math is hard for me

Question

On a certain hot​ summer's day,

324324

people used the public swimming pool. The daily prices are

$ 1.25$1.25

for children and

$ 2.00$2.00

for adults. The receipts for admission totaled

$ 511.50 .$511.50.  

How many children and how many adults swam at the public pool that​ day?

Answer 1

If all swimmers were children, then the receipts would total

324 * $1.25 = $405.00.

Actually, the receipts totalled $511.50, which is

$511.50 - $405.00 = $106.50 more than if all swimmers were children.

That is because some of the swimmers were adults, each paying

$2.00 - $1.25 = $0.75 more than a child.

So, with each replacement of one child with one adult, the total receipts would increase by $0.75.

So, there were

$106.50 / $0.75 = 142 replacements, adults.

Then, there actually were

324 - 142 = 182 children.

Answer: 182 children and 142 adults swam at the public pool that​ day.

Answer 2

Let C = # of children attending

A = # adults

The first equation is sort of obvious:

A + C = 324 (total # of people)

Note that this means A = 324 - C, or C = 324 - A

Now we move on to money spent.

Realize that if each adult spends $2 then 2A represents the money brought in by the adults.

Similarly for children, the amount of money brought in is 1.25C.

So the following equation models the problem:

2A + 1.25C = 511.50

Since I do not particularly like working with decimals I would multiply every term by 100 to clear this equation of decimals. Essentially you are now working with pennies:

200A + 125C = 51150

At this point there are choices to make on how to solve the system of two equations and two unknowns.

I would use the fact that A = 324 - C and substitute that expression into the 2nd equation:

200(324 - C) + 125C = 51150

64800 - 200C + 125C = 51150

64800 - 75C = 51150

64800 - 51150 = 75C

75C = 13650

C = 13650/75 = 182 (this is number of children)

A = 324 - 182 = 142 (this is number of adults)

Check:

1.25(182) = $227.50

2(142) = $284

Total: $511.50