# Ratio and Proportion

## Ratio and Proportion:

(1) Ratio- Antecedent, Consequent.

(A) Definition- The ratio of one quantity to another of the same kind expressed in the same units- is defined to express what multiple part or parts the former is of the latter. It is an abstract number- integral or fractional.

The ratio of one quantity “a” to another quantity “b”- both of the same kind and expressed in the same units is stated as a : b or a/b.

Thus \$25 is five times as big as \$5.

We say the ratio of \$25 to \$5 is as 5 is to 1 and is denoted by 5 : 1 or 5/1.

The quantity “a” is called the Antecedent and “b” the Consequent. The two together are called the terms of the ratio.

Note: Remember that the ratio exists only between two quantities of the same kind. Thus there is no ratio between 14 horses and 8 cows.

(B) Ratio and its Kinds:

(i) If the antecedent of a ratio is equal to the consequent, it is called a ratio of equality; as the ratio 3 : 3.

(ii) If the antecedent is greater than the consequent, the ratio is called a ratio of greater inequality; as the ratio 7 : 5.

(iii) If the antecedent is less than the consequent, the ratio is called a ratio of less inequality; as the ratio 3 : 5.

(iv) If the antecedent and the consequent of a ratio are interchnaged, the ratio so formed is called the inverse of the given one. Thus the ratio c : d is the inverse of the ratio of d : c.

(C) Theorems:

(D) Composition of Ratios:

(i) Compounded Ratio- When the antecedents of two or more ratios are multiplied to form the new antecedent and consequent are multiplied to form the new consequent, the new ratio thus obtained is known as their compounded ratio.

Examples-

• The compounded ratio of a : b and c : d is ac : bd.
• The compounded ratio of a : b, p : q and x : y is apx : bqy.

(ii) Duplicate Ratio- It is the compounded ratio of two equal ratios. Thus, the duplicate ratio of a : b is the ratio of a2 : b2.

(iii) Triplicate Ratio- It is the compounded ratio of three equal ratios. Thus, the triplicate ratio of a : b is the ratio of a3 : b3.

(iv) Sub-Duplicate Ratio- The sub-duplicate ratio of a : b is defined as √a : √b.

(v) Sub-Triplicate Ratio- The sub-triplicate ratio of a : b is defined as ∛a : ∛b.

(vi) Reciprocal Ratio- The reciprocal ratio of the ratio a : b is 1/a : 1/b i.e. b : a.

(E) Illustrative Examples:

(2) Proportion-

(A) Definition- Four quantities a, b, c, d are said to be in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth i.e. when a : b = c : d.

This relation is also expressed as a : b : : c : d or a/b = c/d.

The quantities a and d are called the Extremes, while the quantities b and c are known as the Means. The fourth quantity d is called the Fourth Proportional to a, b, and c.

(B) Fundamental Theorems on Proportion:

(C) Continued Proportions- The quantities a, b, c, d, e, f, …. are said to be in continued proportion if

a/b = b/c = c/d = d/e = e/f = ….

In particular, three non-zero quantities of the same kind a, b, and c are said to be in continued proportion if the ratio of a to b is equal to the ratio of b to c i.e. a : b = b : c.

(D) Mean Proportional and Third Proportional- Let a, b, c be in continued proportion.

Here, “a” is called the first proportional, “b” is called the mean proportional, and “c” is called the third proportional.

(E) Illustrative Examples: