The following figure contains three squares with areas of 100, 16 and 49 lying side by side as shown. By how much should the area of the middle square be reduced in order that the total length PQ of the resulting three squares is 19?

1. Fundamental Properties of Squares and Rectangles (basic)

To master quantitative aptitude, we must start with the most fundamental 2D shapes: the Rectangle and its perfect sibling, the Square. A rectangle is a four-sided polygon where opposite sides are equal and all internal angles are exactly 90°. If we call the longer side the 'length' (L) and the shorter side the 'width' (W), the Area is the product L × W, representing the total space inside. The Perimeter is the total boundary length, calculated as 2(L + W). Understanding these basics is essential as geometry often intersects with other subjects like Geography when calculating land area or territorial spans Geography of India, Transport, Communications and Trade, p.57.

The Square is a special, 'regular' type of rectangle where all four sides are equal (s). This symmetry simplifies its formulas: Area = s² and Perimeter = 4s. For competitive exams, the most critical skill is the 'reverse' calculation: if you are given the area of a square, the side length is always the positive square root of that area (s = √Area). This analytical approach to numbers and shapes has deep roots in Indian history; for example, the mathematician Brahmagupta authored Brahmasphuta-siddhanta, significantly advancing our understanding of mathematical properties during the Gupta period History, class XI (Tamilnadu state board 2024 ed.), The Guptas, p.100.

Feature	Rectangle	Square
Side Relationship	Opposite sides equal	All four sides equal
Area	Length × Width	Side × Side (s²)
Perimeter	2 × (Length + Width)	4 × Side

Sources: History, class XI (Tamilnadu state board 2024 ed.), The Guptas, p.100; Geography of India, Majid Husain (9th ed.), Transport, Communications and Trade, p.57

2. Inverse Relationship: Finding Side Length from Area (basic)

In quantitative aptitude, understanding the relationship between the dimensions of a shape and its total space is fundamental. While we often learn how to calculate the Area of a square by squaring its side ($Side^2 = Area$), the competitive edge comes from mastering the inverse: finding the Side Length when only the area is known. This requires the use of the Square Root ($√$). Mathematically, if a square has an area $A$, its side length $s$ is always $s = √A$. This isn't just a modern trick; ancient civilizations like those in Mesopotamia were using square-root tables as early as 1800 BCE to solve practical problems involving land and volume Themes in world history, Writing and City Life, p.25.

Why is this inverse relationship so vital? In many complex problems, you are presented with a layout or a map where only the total area of specific sections is provided. For instance, when archaeologists study the ruins of Vijayanagara, they look at square platforms and pillar bases to reconstruct the original plan of a temple THEMES IN INDIAN HISTORY PART II, An Imperial Capital: Vijayanagara, p.190. To find the distance between two walls or the length of a path, one must first extract the side lengths from the areas of these square structures. If you encounter a square with an area of 144, your mind should instantly translate that to a side of 12.

To master this, you should memorize the squares of numbers at least up to 25. This allows you to "see through" the area and identify the side length immediately. For example, if you see an area of 49, you know the side is 7; if the area is 225, the side is 15. This speed is crucial when you have to add multiple side lengths together to find a total distance or perimeter. Always remember: the area represents space (two dimensions), while the side length represents distance (one dimension).

Sources: Themes in world history, Writing and City Life, p.25; THEMES IN INDIAN HISTORY PART II, An Imperial Capital: Vijayanagara, p.190

3. Linear Dimensions of Composite Figures (intermediate)

When we talk about composite figures, we are referring to complex shapes created by joining two or more simple geometric forms—such as squares, rectangles, or triangles. In the context of quantitative aptitude, mastering the linear dimensions (length, width, or perimeter) of these figures is often a prerequisite for solving area or volume problems. To analyze these figures, we must first break them down into their individual components and understand how they are connected, much like how three resistors can be connected in a series or parallel circuit to create a larger system Science, Class X NCERT, Electricity, p.182.

The most critical skill here is the ability to translate area into linear distance. For instance, if you are working with a square, the side length is always the square root of its area (s = √A). When multiple squares are placed side-by-side in a row, their combined horizontal length is simply the sum of their individual side lengths. This concept is fundamental when measuring objects in the real world, whether you are calculating the total length of a notebook using a scale Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.145 or determining the spacing between parallel irrigation furrows in a field Indian Economy, Vivek Singh, Agriculture - Part II, p.333.

To solve problems involving the adjustment of composite dimensions, follow these logic steps:

• Step 1: Calculate the current linear dimension of each individual shape.
• Step 2: Find the total current dimension (e.g., Total Length = L₁ + L₂ + L₃).
• Step 3: Determine the "target" dimension required by the problem.
• Step 4: Calculate the difference and adjust the specific component's dimension accordingly to meet the new total.

Remember that changing a linear dimension has a squared effect on the area. If you reduce the side of a square to half its length, its area becomes one-fourth of the original.

Sources: Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.145; Indian Economy, Vivek Singh, Agriculture - Part II, p.333; Science, Class X NCERT, Electricity, p.182

4. Pythagoras Theorem and its Geometric Applications (intermediate)

At the heart of many geometric aptitude problems lies the Pythagoras Theorem, a fundamental principle used to find the relationship between the sides of a right-angled triangle. The theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, this is expressed as a² + b² = c². When we apply this to real-world scenarios, like calculating the diagonal distance across a rectangular playground with a length of 40 m and a width of 30 m, we are essentially finding the hypotenuse of a triangle where the sides are 40 and 30 Exploring Society: India and Beyond. Social Science-Class VI, Locating Places on the Earth, p.10. Using the formula, 30² + 40² = 900 + 1600 = 2500, and the square root of 2500 gives us a diagonal of 50 m.

Beyond triangles, this theorem is deeply linked to the properties of squares. Remember that the Area of a Square = Side². This means if you are given the area of a square, you can find the length of its side by taking the square root (Side = √Area). In complex geometric arrangements—such as squares placed side-by-side—the total horizontal length is simply the sum of the side lengths of each individual square. Understanding this allows you to manipulate shapes: if the total length of a series of squares needs to change, you must adjust the area of at least one square to change its side length accordingly.

In competitive exams like the UPSC CSAT, geometric applications often require you to bridge the gap between 2D shapes. For instance, a rectangle’s diagonal is found using the same logic as the hypotenuse of a triangle. Mastering these Pythagorean Triplets (like 3-4-5, 5-12-13, or 8-15-17) can significantly save time during calculations, as these ratios appear frequently in standardized questions.

Sources: Exploring Society: India and Beyond. Social Science-Class VI, Locating Places on the Earth, p.10

5. Non-Linear Relationship between Length and Area Change (exam-level)

In quantitative aptitude, understanding the relationship between linear dimensions (like side length) and two-dimensional measures (like area) is crucial. For a square, the relationship is quadratic: Area = Side². Conversely, the side length is the square root of the area (Side = √Area). This means that a small, linear change in the side length of a shape results in a disproportionately large (non-linear) change in its area. For instance, in geographical planning, if a settlement is laid out in a rectangular or square pattern Geography of India, Majid Husain, Settlements, p.6, doubling the side of a plot doesn't just double the space; it quadruples it (2² = 4).

When solving problems involving multiple shapes joined together, you must first convert all areas back to their linear dimensions to find the total length or perimeter. Only after adjusting the lengths to meet the new requirements should you square the result to find the new area. This is a common point of failure for many students who mistakenly try to subtract or add area units directly to linear units. Whether we are discussing the total geographical area of 328.75 million hectares Geography of India, Majid Husain, Agriculture, p.1 or small-scale strip cultivation Environment and Ecology, Majid Hussain, Environmental Degradation and Management, p.19, the geometric logic remains constant: linear changes follow a 1:1 ratio, while area changes follow a squared ratio.

To master this, follow a three-step process: Root, Adjust, Square. First, take the square root of the given areas to find the initial lengths. Second, adjust these lengths (add/subtract) to reach the target linear measurement. Third, square the new length to find the resulting area. The difference between the original area and this new area represents the total change required.

Sources: Geography of India, Settlements, p.6; Geography of India, Agriculture, p.1; Environment and Ecology, Environmental Degradation and Management, p.19

6. Solving the Original PYQ (exam-level)

This question perfectly synthesizes the fundamental concepts of square roots and linear measurement you just mastered. In the CSAT, geometry problems often hide simple arithmetic behind visual arrangements. Here, the 'building blocks' are the individual side lengths: because the squares lie side-by-side, the total length PQ is simply the sum of the sides of the three squares. By applying the rule that the side of a square is the square root of its area, you can translate the given areas (100, 16, and 49) into linear dimensions (10, 4, and 7).

To solve this like a seasoned aspirant, first establish the baseline: the current length is 10 + 4 + 7 = 21. The goal is to reduce this total length to 19. Since the outer squares remain unchanged, the 2-unit reduction (21 - 19) must come entirely from the middle square's side. This means the new side length of the middle square must be 4 - 2 = 2. However, the question asks for the reduction in area, not the new side. The new area is 2² = 4, and the difference from the original area (16 - 4) gives us the correct answer, (A) 12.

UPSC frequently uses distractor options based on intermediate steps to catch students who rush. For instance, Option (B) 4 is a classic trap; it represents the new area of the middle square, while Option (C/D) 2 represents the new side length or the amount of side reduction. Always double-check if your final calculation matches the specific metric—area versus length—asked for in the final sentence of the prompt.