Two Numbers with Equal Sum, Product, and Quotient?
Date: 05/07/2007 at 23:14:15 From: Hannah Subject: (no subject) Find two numbers such that when you find the sum, product, and quotient of the two numbers they are all equal. I think working with 0 and possibly negative numbers might help, and I am thinking that the two numbers must be low numbers. But all of my tries come up with two out of the three being equal, but the third not equal.
Date: 05/08/2007 at 10:12:39 From: Doctor Ian Subject: Re: Hi Hannah, Interesting question. Presumably you just mean one of the quotients, right? For example, if you have the numbers 2 and 3, there are two possible quotients: 3/2 and 2/3. I'd probably start the way you're starting--looking for easy and obvious answers, involving small numbers. Of course, this is the sort of case where algebra really comes in handy! Suppose we call our numbers x and y. The condition to be satisfied is x + y = x * y = x / y which translates to three separate conditions, x + y = x * y x + y = x / y x * y = x / y Let's try solving the third equation for x in terms of y: x * y = x / y x * y * y = x y * y = 1 Interesting! This tells us that if such numbers do exist, one of them has to be either 1 or -1. Now we can go back and try the other equations with those values: y = 1: x + 1 = x * 1 x + 1 = x / 1 y = -1: x + -1 = x * -1 x + -1 = x / -1 Can you find a value that makes each pair of equations true? Or can you show that this is impossible? Right away, the equation x + 1 = x * 1 presents a problem: x + 1 = x * 1 x + 1 = x 1 = 0 So y can't possibly be equal to 1, since that leads to a contradiction. So either y is -1, or there is no solution. Can you take it from here? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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