Visualizing Ratios to Scale Least Common Multiples

Date: 10/02/2010 at 08:18:03
From: Louie
Subject: ratio

In my flock of sheep, there are three breeds: Cotsworlds, 
Swaledales and Leicester Longwools. 

The ratio of Cotsworlds to Swaledales is 3:5 and the ratio of
Swaledales to Leicester Longwools is 7:4.

a) What is the ratio of Cotsworlds to Leicester Longwools?

b) There are more than 100 sheep in my flock.  What is the 
smallest number of sheep there could be?

I know that C:S is 5:2, S:L is 7:4, and LCM of 5 and 7 is 35. 

I tried to make it equivalent and got that C:S is 21:35 and 
S:L is 35:20.  Is C:L therefore 21:20?

There are (21 + 35 + 20 =) 76 parts all together; so if there 
are more than 100 sheep, as it says in the second question, 
would I multiply 76 by 2 to get 152 as the answer?

Please help.

Date: 10/03/2010 at 10:45:38
From: Doctor Ian
Subject: Re: ratio

Hi Louie,

The way I think about this is, I imagine that I have some C's and 
some S's next to each other, in the right ratio:

   C C C S S S S S

And I have some S's and L's next to each other, also in the right 
ratio:

   C C C S S S S S
    
         S S S S S S S L L L L

Then I think about how I could scale those to get the same number of 
S's in both groupings.  The smallest number divisible by both 5 and 7 
is 35, as you wrote.  So I could have

  7*3 C's    7*5 S's

             5*7 S's    5*4 L's

or

  21 C's     35 S's     20 L's

And the actual numbers would have to be some multiple of that 
grouping, wouldn't it? 

It's basically the same sort of thing you're doing, but I find that 
the pictures help me keep track of what I know, and how each thing 
relates to everything else. 

Does this make sense? 

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

Date: 10/03/2010 at 11:48:10
From: Louie
Subject: Thank you (ratio)

Dear Dr. Ian,

Thank you so much for your answer. I hadn't thought of using the
pictures before.  Great idea.  Will help me a lot in dealing with
ratio problems in the future. 

Thank you again.
Louie