Ratio and Proportion

                                    Ratio and Proportion

Ratio
The ratio of two quantities of the same kind and in the same unit is the fraction that one quantity is of the other.
The ratio a is to b is the fraction a/b, written as a: b.

Terms of a Ratio
In the ratio a: b, we call a as the first term or antecedent and b the second term or consequent.

Example:
In the ratio 3:4, the first term or antecedent is 3 and the second term or consequent is 4.

An Important Property
A ratio remains unchanged if both of its terms are multiplied by the same nonzero quantity.
Let m is not equal to zero. Then, clearly we have:
i) a/b = ma/mb and therefore, (a:b) = (ma:mb).
ii) a/b =(a/m) / (b/m) and therefore, (a:b) = (a/m : b/m).

Ratio in simplest form
The ratio (a:b) is said to be in simplest form if HCF of a and b is 1.

Proportion
Four numbers a, b, c, d are said to be in proportion, if a : b = c:d and we write, a:b :: c:d.

1. If a:b :: c:d, then
i) a, b, c, d are respectively known as first, second, third and fourth term.
ii) a and d are called extremes while b and c are called means.
iii) (a x d) = (b x c), i.e., product of extremes = product of means.
iv) d is called the fourth proportional to a, b, c.

2. If a:b :: b:c, then
i) a, b, c are said to be in continued proportion.
ii) c is called third proportional to a, b and fourth proportional to a, b, b.
iii) a/b = b/c or b^2 = ac or b = sqrt(ac).
And, b is called the mean proportion or geometric mean between a and c.