# Black and white socks

There are 20 black socks and some white socks in a drawer. Can you work out how many of the socks are white?

There are 20 black socks and $n$ white socks in a drawer.

When a sock is taken from the drawer, the probability that it is white is $\dfrac{1}{n}$.

What is the value of $n$?

*This problem is adapted from the World Mathematics Championships*

**Answer**: 5

There are $20+n$ socks in the drawer altogether, so $\dfrac{n}{20+n}=\dfrac{1}{n}$.

$$\begin{split}&\frac{n}{20+n}=\frac{1}{n}\\

&\Rightarrow n=\frac{1(20+n)}{n}\\

&\Rightarrow n^2=20+n\\

&\Rightarrow n^2-n-20=0\\

&\Rightarrow (n+4)(n-5)=0\\

&\Rightarrow n+4=0\hspace{3mm}\text{or}\hspace{3mm} n-5=0\\&\Rightarrow n=-4\hspace{7mm}\text{or}\hspace{3mm}n=5\end{split}$$ $n$ must be positive as there cannot be a negative number of white socks, therefore there are $5$ white socks in the drawer.