What is the empirical formula of a compound containing 24.56% potassium, 34.81% manganese, and 40.50% oxygen?

 Assume a 100g sample to make things easier; so 24.56% of 100g is 24.56g, for example. The empirical formula is simply the ratio of the moles of each substance. So convert each to moles so that you can compare them: 

24.56g K *(1 mol/39.10g)= .6281 mol K 
34.81g Mn *(1mol/54.94)= .6336 mol Mn 
40.50g O *(1mol/16.00g)= 2.531 mol O 

Remember that we are trying to find the ratio between them. The easiest way to do that is to divide each by the smallest number: 

.6281 mol K /.6281 mol = 1 
.6336 mol Mn /.6281mol = 1.0 
2.531 mol O /.6281 mol =~ 4.0 

So you have a ratio of 1:1:4 (there is some degree of approximation that you have to make) 
So the empirical formula is: 
KMnO4 

The answer is 323 g.