What is a Ratio?

Date: 8/17/96 at 20:13:50
From: Lauretta Hughes
Subject: Introduction to Ratios

I need help understanding ratios.

Thanks.
  Lauretta

Date: 8/30/96 at 16:44:28
From: Doctor Jerry
Subject: Re: Introduction to Ratios

The term "ratio" can mean many things.  It is used, for example, in 
work with similar triangles.  If two triangles are similar, then the 
ratios of corresponding sides are equal. 

It is used when quantities are in proportion.  If the increase in 
length of a person's hair is directly proportional to the time since 
it was last cut, then ratios can be used to calculate various things.  
If I tell you, for example, that after 4 weeks the increase in length 
of my hair is about 1/2 inch, then I can ask for the number of weeks 
required for the increase to be 3.2 inches.  

Ordinarily, the thought process used is that 4 is to 1/2 as W is to 
3.2, where W is the number of weeks required for the increase to be 
3.2 inches.  This statement is equivalent to the equation 

   4/(1/2) = W/(3.2). 

Multiplying by 3.2 gives 

   W = 3.2*(4/(1/2)) = 3.2*(8) = 25.6 weeks.

The reason this works can be explained this way:  The statement that 
the increase in length of a person's hair is directly proportional to 
the time since it was last cut means that I = k*W, where  I  is the 
increase in length,  W  is the number of weeks, and  k  is a 
"proportionality constant."

So, if I1 and W1 are 1/2 and 4, then I1 = k*W1.  So, k = I1/W1.  
If I2 is 3.2 and W2 is the time required, then I2 = k*W2.  
But, k = I1/W1.  So I2 = (I1/W1)W2.  

This equation can be rearranged to give W1/W2 = I1/I2, which is the 
"thought process" mentioned above, that W1 is to W2 as I1 is to I2.

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