Question

A naughty student breaks the pencil in such a way that the ratio of two broken parts is same as that of the original length of the pencil to one of the larger part of the pencil, The ratio of the other part to the original length of pencil is:

• A. $1:2\sqrt{5}$
• B. $2:(3+\sqrt{5})$
• C. $2:\sqrt{5}$
• D. $ca{n}^{\prime }tbedetermined$

Answer 1

Answer: (B)

$2:(3+\sqrt{5})$

Suppose that,

The length of larger part be$=x$

And the smaller part be $=y$

So, their ratio is $x$ is to

$\frac{x}{y}=\frac{kx+ky}{kx}$

$\Rightarrow \frac{x}{y}=\frac{x+y}{x}$

$\Rightarrow x^2=xy+y^2$

$\Rightarrow x^2-y^2=xy$

Let $y=1$

$x^2-1^2=x$

$\Rightarrow x^2-x-1=0$

$\Rightarrow x=\frac{1\pm \sqrt{5}}{2}\left(\text{On}\text{solving}\text{by}\text{second}\text{degree}\text{equation}\text{rule}\right)$

Negative value cannot be considered

So,

$x=\frac{1+\sqrt{5}}{2}$ so, the ratio

$\frac{x}{y}=\frac{\frac{1+\sqrt{5}}{2}}{1}$

$\Rightarrow x:y=(1+\sqrt{5}):2$

Therefore,

$\frac{y}{x+y}=\frac{2}{(1+\sqrt{5}+2)}$

$\Rightarrow \frac{y}{x+y}=\frac{2}{(3+\sqrt{5})}$

Hence, this is the answer.