Lesson Proportions

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Proportions


You probably just learned that the  ratio  is the fraction  (see,  for example,  the lesson  Introduction to Ratios  in this module).
In this lesson we consider  proportions  and their major property.

Definition.

A proportion is an equality of two ratios.

Examples

1.  1%2F2 = 2%2F4  is the proportion.
2.  2%2F4 = 4%2F8  is the proportion.
3.  2%2F7 = 4%2F14  is the proportion.
4.  12%2F7 = 24%2F14  is the proportion.

5.  1%2F2 = 3%2F4  is not a proportion,  because the ratios to the left and to the right are not equal.
6.  2%2F7 = 4%2F7  is not a proportion,  because the ratios to the left and to the right are not equal.

Usually,  proportions are written using four numbers in the following format:
a%2Fb = c%2Fd.

If  a%2Fb = c%2Fd  is a proportion,  then numbers  a  and  d are called  extreme terms of the proportion;  numbers  b  and c  are called  mean terms of the proportion.

As you know,  the necessary and the sufficient condition for two ratios  a%2Fb  and  c%2Fd  to be equal is
ad = bc.

Indeed,  if you have an equality  a%2Fb = c%2Fd,  then multiplying both sides by  bd  you get the equality
ad =  bc.

Inversely,  if you have an equality  ad+=+bc,  then dividing both sides by  bd  you get
a%2Fb = c%2Fd
(provided neither  b  nor  d  are equal to zero).

This is the  major property of the proportion:
the product of extremes is equal to the product of means.

    Thus, this is the same to say   a%2Fb = c%2Fd;

    a%2Fb = c%2Fd  is a proportion;

    ad = bc.


If in a proportion   a%2Fb=c%2Fd   three terms are known and one term is unknown, you can calculate the unknown term via known ones.

For example, if the first term  a  is unknown,  you can calculate it via other terms as  a+=+bc%2Fd.
If the fourth term  d  is unknown,  you can calculate it via other terms as  d+=+bc%2Fa.
This follows from the major property of the proportion  ad = bc.

Thus,  the unknown extreme of the proportion is equal to the product of means divided by the known extreme.

If the mean term  b  is unknown,  you can calculate it via other terms as  b = %28ad%29%2Fc.
If the other mean term  c  is unknown,  you can calculate it via other terms as c = %28ad%29%2Fb.

Thus,  the unknown mean of the proportion is equal to the product of extremes divided by the known mean.


Examples


7.  In the proportion  x%2F9 = 5%2F15  find the unknown  x.

     Solution

     The unknown extreme of the proportion is equal to the product of means divided by the known extreme:

     x = %289%2A5%29%2F15 = 3.


8.  In the proportion  4%2F9 = 20%2Fy  find the unknown y.

     Solution

     The unknown extreme of the proportion is equal to the product of means divided by the known extreme:

     y = %289%2A20%29%2F4 = 45.


9.  In the proportion  4%2Fn = 5%2F15  find the unknown  n.

     Solution

     The unknown mean of the proportion is equal to the product of extremes divided by the known mean:

     n = %284%2A15%29%2F5 = 12.


10.  In the proportion  4.5%2F13.5 = m%2F15  find the unknown  m.

     Solution

     The unknown mean of the proportion is equal to the product of extremes divided by the known mean:
     m = %284.5%2A15%29%2F13.5 = 5.


Summary

        In a proportion, the product of extremes is equal to the product of means.

        The unknown extreme of the proportion is equal to the product of means divided by the known extreme.

        The unknown mean of the proportion is equal to the product of extremes divided by the known mean.


My other lessons on  proportions  in this site are
    - Using proportions to solve word problems
    - Using proportions to solve word problems in Physics
    - Using proportions to solve Chemistry problems
    - Typical problems on proportions
    - Using proportions to estimate the number of fish in a lake
    - HOW TO algebraize and solve these problems using proportions
    - Using proportions to solve word problems in Geometry
    - Using proportions to solve some nice simple Travel and Distance problems
    - Advanced problems on proportions
    - Problems on proportions for mental solution
    - Selected problems on proportions from the archive
    - Entertainment problems on proportions
    - OVERVIEW of lessons on proportions

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