The law of multiple proportions says that when two elements combine with each other to form compounds, the masses of one of the elements which combine with fixed mass of other, bear a simple whole number ratio to one another.For e.g. consider CO and CO2. Both are compounds of carbon and oxygen.
In the carbon monoxide (CO) , 12 parts by mass of the carbon combine to the 16 parts by mass of the oxygen. On the other hand in the carbon dioxide (CO2) , 12 parts by mass of carbon combine to the 32 parts by mass of the oxygen.
Thus the masses of the oxygen which combines with the carbon is in the proportion of 16 : 32 , that is in the ratio 1 : 2 , which is a simple whole number ratio.