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

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