Marble PuzzleDate: 3/10/96 at 0:32:10 From: Cristian Liu Subject: math Here is my question: Jason and Bob have 193 marbles altogether. Bob has 47 marbles less than Jason. If Jason gives Bob 15 marbles, how many more marbles does Jason have more than Bob? Thank you. From Cristian Liu Date: 3/10/96 at 23:6:27 From: Doctor Jodi Subject: Re: math Hi Cristian. Thanks for your question. There are two ways I can think of to solve it. One is that if Jason gives Bob some marbles, he loses 15 and Bob gains 15 at the same time. This means that by giving away 15 marbles, Jason makes a total difference of 30 marbles. So, if before Jason had 47 marbles more than Bob, now he only has 17 marbles more than Bob. Does that make sense? The other way to figure out this puzzle is to figure out how many marbles Jason and Bob each have before and after the exchange. First, let's write some number sentences. If Jason and Bob have 193 marbles altogether, we can write that as j + b = 193 and if Bob has 47 marbles less than Jason, we can write that as b + 47 = j ( or j - 47 = b) Now since j and b + 47 are equal, we can substitute one for the other. So in our first equation, j + b = 193 let's substitute b + 47 for j in the first equation to find out how many marbles Bob has. Our new equation reads (b + 47) + b = 193 We can rewrite this as 2b + 47 = 193 In order to find b, we need to isolate it - to get it alone on one side of the equation. In order to keep this as an equality, we need to do the SAME thing to each side. So if we want to subtract 47 from the lefthand side to find out what 2b is equal to, we need to subtract 47 from the righthand side, too. 2b + 47 = 193 - 47 - 47 ____________________ 2b = 146 Now to get b alone, we need to divide by two, right? 2b = 146 --- ---- 2 2 which means that b = 73 Does this make sense so far? Now, we know that Bob has 73 marbles. We also know that Jason has 47 MORE than Bob has. 73 + 47 = 120 marbles So Bob has 73 marbles and Jason has 120. Let's check to see if the total number of marbles still works. Yes, 120 + 73 = 193, just what we were told at the beginning of the problem. But we still haven't solved the puzzle. What we want to figure out is what will be the difference in the number of marbles each boy has if Jason gives Bob 15 marbles. That means that Jason, who has 120 marbles, will give away 15, leaving 105 marbles. Meanwhile, Bob, who has 73 marbles, will gain another 15, making a total of 88 marbles. The difference between the number of marbles Jason has and the number of marbles Bob has is again 17. 105 - 88 ____ 17 Does this make sense? Write us back if you need a better explanation of some of the steps or if you have any other questions! -Doctor Jodi, The Math Forum Date: 3/12/96 at 0:6:57 From: Eric Liu Subject: math On the question I sent you, I don't understand why it would be 30 Jason makes in the changes because it only said 15 marbles. From Cristian Liu Date: 3/16/96 at 1:29:14 From: Doctor Jodi Subject: Re: math Hi Cristian! Let me try to explain this again. Maybe it would help to try with smaller numbers. I can understand why this problem is confusing! Let me write you a few questions. Ok, say I have 3 marbles and you have 9. You have 6 more than I do. If you give me 2 marbles, then how many do we each have? How many more do you have now? 9 vs. 3 *** *** difference: 6 *** *** give 2 7 vs. 5 *** *** difference: 2 *** ** * difference in differences : 6-2 = 4 But only 2 marbles switched hands! Any idea what is going on here? Write back with more questions... -Doctor Jodi, The Math Forum |
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