Introduction of sample proportion

Introduction of sample proportion:

          Two equal ratios form a proportion. Thus, if a:b=c:d then a,b,c,d are said to be sample proportional. We write this as, a:b::c:d. This is read as “a is to be is as c is to d”A and b are called the extremes and b an c are called the means. The terms d is called the fourth sample proportional to a,b,c.

Types of Sample Proportions:

Continued proportion: quantities a,b,c,d,…. Are said to be in continued proportion if, a/b=b/c=c/d….. i.e., a:b=b:c=c:d=……………….

Mean proportional:  consider a,b, and c are in continued proportion, Proportion will be represented as a:b::b:c. Here, b is called the mean proportional of a and c..

Also c is called the third proportional to a and b.. If a,b, c are in continued proportion, then ac:b² . Hence, if three quantities are in continued proportion, then the product of extremes is equal to the square of the mean.

Note: The product of the extremes is equal to the product of the means.

Inadvertent: If a:b::c:d, then, b:a::d:c. for, a/b=c/d gives a/c=b/d. This operation is called inadvertent. It says that if quantities are proportion they are also proportional when we take it inversely.

Alternendo: if a:b::c:d then a:c::b:c.

For, a/b=c/d gives a/c=b/d. This operation is called alternendo. It says that if four quantities are proportional, they are also proportional when taken alternatively.

Componendo: If a:b::c:d, then (a+b):b:(c+c):d.

Given that, a/b=c/d,

So,(a/b)+1=(c/d)+1

Or, (a+b)/b=(c+d)/d

This operation is called componendo.

Dividendo: If a:b::c:d,

Then (a-b):b::(c-d):d.

Given that, a/b=c/d,

So,(a/b)-1=(c/d)-1

Or, (a-b)/b=(c-d)/d.

This operation is called dividendo.

Componendo and Dividendo:

If a:b::c:d,

Then (a+b):(a-b)::(c+d):(c-d).

Using componendo and dividendo respectively, we get

(a+b)/b=(c+d)/d   -------------------------1

(a-b)/b=(c-d)/d.    -------------------------2

Dividing the respective sides of 1 by those of 2 we get

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

This operation is called componendo and dividendo.

Examples for Finding the Sample Proportion:

Ex 1  : 2/5=4/10

Sol :  2/5=4/10

If we do the cross multiplication, we will get

5*4 = 2 *10

20 = 20 so, : 2/5=4/10 is a proportion.

Ex  2:  2/5=4/10

Sol :    1/5=4/10

If we do the cross multiplication, we will get

5 X4 = 1 X 10

20 = 10 so, : 1/5=4/10 is not a proportion.