*7 + 49a = 504 = 77a + 196

504-196 = 77a

a = 308/77 = 44/11 = 4

Our children equation says c = a+7 so c = 11

I can always verify these answers by plugging into the original expressions.