Ruth Has $2300 More Than KenDate: 04/15/99 at 09:24:21 From: Angie Subject: Math - Whole numbers Ruth has $2300 more than Ken. If Ken gives Ruth $2000, she will have eight times as much money as Ken. How much money does each of them have? I've tried to add both the differences and then divide by 8 but can't seem to get the right answer. I'm really stuck, as I see no other solution! Thanks. Date: 05/26/99 at 17:32:43 From: Doctor Teeple Subject: Re: Math - Whole numbers Dear Angie, Thanks for writing to Dr. Math. Often when we have an amount that we don't know, we represent it with a variable. So for example, we don't know how much money Ruth has, so we'll represent it with an r. Similarly, we don't know how much money Ken has, so we'll represent it with a k. However, we do know some information about Ruth's and Ken's money. We'll use this information to create equations we can use to find out how much money Ruth and Ken have. This is the tricky part - turning the words into equations. So let's do that one equation at a time. The first line is: "Ruth has $2300 more than Ken." In other words, if we were to add $2300 to Ken's money, we would get Ruth's money. This becomes: r = k + 2300 Next: "If Ken gives Ruth $2000, she will have eight times as much money as Ken." We can reword this to be: "If Ken gives Ruth $2000 and we multiply Ken's money by 8, we will have the same amount." Then we reword that to say: "If Ken has $2000 less, and we multiply that by 8, and we add $2000 to Ruth's money, we will have the same amount." Then we can interpret this to be: 8(k-2000) = r + 2000 Now you have two equations with two unknowns, which look like they can be solved by substitution. If you need help with this, or need more explanation on the rewordings or turning them into equations, please write back. - Doctor Teeple, The Math Forum http://mathforum.org/dr.math/ Date: 06/03/99 at 08:16:52 From: Benedict Lim Subject: Re: Math - Whole numbers Hi Dr. Math, Thanks for your answer. It was very clear. However, my problem is that I need to use the model approach to solve this sum and I can't seem to start. Please help! Thanks Angie Date: 06/03/99 at 10:20:10 From: Doctor Teeple Subject: Re: Math - Whole numbers Dear Angie, Thanks for writing back. We'll try our best to figure this out. Unfortunately, I'm not sure what you mean by "model approach." The best I can guess is that the model approach means model real-life situations (such as dealing with money) by math equations that you can solve just with math. This happens all the time with math. In fact, April was Math Awareness Month, which focused on how math can be used to model biological events. (See http://mathforum.org/mam/ for more information.) Does that sound familiar? If it does, we are already doing the model approach. If it's not familiar, write back. - Doctor Teeple, The Math Forum http://mathforum.org/dr.math/ Date: 06/04/99 at 09:09:12 From: Benedict Lim Subject: Re: Math - Whole numbers Dear Dr. Math, Actually, that's not what I meant. How about if I ask for alternate methods to solve the sum, like say breaking it up into units and using multiplication, etc. I' m not very familiar with algebra and prefer an alternative method. I hope it's not too much trouble. Thanks again, Angie Date: 06/04/99 at 10:25:39 From: Doctor Teeple Subject: Re: Math - Whole numbers Dear Angie, Thanks for writing back. Don't get frustrated with us, but unfortunately, I don't see a good non-algebraic way of solving this. Here's why: the numbers that you give in your problem are all relative to the amount of money that Ruth and Ken have now. Without a starting point, the situation becomes more complicated than using only arithmetic. Let me demonstrate. Suppose we know that Ruth has $5000. Then, since we know that Ruth has $2300 more than Ken, we could find, from arithmetic, that Ken has $5000 - $2300 = $2700. But we don't know how much Ruth has. So that's where the algebra comes in. Without algebra, the only alternative I can suggest is to just guess and check. For example, we'll start off by guessing that Ruth has $5000. Then from your first sentence, we'd say that Ken has $2700. We use the second sentence to check. If Ken gives Ruth $2000, Ken has $700 and Ruth has $7000. But since $7000/8 = $875, which does not equal $700. So we know that Ruth doesn't have $5000. The tricky part is figuring out how to adjust our guess for Ruth's money. Suppose we just decide to increase our guess to $6000. Then Ken has $6000 - $2300 = $3700. If Ken gives Ruth $2000, Ken has $1700 and Ruth has $8000. But $8000/8 = $1000, which does not equal $1700. Notice that the $1000 we figured is less than than $1700 and previously, the $875 is more than the $700. So it looks as if we increased our guess too much. Why? Now you have a range to check. Notice that from the second sentence, we know that Ken has to have at least $2000 (because he gives it away), which means that Ruth has at least $4300. That's the lowest you should go with your guesses. The guess and check method can get tedious and tricky to judge how you should change your guesses. So I recommend that you try out the algebra method. It's a good question for learning algebra. (Is this what you're learning now?) If you want to work more on the algebra method, and are having trouble, please write back, and tell us where you're stuck. We'll do our best to help. In the meantime, here are a few archives that might help you: Simple Algebraic Equations http://mathforum.org/dr.math/problems/waters5.25.96.html Isolate the Variable http://mathforum.org/dr.math/problems/jayfer.9.9.96.html Algebra - Solving Equations http://mathforum.org/dr.math/problems/mantha11.3.97.html Remember that the variables can be treated just like numbers because they are acting as placeholders for the numbers we don't know. Also, I'm going to leave this question for other doctors to see, in case they have some other ideas. - Doctor Teeple, The Math Forum http://mathforum.org/dr.math/ Date: 06/04/99 at 12:23:13 From: Doctor Peterson Subject: Re: Math - Whole numbers Hi, Angie. Dr. Teeple left this question for the rest of us to look at, and I think I have the kind of answer you want. Here's your question: Ruth has $2300 more than Ken. If Ken gives Ruth $2000, she will have eight times as much money as Ken. How much money does each of them have? Let's draw a picture of how much Ruth and Ken have to start with: +-------------------------------------------+ |Ruth | +-------------------------------------------+ +------------------------+ |Ken |<------2300-------> +------------------------+ This shows that the difference between the two amounts is $2300. Now if Ken gives Ruth $2000, it will take 2000 from Ken and add it to Ruth: +-------------------------------------------+-------------+ |Ruth | +2000 | +-------------------------------------------+-------------+ +--------+-------------+ |Ken | -2000 |<-------2300-------> <----2000----> +--------+-------------+ Now we know that Ruth's new amount is 8 times as much as Ken's. But that means that the difference between their amounts is 7 times Ken's amount, and this difference is exactly $6300: +-------------------------------------------+-------------+ |Ruth | +2000 | +-------------------------------------------+-------------+ +--------+-------------+ |Ken | -2000 |<-------2300-------> <----2000----> +--------+-------------+ <--1/8--> <-----------------7/8 = 6300--------------------> So Ken's new amount is 1/7 of 6300, or 900. Ken's original amount was 900 + 2000 = 2900, and Ruth's is 2900 + 2300 = 5200. What we're really doing here is algebra in disguise. Rather than using letters R and K, I'm using pictures and words, but I'm still adding known amounts to variables, and so on. When algebra was invented, people found ways to do all this thinking automatically, without having to draw pictures and invent a new way to solve every problem. So when you learn algebra, you'll be saving yourself a lot of work! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 06/06/99 at 02:21:57 From: Benedict Lim Subject: Re: Maths-Whole numbers Your answer was excellent; it's exactly what I was looking for! Thank you. Angie |
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