From Representing Two Bags of Marbles, a Way to Solve All Kinds of ProblemsDate: 04/03/2011 at 22:11:39 From: Sarah Subject: Trying to find original values of algebraic expressions David had 2 bags of marbles. After David sold 44 marbles from Bag X to his friend, the number of marbles in Bag Y was 5/7 of the number of marbles in Bag X. Given that there were 2/5 as many marbles in Bag Y as in Bag X originally, find the number of marbles in Bag X at first. It is confusing to understand the parts of the equation. I get as far as representing the first step as x - 44, and the second step as y = (5/7)x, but I find it REALLY difficult to set up one expression and solve it. Date: 04/05/2011 at 10:30:26 From: Doctor Ian Subject: Re: Trying to find original values of algebraic expressions Hi Sarah, It can often be helpful to make small changes, rather than trying to write down equations in a single step. Here's the problem again: David had 2 bags of marbles. After David sold 44 marbles from Bag X to his friend, the number of marbles in Bag Y was 5/7 of the number of marbles in Bag X. Given that there were 2/5 as many marbles in Bag Y as in Bag X originally, find the number of marbles in Bag X at first. We're asked to find the original number of marbles in bag X. So we might start by giving that quantity a name, like 'x.' And because we'll want to talk about the original number of marbles in bag Y, we can give that a name as well: 'y.' Now we can try rewriting the problem using that notation: David had 2 bags of marbles. Bag X had x marbles originally. Bag Y had y marbles originally. After David sold 44 marbles from Bag X to his friend, the number of marbles in Bag Y was 5/7 of the number of marbles in Bag X. Given that there were 2/5 as many marbles in Bag Y as in Bag X originally, find x. Let's look at that next-to-last paragraph. David is removing 44 marbles from bag X. That means the number of marbles will be reduced by 44, right? So the number of marbles in bag x will be (x - 44). Now we have David had 2 bags of marbles. Bag X had x marbles originally. Bag Y had y marbles originally. After David sold some marbles, y was 5/7 of (x - 44). Given that there were 2/5 as many marbles in Bag Y as in Bag X originally, find x. And that looks like an equation: David had 2 bags of marbles. Bag X had x marbles originally. Bag Y had y marbles originally. After David sold some marbles, y = (5/7)(x - 44). Given that there were 2/5 as many marbles in Bag Y as in Bag X originally, find x. And that last paragraph describes an original relationship between x and y, doesn't it? But it's kind of confusing, so I'd try some concrete numbers to make sure I'm interpreting it correctly. For example, what if there are 5 marbles in bag X? In symbols, if x = 5, then y = ? Now it's clear that there would have to be 2 marbles in bag Y, since 2 is 2/5 of 5. So the relationship between x and y is y = (2/5)x I can put that back in the problem: David had 2 bags of marbles. Bag X had x marbles originally. Bag Y had y marbles originally. After David sold some marbles, y = (5/7)(x - 44). Originally, y = (2/5)x. What is x? So now I have two equations, in two variables, and the details of the problem are irrelevant for purposes of solving the system. That is, I just have y = (5/7)(x - 44) y = (2/5)x Does this make sense? You'll probably never have to solve a problem like this in real life, but there's a valuable lesson to be learned from working through problems like this in such a small-step-by-small-step way, because that method applies to all kinds of problems that you WILL encounter. It's not so much a 'math technique' as a way of thinking about problems in general. What I did here was make small changes to what I had, to get something I preferred, or at least understood better. In spirit, it's the same sort of thing you do when you solve an equation like 12x + 9 = 4x + -1 I'd like that equation better if I had all the x terms on one side, so I can make a small change: -4x + 12x + 9 = -4x + 4x + -1 I can look at those two equations, and see that they mean the same thing. So when I simplify the new equation, to get ... 8x + 9 = -1 ... I know that it has the same solution set as the one I started with. And I just keep doing that, until I get something that is so simple that I can deal with it easily. What I did before was the same sort of thing, but using words and sentences instead of equations. If there is a secret to solving problems, I think this is it: When you're faced with a problem, instead of looking directly for a solution, look for a way to turn it into an easier problem. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 04/06/2011 at 08:57:56 From: Sarah Subject: Thank you (Trying to find original values of algebraic expressions) Thanks! |
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