Rainbow Harbor Cruises charges $16/adult and $8/child for a round-trip ticket. The records show that, on a certain weekend, 900 took the cruise on Saturday and 600 took the cruise on Sunday. The total receipts for Saturday were $12,800, and the total receipts for Sunday were $7,200. Determine how many adults and children took the cruise on Saturday and on Sunday.
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To solve this problem, we need to set up a system of equations. Let's denote the number of adults on Saturday as "A_s," the number of children on Saturday as "C_s," the number of adults on Sunday as "A_u," and the number of children on Sunday as "C_u."
From the given information, we know that:
1. The total number of passengers on Saturday is 900: A_s + C_s = 900.
2. The total number of passengers on Sunday is 600: A_u + C_u = 600.
3. The total receipts for Saturday are $12,800: 16A_s + 8C_s = 12,800.
4. The total receipts for Sunday are $7,200: 16A_u + 8C_u = 7,200.
To solve this system of equations, we can use substitution or elimination method. Let's use the substitution method.
From equation 1, we can express A_s in terms of C_s: A_s = 900 - C_s.
Substituting A_s = 900 - C_s into equation 3, we get: 16(900 - C_s) + 8C_s = 12,800.
Expanding and simplifying, we have: 14,400 - 16C_s + 8C_s = 12,800.
Combining like terms, we get: -8C_s = 12,800 - 14,400.
Simplifying further, we have: -8C_s = -1,600.
Dividing both sides by -8, we find: C_s = 200.
Now, substitute the value of C_s = 200 back into equation 1:
A_s + 200 = 900.
Simplifying, we have: A_s = 900 - 200.
Therefore, A_s = 700.
Using the same approach, we can find the values for the passengers on Sunday.
From equation 2, we can express A_u in terms of C_u: A_u = 600 - C_u.
Substituting A_u = 600 - C_u into equation 4, we get: 16(600 - C_u) + 8C_u = 7,200.
Expanding and simplifying, we have: 9,600 - 16C_u + 8C_u = 7,200.
Combining like terms, we get: -8C_u = 7,200 - 9,600.
Simplifying further, we have: -8C_u = -2,400.
Dividing both sides by -8, we find: C_u = 300.
Now, substitute the value of C_u = 300 back into equation 2:
A_u + 300 = 600.
Simplifying, we have: A_u = 600 - 300.
Therefore, A_u = 300.
To summarize the results:
On Saturday, there were 700 adults (A_s) and 200 children (C_s).
On Sunday, there were 300 adults (A_u) and 300 children (C_u).