Multiplying/Adding Fractions Gives Same AnswerDate: 03/01/2002 at 10:21:49 From: Ryan Subject: Obtaining the same answer when multiplying and adding 2 fractions I don't know how to begin finding out the answer to this. I need to find two fractions that when they are multiplied and added give the same answer. I have two examples I can't use: 13/4 + 13/9 and 13/4 x 13/9 both have the same answer, and 169/30 + 13/15 and 169/30 x 13/15 both have the same answer. Can you find another occurrence of this? Date: 03/01/2002 at 12:32:03 From: Doctor Ian Subject: Re: Obtaining the same answer when multiplying and adding 2 fractions Hi Ryan, If we write the fractions as a/b and c/d, then the condition to be satisfied is that a c a c - + - = - * - b d b d ad cb ac -- + -- = -- bd db bd ad + cb = ac Let's check that with your examples: 1) 13*9 + 13*4 = 13*13 13(9 + 4) = 13*13 13*13 = 13*13 Okay. 2) 13*30 + 169*15 = 13*169 13*30 + 13*13*15 = 13*169 13(30 + 13*15) = 13*169 13*195 = 13*169 195 = 169 Oops! Just to take a completely different approach, let's check that with a calculator: 169/30 = 5.63 13/15 = 0.87 5.63 + 0.87 = 6.50 5.63 * 0.87 = 4.90 With a little thought, we can see that this pair _couldn't_ work, since (if we're working with positive numbers) for the product to be equal to the sum, both fractions would have to be greater than 1. Otherwise, a/b gets bigger when you add c/d to it; but it gets smaller when you multiply it by c/d. Does this make sense? Anyway, let's forget about your second example for now. In particular, let's look at 13*9 + 13*4 = 13*13 13(9 + 4) = 13*13 13*13 = 13*13 Is there a pattern here that we can exploit? Suppose we took a different square, like 17*17, and broke 17 into, say, 5 + 12: 17*(5 + 12) = 17*17 17*5 + 17*12 = 17*17 a d c b a c This would give us the equation 17 17 17 17 -- + -- = -- * -- 12 5 12 5 17*5 + 17*12 17*17 ------------ = ------- 12*5 12*5 17*(5 + 12) 17*17 ------------ = ------- 12*5 12*5 Do you think this would work for 17 17 17 17 -- + -- = -- * -- ? 1 16 1 16 How about 17 17 17 17 -- + -- = -- * -- ? 2 15 2 15 How about 53 53 53 53 -- + -- = -- * -- ? 26 27 26 27 Can you generalize this, to come up with a formula for cranking out more pairs? Note that there are also trivial pairs, like 0 0 0 0 - + - = - * - 1 2 1 2 As you've stated the problem, these are sneaky, but not illegal. I hope this helps. This was an interesting question! Thanks for asking it. Let me know if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 03/01/2002 at 13:24:30 From: Ryan Subject: Obtaining the same answer when multiplying and adding 2 fractions Wow! That was interesting, and you did a great job figuring that out and explaining it. I appreciate your quick response. Ryan |
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