Ruth Has $2300 More Than Ken

Date: 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