Algebra and reasoning

Question

some students on a cross country team attend an outdoor adventure activity with chaperones. the admission cost is $10 for each student and $15 for each chaperone. three students are assigned to each chaperone. there is also a one-time team registration fee of $20. the total cost for this activity is $200.

Part A: write an equation to determine x, the number of students who attend the activity.

Part B: solve the equation you wrote for part A, justifying each step in the process

Answer 1

Part A:

Let x be the number of students and y be the number of chaperones.

The total cost of $200, including the registration fee is:

10x + 15y + 20 = 200

Part B:

Since 3 students are assigned to each chaperone,

x = 3y or y = x/3

Which can be substituted into the first equation and subtract $20 from both sides of the equation.

10x + 15(x/3) = 180

10x +5x=180

15x = 180

Dividing both sides by 15:

x = 180/15 = 12 students

Answer 2

If x represents the number of students on the team, then x/3 represents the number of required chaperones.

For example if there are 3 students, then only one chaperone.

If 6 students then 6/3 means 2 chaperones.

The total cost is:

$10 for each student = 10x

$15 for each chaperone = 15(x/3)

$20 for one time team fee.

That cost is the sum of each of the above, but we know that cost is $200.

10x + 15/(x/3) + 20 = 200 (since 15(x/3) = 15x/3 = 5x, the equation can be written as):

10x + 5x + 20 = 200

To solve:

15x + 20 - 200

15x = 180

x = 180/15 = 12 students

Check:

12 students @$10 = $120

12/3 = 4 chaperones @ $15 = $60

$20 team fee:

120 + 60 + 20 = $200.

Answer 3

10x + 15(x/3) + 20 = 200

15x = 180

x = 12

There were 12 students and 4 chaperones.

Answer 4

Available information:

Student admission cost = $10

Chaperone admission cost = $15

One time registration free = $20

Total cost = $200

x = number of students

Putting our equation together:

10x because $10 per student (x).

15(1/3)x because $15 per chaperone but there's only 1 chaperone for every 3 students, or there's 1/3 chaperone for every student (x). In other words, chaperone = (1/3)x. You pay $15 per chaperone so 15(1/3)x.

+20 because there's a one time $20 fee.

End equation is 200 = 10x + 15(1/3x) + 20

Solving it:

200 - 20 = 10x + 15(1/3)x

180 = 10x + 15(1/3)x

180 = 10x + (15/3)x

180 = 10x + 5x

180 = 15x

180/15 = x

x = 12