Using a Diagram to Find a Formula

Date: 11/13/2002 at 18:03:40
From: Kim
Subject: Comparing ratio data

A bag contains green, yellow and orange marbles. The ratio of green 
to yellow marbles is 2:5. The ratio of yellow to orange marbles is 
3:4. What is the ratio of green marbles to orange marbles?

I tried different formulas and I'm still not sure how to go about 
solving this with a formula. I did solve it by diagramming, but I 
know that there is a way to figure it out quicker without drawing all 
those marbles!  My answer was 3:10. I still want a formula. Please.

My daughter tried to show me but I still need help.  
Thanks,  
Kim

Date: 11/15/2002 at 00:00:26
From: Doctor Ian
Subject: Re: Comparing ratio data

Hi Kim,

The trick is to use the diagram to _find_ an equation. If the ratio of 
green to yellow marbles is 2:5, then they have to come in groups of
2 and 5:  

  (g g y y y y y)

Similarly, if the ratio of yellow to orange marbles is 3:4, then they
have to come in groups of 3 and 4:
 
  (y y y o o o o)

Now, to compare these, you have to have some number of yellow marbles
that is divisible by both 3 and 5. An easy way to do that is to
multiply the first ratio by 3 to get 6:15, and the second one by 5 to
get 15:20. Now we can merge the ratios:

  g  y       y  o
  6:15 <--> 15:20

  g  y  o
  6:15:20

So the ratio of green to orange is 6:20, which reduces to 3:10, which
is what you got. (Whew!)

In the general case, 

  a:b and c:d

we would do this:

  ac:bc  <-->  bc:bd

  ac:bc:bd

so the final ratio would be ac:bd. Let's check that with your example: 
a = 2, b = 5, c = 3, d = 4, so the ratio should be 

  ac:bd = 2*3:5*4

        = 6:20

        = 3:10

Does that make sense? 

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