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.
The proportion is the 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 the 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 the equality a%2Fb+=+c%2Fd, then multiplying both sides by bd you get the equality
ad+=+bc.

Inversely, if you have the equality ad+=+bc, then dividing both sides by bd you get
a%2Fb+=+c%2Fd
(providing 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 the 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+=+ad%2Fc.
If the other mean term c is unknown, you can calculate it via other terms as c+=+ad%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+=+9%2A5%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+=+9%2A20%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 extrems divided by the known mean:
    n+=+4%2A15%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+=+4.5%2A15%2F13.5+=+5.

Summary
In the 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.

For more examples of solved problems using proportions see lessons
Using proportions to solve word problems and
Ratio and Proportion by Rapalje in this module.

Examples of using proportions to solve problems in Physics ans Chemistry are given in lessons
Using proportions to solve word problems in Physics and
Using proportions to solve Chemistry problems in this module.
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