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.