# Properties of Proportions

Before going to properties of proportion we need to know what is a proportion, what are means and extremes. Proportion means balance among the parts of something. In this context it implies that two ratios are equal.

It can be written as

`a / b = c / d`

or  a : b = c : d

Earlier it was written as a: b :: c:d , which has become obsolete these days.

In the proportion a / b = c / d, b and c are means and a and d are extremes.

An example of proportion is   `50 / 40 = 5 / 4`in which 40 and 5 are means and 50 and 4 are extremes.

If two ratios are proportions are in proportion then the product of extremes must be equal to the product of means. That is, ad= bc. The operation of componendo, dividendo and convertendo applied to the proportion a/b= c/d give

(a±b) /b= (c±d)/d

a/(a±b)= c/(c±d)

a/b= c/d= (a±c)/(b±d)

This is also known as simple proportion. There are two kinds of proportions –Direct proportion and inverse proportion.

These are very useful while learning proportions.

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## Properties of Proportion:

• Property 1:
Product of Means = Product of Extremes

Consider 50 / 40 = 5 / 4 in which 40 and 5 are means and 50 and 4 are extremes in proportion.

so 40 × 5 = 50 × 4

• Property 2:

If a / x = x / b

then x2 = a.b

Here x is mean proportion of a and b.

• Property 3:

Duplicate ratio of a:b is a2: b2 and sub duplicate ratio of a:b is ?a : ?b

• Property 4:

Triplicate ratio of a:b is a3:b3  and sub-triplicate ratio of a:b is 3?a : 3?b.

• Property 5 (Alternando Rule):

If a:b = c:d then a :c = b :d

• Property 6(Componendo Dividendo Rule):

a/b = c/d then (a+b / a-b) = (c+d / c-d)

• Property 7:

If a/b = c/d = e/f then a/b = c/d = e/f= (a+b+e / c+d+f)

These are the seven properties of proportions.

Students can also get help on solving proportions with variables involving different topics from the online tutors.

## Example on Proportion:

Example 1: Find a if a/4= 3/2.

Solution:

By using property 1:

a/4=3/2

(a) (2) = (4) (3)

2a = 12

Dividing both side by 2

a=6

Therefore the value of a=6

Example 2: Is 6: 4 = 3: 2 a proportion ?

Yes. as Property 1 satisfied

(6) (2) = (4) (3)

12   = 12, this is true. It is a proportion.

Example 3: A map is scaled so that 10 cm on the map is equal to 14 actual miles. If two cities on the map are 20 cm apart, what is the actual distance the cities are apart?

Let = the actual distance.

Map/actual= 10/14=20/x

Apply the Cross-Products Property.

10x= 280

x=28

The cities are 28 miles apart.