Welcome to this module designed to equip you with a strong understanding of Ratio and Proportion, and Averages, two fundamental quantitative aptitude topics crucial for the UPSC Civil Services Preliminary Examination (CSAT Paper II). These concepts are frequently tested and form the basis for solving many complex problems.

A ratio is a comparison of two quantities by division. It tells us how many times one quantity contains another. For example, if there are 5 apples and 3 oranges, the ratio of apples to oranges is 5:3.

A proportion is an equation stating that two ratios are equal. For instance, if the ratio of boys to girls in one class is 2:3, and in another class it's 4:6, these ratios are proportional because 2/3 = 4/6. This can be written as 2:3 :: 4:6.

Understanding different types of ratios helps in tackling diverse problems:

• Simple Ratio: A direct comparison (e.g., 5:3).
• Compound Ratio: Formed by multiplying the numerators and denominators of two or more ratios (e.g., the compound ratio of a:b and c:d is ac:bd).
• Duplicate Ratio: The ratio obtained by squaring the terms of a given ratio (e.g., the duplicate ratio of a:b is a²:b²).
• Sub-duplicate Ratio: The ratio obtained by taking the square root of the terms (e.g., √a:√b).
• Triplicate Ratio: The ratio obtained by cubing the terms (e.g., a³:b³).
• Sub-triplicate Ratio: The ratio obtained by taking the cube root of the terms (e.g., ³√a:³√b).

The average, or arithmetic mean, is the sum of a set of numbers divided by the count of those numbers. It represents a typical or central value in a dataset.

Understanding the properties of averages can simplify calculations:

• Property 1: The sum of observations is equal to the average multiplied by the number of observations. (Sum = Average × Number).
• Property 2: If each observation is increased or decreased by a constant, the average also increases or decreases by that constant.
• Property 3: If each observation is multiplied or divided by a constant, the average is also multiplied or divided by that constant.
• Property 4: The sum of deviations of observations from the average is always zero.

These concepts are often intertwined in CSAT problems. For instance, you might need to find the average of quantities that are in a given ratio, or determine a ratio based on average values. Understanding how to manipulate ratios and apply average formulas in conjunction is key to solving these integrated problems.

When tackling mock tests:

1. Understand the Question: Read carefully to identify what is being asked and what information is provided.
2. Identify the Concept: Determine if it's a ratio, proportion, average, or a combination.
3. Apply the Right Formula/Method: Use the appropriate formulas and properties.
4. Check Your Work: If time permits, quickly re-verify your calculations.
5. Learn from Mistakes: Analyze incorrect answers to understand where you went wrong.