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there are Indians and Europeans in an army of 12,000. the average height of Europeans is5'10" and that of Indians is 5'9" and the average height of entire army is 5'9,3/4. find out the number of Indians in Indian army

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This is, I think, a way of doing things that allows you to skip through a lot of the formulas: notice that the average is $3/4$ of the way towards the Europeans, so that there are $3/4: 1/4$ times more Europeans than Indians. If the average was $69.5$, i.e., halfway between the two averages, then there would be as many Indians as Europeans.