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Chapter 2: Problem 38

Rainbow Harbor Cruises charges \$16/adult and \(\$ 8 /\) child for a round-trip ticket. The records show that, on a certain weekend, 1000 people took the cruise on Saturday and 800 people took the cruise on Sunday. The total receipts for Saturday were \(\$ 12,800\) and the total receipts for Sunday were \(\$ 9,600\). Determine how many adults and children took the cruise on Saturday and on Sunday.

Short Answer

Expert verified
In conclusion, on Saturday there were 600 adults and 400 children, and on Sunday, there were 400 adults and 400 children who took the Rainbow Harbor Cruises.

Step by step solution

01

Saturday: Set up the system of equations

Let A be the number of adults and C be the number of children who took the cruise on Saturday. We have the following information: 1. The total number of people on Saturday is 1000, so A + C = 1000. 2. The ticket prices are \(16 for adults and \)8 for children, amounting to a total receipt of $12,800 for Saturday. Therefore, 16A + 8C = 12800. We now have a system of equations: \( \begin{cases} A + C = 1000 \\ 16A + 8C = 12800 \end{cases} \)
02

Saturday: Solve the system of equations

We can solve the system of equations by using substitution or elimination. Let's use the elimination method. Multiply the first equation by -8: -8(A + C) = -8(1000) -8A - 8C = -8000 Now, add this to the second equation to eliminate C: (16A + 8C) + (-8A - 8C) = 12800 + (-8000) 8A = 4800 Divide both sides by 8: A = 600 Now, substitute the value of A back into the first equation to find the value of C: 600 + C = 1000 C = 400 So, there were 600 adults and 400 children on Saturday.
03

Sunday: Set up the system of equations

Let X be the number of adults and Y be the number of children who took the cruise on Sunday. We have the following information: 1. The total number of people on Sunday is 800, so X + Y = 800. 2. The ticket prices are \(16 for adults and \)8 for children, amounting to a total receipt of $9,600 for Sunday. Therefore, 16X + 8Y = 9600. We now have a system of equations: \( \begin{cases} X + Y = 800 \\ 16X + 8Y = 9600 \end{cases} \)
04

Sunday: Solve the system of equations

Again, let's use the elimination method. Multiply the first equation by -8: -8(X + Y) = -8(800) -8X - 8Y = -6400 Now, add this to the second equation to eliminate Y: (16X + 8Y) + (-8X - 8Y) = 9600 + (-6400) 8X = 3200 Divide both sides by 8: X = 400 Now, substitute the value of X back into the first equation to find the value of Y: 400 + Y = 800 Y = 400 So, there were 400 adults and 400 children on Sunday. In conclusion, on Saturday there were 600 adults and 400 children, and on Sunday, there were 400 adults and 400 children who took the Rainbow Harbor Cruises.

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