Ratios as Fractions

Date: 01/26/97 at 01:06:06
From: Michelle Gerard
Subject: Ratios 

I am in grade seven and I always have trouble with my math. I am 
particularly stuck in the area of ratios. For example, last month we 
started by trying to put ratios into simplest form, like 64:12 = ? 
Please help, Dr. Math!

Thank you very much, 
Michelle 

Date: 01/26/97 at 19:17:41
From: Doctor Mike
Subject: Re: Ratios 

Hello Michelle,
  
One way to look at 64:12 is as the fraction 64/12.  This is what is 
called an "improper" fraction because its numerator is bigger than its 
denominator, but it is still a perfectly good fraction.  You have 
probably had problems where you are supposed to reduce a fraction to 
its lowest terms.  An example is that the fraction 7/14 equals 1/2.  I 
wanted to say this so you will know this is not a completely new thing 
for you to learn how to do.  It's just the same old fraction problem 
in different words.
  
Back to your problem in fraction form: 64/12 is the same as 32/6, and 
that is the same as 16/3.  We cannot go any farther than that.  In 
terms of ratios, 64:12 equals 16:3. I hope this helps on the mechanics 
of what you are supposed to do for this kind of problem. 
  
Perhaps you need a little more info on what ratios are all about.  
Sometimes when a teacher is rushed she/he will do a quick explanation 
that everybody may not absorb completely.  A few examples might help.  
Suppose the fast-track advanced math class has 26 girls in it and 13 
boys.  That is a ratio of 26:13 but it is also a ratio of 2:1.  The 
ratio of 2:1 is easier to understand because it captures more simply 
the idea of "twice as many".  There are twice as many girls as boys.
By the way 26 + 13 = 39 students is too many for one math class!
  
Here is another one.  At age 12 you have probably spent 8 years in 
grades K through 7.  There is a ratio there, namely, the ratio of 
"number of years in school" to "number of years in your whole life".  
For you this ratio is 8:12 or 8-to-12. You could simplify that to 4:6 
and further to 2:3.  The same thing can be done with the fraction 8/12 
which is the fraction of your life you have spent in school (including 
summers). You reduce that fraction to 4/6 or 2/3.  Either way, as the 
ratio 2:3 or the fraction 2/3, it says you have spent two thirds of
your life as a person of school age.
  
One more little warning about how you use these words.  When you do 
ratios it is important for you (and your teacher) to be clear about 
WHAT is in a ratio to WHAT ELSE.  In the last example I gave there are 
lots of ratios we could talk about:
  
   8:12 -- school years to total years
   12:8 -- total years to school years
   8:4  -- school years to preschool years
   4:8  -- preschool years to school years
   12:4 -- total years to preschool years
   4:12 -- preschool years to total years
  
This is not something else for you to worry about learning. It's just 
an encouragement for everybody to try to make sense of what the 
numbers mean.  

I hope what I have said makes sense to you, and that on your next test 
the ratio of "your score : total points" will be 40:40 = 100:100 = 1:1 
(these are all different forms of the same ratio). 

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/