Section Topics

 Fraction, Fraction with Units, Fractions &  Ratios; and Proportionality forwards & backwards.

Section Pages

1. Two Term Ratios
2. New Fractions/Ratios from Old
3. Triple Term Ratios

Would you like to show yourself or others how to be  algebra power users? 

Ratios And Fractions
(or ratios versus fractions)

The following lesson cover the properties of two term and multiple term ratios.  Fractions may be identified with two term ratios.  Two fraction are equal or equivalent when and only when the corresponding two term ratios are equal or equivalent.  But fractions can be added, subtracted, multiplied and divided while the same operations are not defined  for two- and multiple term ratios.  While we may call a fraction, a ratio or a rational number, ratios are different. Triple term ratios exist, but  triple term fractions do no exist. 

1. Fractions As (two term) Ratios and Fractions Versus Ratios: Fractions are often called ratios, and vice-versa.  But the vice-versa only holds for two term ratios. This lesson identifies fractions with two-term ratios and contrasts the properties of fractions and two-term ratios.   (Ratios cannot be added, subtracted or compared, but like fractions, the terms in ratios can be raised or lowered).
   
2. Implied or Derived Ratios - New Fractions and Ratios from Old:  If two fractions are equivalent then their reciprocals are also equivalent.  Likewise if a pair of two-term ratios are equivalent, interchanging the first and second terms of each ratio in the pair leads to a pair of equivalent ratios.  Beyond that, more equivalent ratios can also be generated from a pair of ratios.    Food for thought:   How may equivalent fractions or ratios may be formed from the relations   ad = bc?
   
3. Multiple Ratios: Multiple Term Ratios - Three Term Ratios to be precise. We read the triple ratio  a : b :c as a to b to c. We further write 

   a : b: c  ::  A: B: C

   to  say two triple ratios  a : b: c  and   A: B: C are equal or equivalent when and only when 

   a b

   =

   A B

   and

   b c

   =

   B C

   there are other ways to say when two triple ratios are equal or equivalent. 

   Note:  Triple ratios or triple proportionalities occur between the sides of similar triangles.  More generally, multiple ratios or proportionalities occur between the sides of similar triangles. 

The discussion of ratios or multiple ratios  is best understood besides a discussion of proportionality. 

Inner Versus Outer Terms - small point:  In the discussion of equality of ratios a : b =  A: B written in that order, the inner terms are small b and big A while the outer terms are small a and big B.  In contrast, if we rewrite the equality as   A: B = a : b, we find the inner and outer terms are interchanged. However, the equality requires the product of the inner and outer terms be equal, that is  aB = Ab.  That equality is not affected by rewriting a : b =  A: B as A: B = a : b, and the resulting swap of inner and outer terms 

 [ Back ] [ Next ] [Top of this Page]  
www.whyslopes.com?

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  All Rights Reserved.