Maths Revision Question

[starsy1]

Hi all,

A friend has asked for some help with a question on a maths mock paper for GCSE revision. It's a bit beyond me, well any recollection of being able to do anything similar is too far out of my grasp to make effective use! Any help greatly appreciated

Question:
A rectangular sheet of paper can be cut into two identical rectangular pieces in two different ways.

When the original rectangle of paper is cut one way the perimeter of each of the two pieces is 50 cm ( - I can't copy the diagram here, but the rectangle is cut equally into two pieces lengthways).

When the original rectangle of paper is cut the other way the perimeter of each of the two pieces is 64 cm (- again, there's a diagram showing the rectangle cut widthways to make two equal sized rectangles).

What is the perimeter of the original sheet of paper?

[Bibliovore]

When you cut the piece of paper length-wise, the new shapes each have two sides of length equal to the original, but the total of the widths is the same of the original. So adding up the perimeters of these you have in effect doubled the total length.
Cutting the pieces of paper width-wise has the same effect, but creates two shapes each with two sides of width equal to the original, but the lengths are the same.

Expressing this algebraically you have

(cut Length-wise)
4 times length + 2 times width = 64cm

(cut Width-wise)
2 times length + 4 times width = 50cm

Multiple the bottom equation by 2 and you get
4L + 8 W = 100 cm
Subtract the top equation and you get
4L-4L+8W-2W= 100-64
Or 6W = 36
So the original width is 6cm

Substitute the width back into one of the original two equations and you get
4L + (2 X6) = 64
4L = 52
So the length is 13

This then makes the original perimeter (13 X2) + (6 X2) or 42cm

[Sorry, I read this as the total perimeter of the two new shapes was 50cm or 64cm, rather than the perimeter of each of the new shapes, so my solution doesn't work, while Guest 55's does]

[Guest55]

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

When the original rectangle of paper is cut one way the perimeter of each of the two pieces is 50 cm ( - I can't copy the diagram here, but the rectangle is cut equally into two pieces lengthways).

When the original rectangle of paper is cut the other way the perimeter of each of the two pieces is 64 cm (- again, there's a diagram showing the rectangle cut widthways to make two equal sized rectangles).

What is the perimeter of the original sheet of paper?

Let width be w and length l

......... w .........
.
.
l
.
.......... w .........

If a perimeter is 50 then the distance around half the shape is 25 cm. So cut lengthwise [top to bottom?]

0.5 w + l = 25 equation1

Cut across

w + 0.5 l = 32 [If I've interpreted the direction of cuts correctly]

so, doubling the first equation, w + 2l = 50 and from the second we know w + 0.5 l = 32

subtracting gives, 1.5 l = 18, so l = 12 cm

Substituting l = 12 into equation 1, 0.5 w + 12 = 25, so w = 26 cm

So paper is 12 cm by 26 cm and perimeter is ....

[solimum]

Don't even need to calculate the individual dimensions

Original perimeter -= 2w + 2l

Cut one way perimeter = 2w+2 *(0.5l) = 2w + l = 50

Cut the other way perimeter = l + 2w = 64

Add two equations together 3w + 3l = 114

So w+l = 114/3 = 38

So 2w + 2l = 76

original perimeter = 76cm (Phew - that's the same as Guest55!)