The Method of False PositionDate: 02/06/2004 at 23:37:09 From: Lynne Subject: Math Riddle There is a quantity such that 2/3 of it, 1/2 of it, and 1/7 of it added together becomes 33. What is the quantity? Solve the problem by the method of false position. I know that using the method of false position, we are looking for the root of an equation, and need a way of making a guess that is better than our previous guess. Date: 02/07/2004 at 14:59:35 From: Doctor Douglas Subject: Re: Math Riddle Hi Lynne - Thanks for writing to the Math Forum. Let's let 'x' be the unknown quantity, and we can write the following equation: x*(2/3) + x*(1/2) + x*(1/7) = 33 Assume x = 42. This is a convenient first guess because it is divisible by all of the denominators {3,2,7}, which makes our life a bit easier on the first step. 42*(2/3) + 42*(1/2) + 42*(1/7) = 28 + 21 + 6 = 55. So our first guess of 42 is approximately too big by a factor of 55/33 = 5/3. Our next guess is therefore 42*(3/5) = 126/5 or 25.2. We plug this, our second guess, in for x in the equation above to obtain (126/5)*(2/3) + (126/5)(1/2) + (126/5)(1/7) = 84/5 + 63/5 + 18/5 = 165/5 = 33 which is exactly what we want it to be, and we've therefore found the value of x for which the equation is true. There is of course a more direct method to solve the original equation using algebra techniques, but I think that this problem shows you how the method of false position works so that you can also apply it to cases where you cannot simply solve for x. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ Date: 02/07/2004 at 15:53:54 From: Lynne Subject: Thank you (Math Riddle) Thank you! Date: 02/09/2004 at 09:46:13 From: Doctor Peterson Subject: Re: Thank you (Math Riddle) Hi, Lynne. Just a quick note to add on to Dr. Douglas's reply. I assume you are aware that this method was used in ancient Egypt. See this page: The Egyptians' Method of False Position http://mathforum.org/library/drmath/view/62036.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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