Ratio and Proportion – Logical Reasoning Study Notes

1. Understanding Ratios: The Logic of Comparison

In competitive examinations like the UPSC CSAT, SSC CGL, and CGPSC, questions on ratios are rarely purely computational. Instead, they test your conceptual understanding of how quantities scale relative to one another. A ratio is denoted as a : b (read as “a is to b”) and can be written as a fraction a/b. Here, ‘a’ is known as the antecedent, and ‘b’ is known as the consequent.

One of the most critical properties of a ratio is that it is a dimensionless quantity. Because it compares two quantities of the same unit, the units cancel out. For example, the ratio of 5 kg to 15 kg is simply 1:3, not 1:3 kg. If you are given quantities in different units, you must convert them to a common unit before calculating the ratio. For instance, to find the ratio of 2 km to 500 meters, you must convert 2 km to 2000 meters first, yielding a ratio of 2000:500, which simplifies to 4:1.

Another fundamental concept is the Compound Ratio. When we multiply the corresponding terms of two or more ratios, we get a compound ratio. If the given ratios are a : b and c : d, their compound ratio is expressed as:

Compound Ratio = (a × c) : (b × d) = ac : bd

2. Proportions and Continued Proportions

When two ratios are equal, they are said to be in proportion. We represent this as a : b :: c : d (read as “a is to b as c is to d”) or mathematically as a/b = c/d. In this expression, the terms ‘a’ and ‘d’ are called the extremes, while ‘b’ and ‘c’ are called the means.

The foundational rule of proportion states that the product of the extremes must always equal the product of the means:

Product of Extremes = Product of Means ⇒ a × d = b × c

If three quantities a, b, and c are such that a : b :: b : c, they are said to be in continued proportion. In this scenario, ‘b’ is called the Mean Proportion between ‘a’ and ‘c’, and ‘c’ is called the Third Proportion to ‘a’ and ‘b’. Let’s define these mathematically:

• Mean Proportion: Since a/b = b/c, we get b² = ac, which means b = √(ac).
• Third Proportion: From the same relation, the third proportion c is calculated as c = b² / a.
• Fourth Proportion: If four quantities are in proportion (a:b :: c:d), the fourth proportion d is calculated as d = (b × c) / a.

3. Time-Saving Operations on Proportions

When solving complex algebraic or logical reasoning problems under strict time constraints, you can manipulate proportions using standard mathematical properties. These operations help bypass tedious calculations:

• Invertendo: If a/b = c/d, then b/a = d/c.
• Alternendo: If a/b = c/d, then a/c = b/d.
• Componendo: If a/b = c/d, then (a + b)/b = (c + d)/d.
• Dividendo: If a/b = c/d, then (a – b)/b = (c – d)/d.
• Componendo & Dividendo (C&D Rule): If a/b = c/d, then (a + b)/(a – b) = (c + d)/(c – d). This rule is highly effective in simplifying equations involving fractions and roots.

4. Distribution Ratios and Partitioning

A highly common question type in competitive exams involves dividing a total sum or quantity into parts based on a given ratio. Suppose a total quantity S needs to be distributed among three individuals A, B, and C in the ratio a : b : c.

Rather than relying on complex equations, we use the concept of a ratio constant (x). The shares of A, B, and C can be represented as ax, bx, and cx respectively. Since the sum of