25. A gold coin is worth $x \%$ more than a silver coin. The silver coin is worth $y \%$ less than the gold coin. Both $x$ and $y$ are positive integers. How many possible values for $x$ are there?
A 0
B 3
C 6
D 9
E 12
25. E Let the costs of a gold coin and a silver coin be $£ g$ and $£ s$ respectively.
Then $g=\left(1+\frac{x}{100}\right) s$ and $s=\left(1-\frac{y}{100}\right) g$.
Hence $g / s=(100+x) / 100$ and $s / g=(100-y) / 100$. Therefore $(100+x)(100-y)=10000$.
So $x=\frac{10000}{100-y}-100$. Therefore $100-y$ is a factor of 10000 and we need consider only values of $100-y$ less than 100 as $y$ is a positive integer.
Hence the possible values of $100-y$ are $1,2,4,5,8,10,16,20,25,40,50,80$. The corresponding values of $x$ are 9900, 4900, 2400, 1900, 1150, 900, 525, 400, 300, 150, 100, 25.
So there are twelve possible values of $x$.