Ratio

1. Ratio:
A ratio is simply a fraction. The following notations all express the ratio of x to y
= > x:y , x÷y , or x/y.

In the ratio x:y, we call x as the first term or antecedent and y the second term or consequent.
Writing two numbers as a ratio provides a convenient way to compare their sizes. For example, since 3A ratio compares two numbers. Just as you cannot compare mangoes and pineapples, so the numbers you are comparing must have the same units.
For example, you cannot form the ratio of 2 feet to 4 meters because the two numbers are expressed in different units. feet vs. meters.

Example 1:
4 : 5 = 8 : 10 = 12 : 15.
Also, 4 : 6 = 2 : 3.

2. Proportion:
The equality of two ratios (fractions) is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
Product of means=Product of extremes
So,
a:b::c:d?(bxc)=(axd)

2.1 Fourth Proportional:
If a:b=c:d, then d is called the fourth proportional to a,b,c.

2.2 Third Proportional:
a:b=c:d, then c is called the third proportional to A and B.

2.3 Mean Proportional:
Mean proportional between a and b is √ab.

2.4 Comparison Of Ratios:
We say that (a:b)>(c:d) we say ad>bc

2.5 Compounded Ratio:
The compounded ratio of the ratios: (a:b),(c:d),(e:f) is (ace:bdf)

2.6 Duplicates Ratios:
Duplicates ratio of (a:b) is (a2:b2)

2.7 Subduplicates Ratios:
Sub-duplicates ratio of (a:b) is (a1/2:b1/2)

2.8 Triplicate Ratios:
Triplicates ratio of (a:b) is (a3:b3)

2.9 Subtriplicate Ratios:
Sub-triplicates ratio of (a:b) is (a>1/3:b1/3)
If ab=cd then, a+b/a-b=c+d/c?d [Componendo and Dividendo]