# A rectangular property  is to be fenced? The property is 77.3 m long and49.3 m wide. The fence posts will be installed on 3.2 m centre-to-centre spacing. The fence will run along the property lines on three sides. On the fourth side, closest to John Street, the fence will be inset 4.5 m from the property line. Also on the fourth side, a gate will be installed. The opening for the gate will be 8.0 m wide with one of the posts at the gate opening located 7.6 m from the corner of the fence that is closest to manchester Avenue. How many posts will be located on the side closest to Manchester Avenue? Include the corner posts? How many posts will be located on the side furthest from John Street? Include the corner posts? How many posts will be located on the side closest to John Street? Include the corner posts. ?? How many fences will be needed in total??

The property is 49.3 meters wide.  The fence is inset 4.5 meters from the property line on the side closest to John Street.  This makes the fenced property 44.8 meters wide.  The fence posts are installed 3.2 meters apart.  Divide 44.8 / 3.2 = 14.  This means there will need to be 15 fence posts (counting the corner post) on the side closest to Manchester Avenue, as well as 15 fence posts (counting the corner post) on the side furthest from Manchester Avenue.  Each side will contain 14 pieces of fencing, each measuring 3.2 meters long.  14 * 3.2 = 44.8 m of fencing + 4.5 m inset from property line = 49.3 m.

The length of the property does not divide evenly into 3.2 sections, so there will be one section that is not evenly spaced.  The back property line (furthest from John Street) is 77.3 meters.  Divide 77.3 / 3.2 `~~` 25.  This means there will need to be 26 fence posts (counting the corner post) on the side furthest from John Street.  This side will contain 25 pieces of fencing.  24 of the pieces will be 3.2 meters long.  1 piece will be 0.5 meters long.  24 * 3.2 + 0.5 = 77.3 m.

Now we divide up the fourth side, the side closest to John Street containing the gate.  Beginning on the corner of Manchester Avenue and John Street, there is a corner post, 3.2 meters of fencing, a 2nd post, 3.2 meters of fencing, a 3rd post, 1.2 meters of fencing, and then the post at the gate opening.  The third section of fencing must be 1.2 meters because the problem states there is 7.6 meters from the corner post the the gate post.  Next, there is a gate 8 meters long, followed by another post.  If you add the 7.6 meters already accounted for plus the 8 meters for the gate, that makes 15.6 meters.  The problem states the length of the property is 77.3 meters.  77.3 - 15.6 = 61.7 meters of property remaining on the side closest to John Street.  This length also does not divide evenly into 3.2 sections, so there will be one section that is not evenly spaced.  Divide 61.7 / 3.2 `~~` 20.  This means there will need to be 21 fence posts (counting the corner post) from the gate to the corner post. This section will contain 20 pieces of fencing. 19 of the pieces will be 3.2 meters long. 1 piece will be 0.9 meters long.  19 * 3.2 + 0.9 = 61.7 m.  7.6 m (before gate) + 8 m (gate) + 61.7 m (after gate) = 77.3 m.

Summary:

Side closest to Manchester Avenue:
15 fence posts (including corner posts)
14 fence pieces, each with a length of 3.2 meters

Side furthest from Manchester Avenue:
15 fence posts (including corner posts)
14 fence pieces, each with a length of 3.2 meters

Side closest to John Street:
25 fence posts (including corner and gate posts)
21 fence pieces, each with a length of 3.2 meters
1 fence piece with a length of 1.2 meters
1 fence piece with a length of 0.9 meters
1 gate with a length of 8 meters

Side furthest from John Street:
26 fence posts (including corner posts)
24 fence pieces, each with a length of 3.2 meters
1 fence piece with a length of 0.5 meters

Total number of fence posts:
77 fence posts (including corner and gate posts)

Total number and length of fence pieces:
73 fence pieces, each with a length of 3.2 meters
1 fence piece with a length of 0.5 meters
1 fence piece with a length of 1.2 meters
1 fence piece with a length of 0.9 meters
1 fence gate with a length of 8 meters
Total fence = 244.2 meters

Check work:
Property length = 77.3 m
Property width = 49.3 m - 4.5 m inset = 44.8 m
Perimeter = 2 * 77.3 + 2 * 44.8 = 244.2 m

Approved by eNotes Editorial Team