Visualizing Two Word Problems

Date: 03/12/98 at 23:54:18
From: Sherilyn
Subject: word problem

Sandy has thrice as many beads as Wendy. Ann has half as many beads as 
Wendy. If Sandy has 180 beads more than Ann, how many beads do the 3 
girls have in all ?

My solution comes in this way :

Sandy o........o........o........o........o

Wendy o........o

Ann   o....o

I take 180, divide by 5 parts (Sandy has 4 parts plus 1 part from 
Wendy). It will give the quotient of 36 for each part.

Looks like I am unable to proceed further. Please help!

Date: 03/13/98 at 12:23:30
From: Doctor Lim
Subject: Re: word problem

Hi there,

You were almost there. Let us go over your problem.

>Sandy has thrice as many beads as Wendy. 

Sandy [____________][____________][____________]
Wendy [____________]

>Ann has half as many beads as Wendy. 

Sandy [____________][____________][____________]
Wendy [____________]
Ann   [_____]

This means that: 

Wendy [____________]
Wendy [_____][_____]
Ann   [_____]

This also means that:
 
Sandy [____________][____________][____________]
Sandy [_____][_____][_____][_____][_____][_____]
Wendy [_____][_____]
Ann   [_____]

We finally get:

Sandy [_____][_____][_____][_____][_____][_____]    ---> 6 units
Wendy [_____][_____]                                ---> 2 units 
Ann   [_____]                                       ---> 1 unit

>If Sandy has 180 beads more than Ann, 

Sandy [_____][_____][_____][_____][_____][_____]
Ann   [_____] <----------- 180 ---------------->

Do you see the relationship now? 

Yes that is right. 5 units ===> 180 beads.

The total number of units that the girls have are 9 units. So

5 units ===> 180 beads
9 units ===> 180/5 * 9 beads
             = ??

I am sure that you will be able to get the answer now.

Now change the problem a little and see if you can get the method 
right:

Sandy has four times as many beads as Wendy. Ann has half as many 
beads as Wendy. If Sandy has 210 beads more than Ann, how many beads 
do the 3 girls have in all?

If a word problem looks difficult, read it sentence by sentence. You 
will find that the problem is not that difficult after all.

-Doctor Lim, The Math Forum
 http://mathforum.org/dr.math/   

Date: 03/14/98 at 22:51:36
From: Sherilyn
Subject: word problem

Thank you for helping me in my word problem which you have answered on 
14 Mar '98. Your explanation is very clear and this gives me more 
confidence in tackling such sums in future. Meanwhile, I have this 
problem sum; please tell me if my answer is correct.

The question is:
If Paul gives 20 of his marbles to Simon, he will have three times as 
many marbles as Simon. If he gives 10 of his marbles to Simon, he will 
have five times as many marbles as Simon. How many marbles does Paul 
have?

My working is :

[10][10][10][10][10][10][10][10]    Paul's original no. of marbles
[10][10]                            That's for Simon

80 minus 20 = 60

60 minus 10 = 50

Meaning Paul will have 50 marbles. (Is this the answer?)

Please help!!

Date: 03/16/98 at 16:12:16
From: Doctor Lim
Subject: Re: word problem

Hi there,

I am glad to be of help. Let us try to work out this one.

> If Paul gives 20 of his marbles to simon, he will have three times 
as many marbles as Simon. 

Model A

At first, Simon has [___]

When Paul gives 20 marbles to Simon,
Paul [_______][_______][_______]   
Simon[_______]     

To make this in relation to Simon,
Paul [___][20][___][20][___][20]
Simon[___][20]

>If he gives 10 of his marbles to Simon, he will have five times as 
many marbles as Simon. How many marbles does Paul have?

Model B

At first, Simon has [___]

When Paul gives 10 marbles to Simon,

Paul [______][______][______][______][______]
Simon[______] 

To make this in relation to Simon,
Paul [__][10][__][10][__][10][__][10][__][10]
Simon[__][10]

Your working is not right, as the two models you drew are not of the 
same unit, so you cannot just take away from them.

From my two sets of models, I still cannot find an identical unit, so 
I must redraw them.

Instead of having: 

Model A
Simon[___][20]

I have to change it to the following to suit the second statement:

Model A
Simon [__][10][10]

This is to make this model the same unit as the first and second 
statement.

==================
Model A

Paul  [__][10][10][__][10][10][__][10][10]
Simon [__][10][10]

Paul  [__________][__________][__________]
Simon [__________]

==================
Model B

Paul [______][______][______][______][______]
Simon[______] 

Paul [__][10][__][10][__][10][__][10][__][10]
Simon[__][10]

==================
Now I have to compare the two models. I'm going to call Paul and Simon 
"Paul A" and "Simon A" when working with Model A, and "Paul B" and 
"Simon B" when working with Model B.

Paul A  [__][10][10][__][10][10][__][10][10]
Simon A [__][10][10]

Paul B  [__][10][__][10][__][10][__][10][__][10]
Simon B [__][10]

I shall return [10] from Simon A to Paul A. This is to make Simon A 
and Simon B the same.

Paul A  [__][10][10][__][10][10][__][10][10][10]
Simon A [__][10]

Paul B  [__][10][__][10][__][10][__][10][__][10]
Simon B [__][10]

==================
Paul A  [__][10][10][__][10][10][__][10][10][10]
Paul B  [__][10][__][10][__][10][__][10][__][10]

==================
We rearrange them:

Paul A  [__][__][__][10][10][10][10][10][10][10]
Paul B  [__][__][__][__][__][10][10][10][10][10]

Paul A has 3 [__] and 7 [10]
Paul B has 5 [__] and 5 [10]

We take away from those that are the same:

Paul A  [__][__][__][10][10][10][10][10]  [10][10]
Paul B  [__][__][__][10][10][10][10][10]  [__][__]

We now know that [10][10] is the same as [__][__].

That means that [__] is the same as [10]

==================
Simon has [__]. This means that Simon has [10] or 10 marbles.

Let us go back to Model A:
Paul  [__][10][10][__][10][10][__][10][10]
Simon [__][10][10]

If [__] means 10 marbles, then Paul has 
[__][__][__] + [10][10][10][10][10][10].
So Paul has 10 + 10 + 10 + 60 = 90.
But Paul gave 20 to Simon.

That means that Paul has 90 + 20 = 110 marbles at first.

==================
This problem is difficult, as you have to compare both the models that 
you have drawn. You have to make sure that the units you draw are 
identical. Otherwise, you cannot compare them.

The difference in my m models is that I had broken up the 20 marbles 
that Paul gave to Simon as 10 and 10. In this way, I can break up the 
models easily.

Don't worry about these kind of word problems. You have to play around 
with the statements and think of the possibilities of the type of 
models that you get. Once you have understood these concepts, word 
problems are no longer a "problem."

Have fun working it out. How about trying it with other numbers or 
working your problem backwards? You can then give that word problem to 
your friends and get them to solve it. This will be another way to 
revise. It will be fun for you and your friends to think that you, not 
just your teacher, can make difficult word problems.

-Doctor Lim, The Math Forum
 http://mathforum.org/dr.math/