# Math Help - Scale problem

1. ## Scale problem

When you cut a piece of A4 paper in half (portraiot way up, cut across) youget two pieces of A5 paper.

Ignoring its thickness, A5 paper is similar to A4 paper.

Express the length of A5 paper as a percentage of the length of A4 paper.

OK, I get that A5 is 1/2 of A4. So, that would make it 50% of A4. But this is wrong.

Could someone please explain this step by step please?

2. Originally Posted by GAdams
When you cut a piece of A4 paper in half (portraiot way up, cut across) youget two pieces of A5 paper.

Ignoring its thickness, A5 paper is similar to A4 paper.

Express the length of A5 paper as a percentage of the length of A4 paper.

OK, I get that A5 is 1/2 of A4. So, that would make it 50% of A4. But this is wrong.

Could someone please explain this step by step please?
As you get two A5 pieces from an A4 sheet, the area of an A5 is half
that of A4. The areas scale as the square of the linear dimensions, so
the length of an length(A5) sheet is length(A4)/sqrt(2).

RonL

3. OK. So if the area is 1/2, then the length will be the square root of 1/2.

Which is 0.707

If I multiply this by 100, I should get the length of A5 as a percentage of A4

Which is rounded to 71%.

4. Originally Posted by GAdams
OK. So if the area is 1/2, then the length will be the square root of 1/2.

Which is 0.707

If I multiply this by 100, I should get the length of A5 as a percentage of A4

Which is rounded to 71%.
Yes. And the great secret of many problems like this is you can actualy
measure a peice of A4. The width will be the height of the A5 and its height
is of course the height of a piece of A4.

RonL

When you cut a piece of A4 paper in half (portrait, cut across)
you get two pieces of A5 paper.

A5 paper is similar to A4 paper.

Express the length of A5 paper as a percentage of the length of A4 paper.

Another approach (to me, the obvious one) . . . make a sketch!
Code:
    - *-----------* -
: |           | :
: |           | ½L
: |           | :
L * - - - - - * -
: |           | :
: |           | ½L
: |           | :
- *-----------* -
:     W     :

A4 has length $L$, width $W$.

A5 has length $W$, width $\frac{1}{2}L$.

We want the ratio: . $\frac{W}{L}$

Since the rectangles are similar: . $\frac{W}{\frac{1}{2}L} \:=\:\frac{L}{W}\quad\Rightarrow\quad W^2\:=\:\frac{1}{2}L^2\quad\Rightarrow\quad\frac{W ^2}{L^2} \:=\:\frac{1}{2}
$

Hence: . $\left(\frac{W}{L}\right)^2 \:=\:\frac{1}{2}\quad\Rightarrow\quad \frac{W}{L} \:=\:\frac{1}{\sqrt{2}} \:=\:0.707106781$

Therefore: $W$ is about 70.7% of $L$.