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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.
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The law of multiple proportion is one of the fundamental laws of stoichiometry and was first discovered by the english chemist John Dalton in 1803. The law states that when chemical elements combine, they do so in a ratio of small whole numbers. For example, carbon and oxygen react to form carbon monoxide (CO) or carbondioxide (CO2), but not CO1.3. Further, it states that if two elements form more than one compound between them, the ratios of the masses of the second element to a mass of the first element will also be in small whole numbers.