Lesson Proportions

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<H2>Proportions</H2>

You probably just learned that the &nbsp;ratio&nbsp; is the fraction &nbsp;(see, &nbsp;for example, &nbsp;the lesson &nbsp;<A HREF= http://www.algebra.com/algebra/homework/proportions/lessons/ratios.lesson>Introduction to Ratios</A>&nbsp; in this module).
In this lesson we consider &nbsp;proportions&nbsp; and their major property.

<BLOCKQUOTE>Definition.

A proportion is an equality of two ratios.
</BLOCKQUOTE>

Examples

1. &nbsp;{{{1/2}}} = {{{2/4}}} &nbsp;is the proportion.
2. &nbsp;{{{2/4}}} = {{{4/8}}} &nbsp;is the proportion.
3. &nbsp;{{{2/7}}} = {{{4/14}}} &nbsp;is the proportion.
4. &nbsp;{{{12/7}}} = {{{24/14}}} &nbsp;is the proportion.

5. &nbsp;{{{1/2}}} = {{{3/4}}} &nbsp;is not a proportion, &nbsp;because the ratios to the left and to the right are not equal.
6. &nbsp;{{{2/7}}} = {{{4/7}}} &nbsp;is not a proportion, &nbsp;because the ratios to the left and to the right are not equal.

Usually, &nbsp;proportions are written using four numbers in the following format: 
{{{a/b}}} = {{{c/d}}}.

If &nbsp;{{{a/b}}} = {{{c/d}}} &nbsp;is a proportion, &nbsp;then numbers &nbsp;{{{a}}} &nbsp;and &nbsp;{{{d}}} are called &nbsp;extreme terms of the proportion; &nbsp;numbers &nbsp;{{{b}}} &nbsp;and {{{c}}} &nbsp;are called &nbsp;mean terms of the proportion.

As you know, &nbsp;the necessary and the sufficient condition for two ratios &nbsp;{{{a/b}}} &nbsp;and &nbsp;{{{c/d}}} &nbsp;to be equal is
{{{ad}}} = {{{bc}}}.

Indeed, &nbsp;if you have an equality &nbsp;{{{a/b}}} = {{{c/d}}}, &nbsp;then multiplying both sides by &nbsp;{{{bd}}} &nbsp;you get the equality
{{{ad}}} = &nbsp;{{{bc}}}.

Inversely, &nbsp;if you have an equality &nbsp;{{{ad = bc}}}, &nbsp;then dividing both sides by &nbsp;{{{bd}}}&nbsp; you get
{{{a/b}}} = {{{c/d}}}
(provided neither &nbsp;{{{b}}} &nbsp;nor &nbsp;{{{d}}} &nbsp;are equal to zero).

<BLOCKQUOTE>This is the &nbsp;major property of the proportion: 
the product of extremes is equal to the product of means.
</BLOCKQUOTE>

<BLOCKQUOTE>&nbsp;&nbsp;&nbsp;&nbsp;Thus, this is the same to say &nbsp;&nbsp;{{{a/b}}} = {{{c/d}}};

&nbsp;&nbsp;&nbsp;&nbsp;{{{a/b}}} = {{{c/d}}} &nbsp;is a proportion;

&nbsp;&nbsp;&nbsp;&nbsp;{{{ad}}} = {{{bc}}}.
</BLOCKQUOTE>

If in a proportion &nbsp;&nbsp;{{{a/b=c/d}}} &nbsp;&nbsp;three terms are known and one term is unknown, you can calculate the unknown term via known ones.

For example, if the first term &nbsp;{{{a}}} &nbsp;is unknown, &nbsp;you can calculate it via other terms as &nbsp;{{{a = bc/d}}}.

If the fourth term &nbsp;{{{d}}} &nbsp;is unknown, &nbsp;you can calculate it via other terms as &nbsp;{{{d = bc/a}}}.
This follows from the major property of the proportion &nbsp;{{{ad}}} = {{{bc}}}.

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

If the mean term &nbsp;{{{b}}} &nbsp;is unknown, &nbsp;you can calculate it via other terms as &nbsp;{{{b}}} = {{{(ad)/c}}}.
If the other mean term &nbsp;{{{c}}} &nbsp;is unknown, &nbsp;you can calculate it via other terms as {{{c}}} = {{{(ad)/b}}}.

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

Examples

7. &nbsp;In the proportion &nbsp;{{{x/9}}} = {{{5/15}}} &nbsp;find the unknown &nbsp;{{{x}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Solution

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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x}}} = {{{(9*5)/15}}} = {{{3}}}.

8. &nbsp;In the proportion &nbsp;{{{4/9}}} = {{{20/y}}} &nbsp;find the unknown {{{y}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Solution

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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = {{{(9*20)/4}}} = {{{45}}}.

9. &nbsp;In the proportion &nbsp;{{{4/n}}} = {{{5/15}}} &nbsp;find the unknown &nbsp;{{{n}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Solution

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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{n}}} = {{{(4*15)/5}}} = {{{12}}}.

10. &nbsp;In the proportion &nbsp;{{{4.5/13.5}}} = {{{m/15}}} &nbsp;find the unknown &nbsp;{{{m}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Solution

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The unknown mean of the proportion is equal to the product of extremes divided by the known mean:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{m}}} = {{{(4.5*15)/13.5}}} = {{{5}}}.

Summary

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

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

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

My other lessons on &nbsp;proportions&nbsp; in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/proportions/lessons/Using-proportions-to-solve-word-problems.lesson>Using proportions to solve word problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/proportions/lessons/Using-proportions-to-solve-word-problems-in-Physics.lesson>Using proportions to solve word problems in Physics</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/proportions/lessons/Using-proportions-to-solve-Chemistry-problems.lesson>Using proportions to solve Chemistry problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/proportions/lessons/Typical-problems-on-proportions.lesson>Typical problems on proportions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/proportions/lessons/Using-proportions-to-estimate-the-number-of-fish-in-a-lake.lesson>Using proportions to estimate the number of fish in a lake</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/proportions/lessons/HOW-TO-algebreze-and-solve-this-problem-using-proportions.lesson>HOW TO algebraize and solve these problems using proportions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/proportions/lessons/Using-proportions-to-solve-word-problems-in-Geometry.lesson>Using proportions to solve word problems in Geometry</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/proportions/lessons/Using-proportions-to-solve-some-nice-simple-Travel-and-Distance-problems.lesson>Using proportions to solve some nice simple Travel and Distance problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/proportions/lessons/Miscellaneous-problems-on--proportions.lesson>Advanced problems on proportions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/proportions/lessons/Problems-on-proportions-for-mental-solution.lesson>Problems on proportions for mental solution</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/proportions/Selected-problems-on-proportions-from-the-archive.lesson>Selected problems on proportions from the archive</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/proportions/lessons/Entertainment-problems-on-proportions.lesson>Entertainment problems on proportions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/proportions/lessons/OVERVIEW-of-lessons-on-proportions.lesson>OVERVIEW of lessons on proportions</A>