a rectangular sheet of paper can be cut into two identical rectangular pieces in two different ways.

Question

a rectangular sheet of paper can be cut into two identical rectangular pieces in two different ways. when the original sheet of paper is cut one way, the perimeter of each of the two pieces is 50cm. when the original sheet of paper is cut the other way, the perimeter of each of the two pieces is 64cm. what is the perimeter of the original sheet of paper?

Answer 1

A = smaller edge of the paper

B = larger edge

 

If the paper is cut by bisecting the b edges, we have 2 A edges and 2 edges equal to half of B or

 

1) 2A + .5B + .5B = 50

 

If we cut it by bisecting the A edges, we have

 

2) A + 2B = 64

 

Add the two equations:

 

1) 2A + B = 50

2) A + 2B = 64

_____________

 

3A + 3B = 114

 

3 (A + B) = 114

A + B = 114 / 3

 

A + B = 38

 

Now we express A in terms of B

 

A = 38 - B

 

Substitute into the second equation:

 

2) 38 - B + 2B = 64

B = 64 - 38 = 26

 

1) 2A + 26 = 50

2A = 24

A = 12

 

The original sheet is 12 x 26 with a total perimeter of 76 cm

 

 

 

Answer 2

You can cut the original rectangular sheet into two identical pieces in two ways: (a) parallel to its length (l) , or, (b) parallel to its breadth (b). For way (a) the two new rectangles each have the same length as the original rectangle, but half its breadth. Therefore, the perimeter of each piece is 2*l + 2*1/2*b = 2*l + b.  For way (b), the two new rectangles have the same breadth as the original rectangle, but half its length. Thus the perimeter of each of the two pieces is 2*b + 2*1/2*l = 2b + l.

 

Using the information in the problem 2l + b = 50, 2b+l = 64, hence, l = 12 cm and b = 26 cm. The perimeter of the original rectangular sheet is 2l + 2b = 76 cm.