Average Age at a PartyDate: 10/27/1999 at 10:50:39 From: Rob Amure Subject: Discrete math problems At the classroom costume party the average age of the (b) boys is g and the average age of the (g) girls is b. If the average age of everyone, including the 42-year-old teacher, is b+g, what is the value of b+g? Date: 10/28/1999 at 10:33:31 From: Doctor Rob Subject: Re: Discrete math problems Thanks for writing to Ask Dr. Math, Rob. The total age of the b boys is b*g, and the total age of the g girls is g*b. The total age of everyone is b*g + g*b + 42, and there are b+g+1 people. This gives you the equation (2*b*g + 42)/(b+g+1) = b+g 2*b*g + 42 = (b+g)^2 + (b+g) 42 = b^2 + g^2 + b + g. Now there are infinitely many solutions to this equation in real numbers. In integers, however, the situation is different. Let b and g be positive integers. Then multiply both sides by 4 and complete the square on both b and g by adding 1 + 1. This will lead to 170 = (2*b + 1)^2 + (2*g + 1)^2 It's not hard to find the solutions of this (there are two with b > 0, g > 0), and to compute b+g. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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