# Math Help - Surds

1. ## Surds

A large rectangular piece of card is square root of 5 + square root of 20 cm long and square root of 8 cm wide. A small rectangle square root of 2 cm long and square root of 5 cm wide is cut out of the piece of card. Express the area of the card that is left in a percentage of the area of the large rectangle?

2. Let $A_1$ be the large piece and $A_2$ the small piece

The ratio of the card left is therefore given by $\dfrac{A_1-A_2}{A_1} \times 100\%$

$A_1 = (\sqrt{5} + \sqrt{20}) \times \sqrt{8}$

Since $\sqrt{20} = \sqrt{4\cdot 5} = 2\sqrt{5}$ and that $\sqrt{8} = \sqrt{2^2 \cdot 2} = 2\sqrt{2}$ we can rewrite the above equation:

$A_1 = (\sqrt{5}+2\sqrt{5}) \ times 2\sqrt{2} = 3\sqrt{5} \times 2\sqrt{2} = 6\sqrt{10}$

Note I have used the rule of surds which says that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\: \: a,b \geq 0$

Can you find $A_2$ using the same principles?

3. A2 = square root of 2 x square root of 5 = Square root of 10

How do we express the area of the card that is left as a percentage of the area of the big rectangle?

Square root of 10 divided by 6 Square root of 10???

4. Originally Posted by Natasha1
A2 = square root of 2 x square root of 5 = Square root of 10

How do we express the area of the card that is left as a percentage of the area of the big rectangle?

Square root of 10 divided by 6 Square root of 10???
No, " $\sqrt{10}$" is NOT the "area of the card that is left". It is the area that is removed.

5. Is the answer 83.3 percent?

6. Originally Posted by Natasha1
$\dfrac{6\sqrt{10} - \sqrt{10}}{6\sqrt{10}} = \dfrac{5}{6} \approx 83.3\%$