Solving a Ratio with and without a Diagram

Date: 05/12/2003 at 12:56:28
From: Winona
Subject: Ratio

In an auditorium, the ratio of the number of girls to the number of 
boys was 5:9.  When 203 girls entered the auditorium, the new ratio of 
the number of girls to the number of boys became 4:3. How many pupils 
were in the auditorium at first?

How can I solve this without using a diagram?

Date: 05/12/2003 at 18:30:13
From: Doctor Ian
Subject: Re: Ratio

Hi Winona,

I'd probably use a diagram, but if you don't want to, you might 
approach it this way. Suppose the number of boys is B, and the number
of girls is G. Then we know that 

  G   5
  - = -
  B   9

After 203 girls enter the auditorium, we have

  G+203   4
  ----- = -  
  B       3

How does this help?  Well, from the orignal situation, we know that 

      5
  G = - * B
      9

Do you see why?  So we can substitute for G in the modified situation:

  (5/9)B + 203   4
  ------------ = -  
  B              3

Can you take it from here? 

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

Date: 05/13/2003 at 09:58:51
From: Winona
Subject: Thank you (Ratio)

Thanks Dr. Ian,

My mom wants to know which way is easier, using a diagram or the way 
that you showed me?
 
Bye,
Winona

Date: 05/13/2003 at 10:50:33
From: Doctor Ian
Subject: Re: Thank you (Ratio)

Hi Winona,

It's sort of like asking: Which is easier, tennis or golf? It depends 
on who you are, and what you find easy. It also depends on what kind 
of diagram you draw. I didn't draw one, because you told me not to use 
one; and I didn't see the one that you drew. 

I like diagrams because they help me keep my information straight, and 
a good one can help me solve a problem without having to set up and 
solve equations. But every problem is different, and there's no point 
in making a diagram unless it's going to be helpful. 

Let's look at the problem again:

  In an auditorium, the ratio of the number of girls to the 
  number of boys was 5:9. When 203 girls entered the auditorium, 
  the new ratio of the number of girls to the number of boys 
  became 4:3. How many pupils were in the auditorium at first?

The first thing I notice here is that I'd like both the ratios to be
'something:9', since that makes them easier to compare. So I'd change
the second ratio from 4:3 to 12:9, which has the same meaning:

  In an auditorium, the ratio of the number of girls to the 
  number of boys was 5:9. When 203 girls entered the auditorium, 
  the new ratio of the number of girls to the number of boys 
  became 12:9. How many pupils were in the auditorium at first?

Now, without doing anything more, I can see that I must have added 7
girls for each group of 9 boys. This means that 203 must be a multiple 
of 7, and in fact, 203 = 7 * 29. So I must have had 29 groups of 9 
boys to begin with, and 29 groups of 5 girls.  

Would a diagram help? Originally, I can divide the students into 
groups, where each group has 5 girls and 9 boys:

   ggggg
   bbbbbbbbb

   ggggg
   bbbbbbbbb

    .
    .

   ggggg
   bbbbbbbbb

Afterward, each group has 12 girls; that is, there are 7 extra girls
in each group:

   gggggGGGGGGG
   bbbbbbbbb

   gggggGGGGGGG
   bbbbbbbbb

    .
    .

   gggggGGGGGGG
   bbbbbbbbb

Now, there are 203 new girls, which is 29 groups of 7 girls:

       7    
    ------- *
    GGGGGGG |
    GGGGGGG |
     .      | 29
     .      |
    GGGGGGG |

So I must have started with 29 of these:

   ggggg
   bbbbbbbbb

In this case, the diagram didn't add anything. It just took extra time 
to draw. In other cases, a diagram can make the solution very easy to 
see without doing any calculations:

   Changing the Concentration of a Solution
   http://mathforum.org/library/drmath/view/60742.html 

But even when diagrams would he useful, some people have a hard time
making them, and prefer to use equations whenever possible. There are
lots of ways to solve any problem, and which one is 'best' depends on
who you are. 

Does this help?  

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