1. A farmer had several pens out by the barn. He had used 17 lenghts of fence to make the pens, but now he needs to use six of those fence lengths for another project. He wants to remove six fence lenghts but still have two square pens remaing. Which lenghts should he remove?

2. Mrs. young has her 18 students seated in a circle. They are evenly
spaced and numbered in order. which student is directly opposite these three students?
.student#1?
.student#5?
.student#18?

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1. To solve this problem, let's assume that each square pen requires the same length of fence. Since there are two square pens remaining, each pen would require 17/2 = 8.5 lengths of fence.

To find out which lengths of fence the farmer should remove, we need to determine how many fence lengths are needed for each pen without exceeding 8.5 lengths.

Let's try removing 6 fence lengths from the total of 17 lengths.

- With 17 - 6 = 11 lengths remaining, divide it by 2 (for the two pens) to get 11/2 = 5.5 lengths per pen.

This means that if the farmer removes 6 lengths of fence, the resulting pen sizes would be 5.5 lengths each, which is not a whole number.

Therefore, removing 6 lengths of fence would result in uneven pen sizes, and it is not possible to have two square pens remaining.

2. Since the students are evenly spaced and numbered in order, we can determine the student directly opposite any given student by subtracting the student's number from the total number of students and adding one.

a) To find the student directly opposite Student #1:
Total number of students = 18
Opposite student = 18 - 1 + 1 = 18

Therefore, Student #18 is directly opposite Student #1.

b) To find the student directly opposite Student #5:
Total number of students = 18
Opposite student = 18 - 5 + 1 = 14

Therefore, Student #14 is directly opposite Student #5.

c) To find the student directly opposite Student #18:
Total number of students = 18
Opposite student = 18 - 18 + 1 = 1

Therefore, Student #1 is directly opposite Student #18.

1. To solve this problem, we need to find the lengths of fence that can be removed while still maintaining two square pens. Let's break down the information given:

- The farmer initially used 17 lengths of fence to make the pens.
- He needs to remove 6 fence lengths for another project.
- He wants to keep two square pens.

To find out which lengths should be removed, we need to determine the number of fence lengths that make up one square pen and then subtract that from the total number of lengths used.

A square has four sides of equal length, so if a pen is square, two lengths of fence make up one side. Therefore, two square pens would require 2 x 4 = 8 lengths of fence.

To have two square pens remaining after removing 6 lengths, the farmer should subtract 6 lengths from the total used lengths: 17 - 6 = 11 lengths.

However, this is more than 8 lengths needed for two square pens. So, we cannot have two square pens remaining while removing exactly 6 lengths.

It seems there might be a mistake or missing information in the problem statement. The farmer can either remove 8 lengths to have two square pens or remove 6 lengths to have only one square pen remaining.

2. To find the student directly opposite another student in a circle, you can use modular arithmetic.

- The total number of students is given as 18.
- The students are seated in a circle and evenly spaced, numbered in order.

To find the student directly opposite student #1, we need to find the number that is halfway around the circle from student #1. Since there are 18 students, we divide that by 2 to find the halfway point: 18 / 2 = 9.

Therefore, student #9 would be directly opposite student #1.

To find the student directly opposite student #5, we can add 5 to the halfway point found earlier: 9 + 5 = 14.

Therefore, student #14 would be directly opposite student #5.

To find the student directly opposite student #18, we need to subtract the halfway point from 18: 18 - 9 = 9.

Therefore, student #9 would be directly opposite student #18.

In summary, the answers are:
- Student #1 is directly opposite student #9.
- Student #5 is directly opposite student #14.
- Student #18 is directly opposite student #9.