An army of 12000 consists of Europeans and Indians. The average height of a European is 5 ft 10 inches and that of an Indian 5 ft 9 inches and that of the whole army is 5 ft 9 3/4 inches. Find the numbers of Indians in the army?
Question
An army of 12000 consists of Europeans and Indians. The average height of a European is 5 ft 10 inches and that of an Indian 5 ft 9 inches and that of the whole army is 5 ft 9 3/4 inches. Find the numbers of Indians in the army?
Solution
To solve this problem, we can use the concept of weighted averages. Here are the steps:
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Convert all heights to inches for simplicity. The average height of a European is 70 inches (5 ft 10 inches), the average height of an Indian is 69 inches (5 ft 9 inches), and the average height of the whole army is 69.75 inches (5 ft 9 3/4 inches).
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Let's denote the number of Europeans as E and the number of Indians as I. We know that E + I = 12000.
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We also know that the total height of the army is the sum of the heights of all Europeans and Indians. This can be expressed as 70E + 69I = 69.75 * 12000.
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Now we have a system of two equations, and we can solve it. First, express E from the first equation: E = 12000 - I.
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Substitute E in the second equation: 70(12000 - I) + 69I = 69.75 * 12000.
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Simplify the equation and solve it for I: 840000 - 70I + 69I = 837000.
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This simplifies to I = 840000 - 837000 = 3000.
So, there are 3000 Indians in the army.
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