Consecutive Odd IntegersDate: 08/28/97 at 18:32:11 From: Heather Mahurin Subject: Addition-Word Problem I need to know three consecutive odd integers that equal 150. Date: 08/28/97 at 19:45:13 From: Doctor Barney Subject: Re: Addition-Word Problem First, we should approach this problem with the understanding that we have no guarantees that there are three consecutive odd integers that add up to 150, but let us assume that there are, and in this manner we may either find three of them or prove that there are no such three numbers. Let us name these three unidentified integers X, Y, and Z, for convenience, until we find their real values. Since they are consecutive odd integers, we know that the largest is 2 more than the middle one, and that the middle one is 2 more than the smallest. Arbitrarilly assuming that X is the smallest and Z is the largest, we may write: X+2 = Y and Y+2 = Z. We also know that X+Y+Z = 150. Now I am going to eliminate variables from this equation by "substitution". If Z = Y+2, that means that Z is the same number as Y+2, whatever numbers Z and Y really are, so anywhere there is a Z I can put in a Y+2 instead without changing the truth of the equality. X+Y+(Y+2) = 150. Now I will do the same thing using Y = X+2. X+(X+2)+((X+2)+2) = 150 Next add up all the X's and all the 2's 3X+6 = 150 Now subtract 6 from both sides of the equation 3X = 144 Finally solve for X (have you had algebra yet?) by dividing by 3 X = 48 But X is even! This means that there are not any three consecutive odd integers that add up to 150. Actually, it is fairly easy to prove that any three odd integers can never add up to an even number: any two odd integers will add up to an even number, and adding a third odd number will always create an odd number for the final sum. Can you find three consecutive even integers that add up to 150? -Doctor Barney, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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