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?
Age
14 to 16
Challenge level
Problem
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
Student Solutions
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.