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A boy named Bechton would like to celebrate his birthday at a private beach. He has saved $120 for admission to the beach. Admission price is $6 for adults (x) and $3 for children (y). The beach has a rule that for large parties there must be one adult (x) for every two children (y). Explain how you would use the following strategies to help Bechton invite enough people to spend exactly $120 for admission and maintain the proper ratio of children and adults. Explain your work.*

*Write the equations you would use to figure out how many people to invite.

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Answer: To help Bechton invite enough people to spend exactly $120 for admission while maintaining the proper ratio of children and adults, we need to use a system of equations.

Let's assume Bechton invites x adults and y children to the private beach.

Strategy 1: Total Cost Strategy

Since Bechton has saved $120 for admission, the total cost of inviting x adults and y children should be equal to $120.

The equation for the total cost (C) would be:

C = 6x + 3y

Strategy 2: Proper Ratio Strategy

The beach has a rule that for large parties, there must be one adult for every two children. This means the ratio of adults to children should be 1:2.

The equation for the proper ratio would be:

x = 2y

Now, we have a system of two equations:

C = 6x + 3y

x = 2y

We can now solve this system of equations to find the values of x and y (the number of adults and children Bechton should invite).

Step 1: Substitute the value of x from equation (2) into equation (1):

C = 6(2y) + 3y

Step 2: Simplify the equation:

C = 12y + 3y

C = 15y

Step 3: Now, we know that C (total cost) should be $120, so we can set up the equation:

15y = 120

Step 4: Solve for y:

y = 120 / 15

y = 8

Step 5: Now that we have the value of y, we can find the value of x using equation (2):

x = 2y

x = 2 * 8

x = 16

So, Bechton should invite 16 adults (x) and 8 children (y) to spend exactly $120 for admission, maintaining the proper ratio of one adult for every two children.

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