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Ratio and Proportion Formulas - Aptitude Questions and Answers Formulas Page

Ratio and Proportion - Formulas

Ratio

A ratio compares two quantities of the same unit, expressed as a fraction a/b or a:b. The first term (a) is the antecedent, and the second term (b) is the consequent. The ratio remains unchanged if both terms are multiplied or divided by the same non-zero number.

Ratio of a to b is written as a:b or a/b
Antecedent is the first term (a)
Consequent is the second term (b)
Rule: Multiplying or dividing each term by the same non-zero number does not change the ratio
Example: 4:5 = 8:10 = 12:15; 4:6 = 2:3

Proportion

Proportion is the equality of two ratios. When four quantities a, b, c, d are in proportion, the product of the means equals the product of the extremes.

If a:b = c:d, we write a:b :: c:d
Extremes: a and d (first and last terms)
Means: b and c (middle terms)
Product of means = Product of extremes: (b × c) = (a × d)

Types of Proportionals

Different proportional relationships exist when comparing three or four quantities in proportion.

Fourth Proportional: In a:b = c:d, d is the fourth proportional to a, b, c
Third Proportional: In a:b = b:c, c is the third proportional to a and b
Mean Proportional: Between a and b is √(ab) — [geometric mean: square root of the product of the two numbers]

Comparison of Ratios

Ratios can be compared by converting them to fractions and comparing the fractional values.

(a:b) > (c:d) when a/b > c/d

Special Ratios

Certain operations on ratios produce specific types of derived ratios with defined mathematical relationships.

Compounded Ratio: For ratios (a:b), (c:d), (e:f), the compounded ratio is (ace:bdf)
Duplicate Ratio: of (a:b) is (a²:b²)
Sub-duplicate Ratio: of (a:b) is (√a:√b)
Triplicate Ratio: of (a:b) is (a³:b³)
Sub-triplicate Ratio: of (a:b) is (∛a:∛b)

Important Properties

Key mathematical properties and theorems related to ratios and proportions, including the componendo and dividendo rule.

If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d) — [componendo and dividendo: used to simplify proportional relationships]

Variations

Variation describes how one quantity changes in relation to another, either directly or inversely.

Direct Proportion: x ∝ y means x = ky for constant k
Inverse Proportion: x ∝ 1/y means xy = k for constant k

Chapter 3