Find the Numbers
Date: 08/28/2003 at 22:05:14 From: Jen Subject: Math Logic? One number is 61 more than a second number. The sum of the two numbers is 127. Find the numbers. I think you have to have an equation but I don't know how to set it up. I think it may be 61 + x = 127, but I am not sure. Please help!
Date: 08/29/2003 at 09:19:29 From: Doctor Ian Subject: Re: Math Logic? Hi Jen, When I don't know what else to do, I make a guess, and then try to check it. Often, the process of checking can lead me to an equation. Here's how that might work. I know I'm looking for a couple of numbers. What about 1 and 2? Let's check that: 1) One number is 61 more than a second number. Okay, so 1 and 2 aren't going to do it. What this tells me, though, is that once I choose a first number, the second number is determined: I just add 61 to the number I choose. So how about 1 and 62? That will clearly meet the first test. What about the second? 2) The sum of the two numbers is 127. The sum of 1 and 62 is 63, which is way too low. But note that I can write my sum this way: ? <---- The '?' reminds me that the 1 + (1 + 61) = 127 equation might not be true! If I choose a different starting number, like 10, I get ? 10 + (10 + 61) = 127 And at this point, there are a couple of ways we might go. One would be to get smarter about our guesses, using each guess to help us figure out how much larger or smaller our next guess should be. Sometimes that's the smartest way to proceed, if checking is easy enough. But we can also look at our two equations, and notice that only one thing is changing, which is the number that we choose as a guess. That's a clue that we should consider using an uknown to represent it. Using 'g' to stand for 'guess', we could write g + (g + 61) = 127 Before rushing to solve this, we want to check it against the problem description. 1) One number is 61 more than another. Well, (g+61) is clearly 61 more than g, whatever g turns out to be. 2) The sum of the numbers is 127. We add the two numbers and get 127, so we're good to go. I find that letting an example or two lead me to an equation, and then checking the equation against the problem statement, is often easier than trying to 'translate' the problem directly into an equation. Let me note here that checking your equations this way is a good habit to get into. Let's look at the equation you came up with: 61 + x = 127 You have two numbers that add up to 127, so you've nailed the second condition. But what about the first? Which of the two numbers is 61 more than the other? 61 is 61 more than x only when x is zero. And x is 61 more than 61 only when x is 122. It's easy to make mistakes like this when trying to go directly from words to equations. But as long as you're careful to check your equations against the original description, you can catch errors like this on your own. Anyway, so now we have an equation, which seems to represent the problem statement, and which has a single uknown. Can you take it from here? I hope this helps. Write back if you have more questions, about this or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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