If X + 2Y = 2X + Y, then X^2 / Y^2 is equal to

1. Basics of Linear Equations and Variables (basic)

At the heart of quantitative aptitude lies the concept of a variable—a symbol, usually a letter like X or Y, that serves as a placeholder for a numerical value we don't yet know. A linear equation is simply a mathematical statement of equality involving these variables, where the highest power of any variable is 1. We treat the equal sign (=) as a pivot point of a balance scale; to keep the equation 'true,' any operation performed on one side must be performed on the other. This algebraic balancing is used in various fields, such as determining market equilibrium where supply and demand functions are set equal to one another to find a stable price Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74.

To master these equations, we use a process called isolation. This involves moving all terms containing our target variable to one side of the equation and the remaining terms to the opposite side. When a term 'crosses' the equal sign, its mathematical operation reverses: addition becomes subtraction, and multiplication becomes division. For example, if we have terms on both sides, we systematically subtract the smaller term from the larger one to simplify the expression. This logic is a prerequisite for more complex operations, such as the substitution method, where we solve for one variable in terms of another from a single equation before plugging it into a second equation to find a complete solution Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.53.

In many theoretical exercises, we simplify our focus by using the assumption of ceteris paribus, which means 'other things remaining equal.' As explained in Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.53, this allows us to concentrate on how a specific variable changes while holding others constant. Understanding how to rearrange these variables is not just about finding a number; it is about uncovering the relationship between different entities, such as the ratio between two variables that must remain in balance within a system.

Sources: Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74; Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.53

2. Transposition and Balancing Equations (basic)

At its heart, an algebraic equation is like a perfectly balanced weighing scale. The equals sign (=) is the fulcrum, and for the scale to stay level, any operation performed on one side must be performed on the other. This is the Balancing Principle. For example, when economists determine market equilibrium, they set demand equal to supply and manipulate both sides of the equation to find the equilibrium price (Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.73). If you add 10 to the left side, you must add 10 to the right to maintain equality.

Transposition is the logical shortcut we use to apply this balancing principle quickly. Instead of writing out "subtract 5 from both sides," we simply "move" the 5 across the equals sign. However, the golden rule of transposition is that the mathematical operation must reverse when a term crosses the border:

• Addition becomes Subtraction (and vice versa).
• Multiplication becomes Division (and vice versa).

For instance, in the linear equation Y = a + bX, if we want to isolate the constant 'a' (the Y-intercept), we transpose 'bX' to the other side, changing its sign to get a = Y - bX (Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58).

When dealing with equations that have variables on both sides, such as 3X + 4 = X + 10, the strategy is to group like terms. We transpose all the 'X' terms to one side and all the constants to the other. This systematic rearrangement is a fundamental tool used even in complex fields like Geography to solve for unknown values in pressure systems or climatic data via the elimination method (Physical Geography by PMF IAS, Pressure Systems and Wind System, p.324).

Sources: Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.73; Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.324

3. Introduction to Ratios and Proportions (basic)

At its heart, a ratio is a way of comparing two quantities of the same kind to see how many times one value contains the other. Think of it as a tool for relative measurement rather than absolute counting. For instance, when analyzing regional dynamics, we might compare the sizes of different countries to understand their relative influence (Contemporary World Politics, Contemporary South Asia, p.40). A proportion, on the other hand, is simply an equation that states that two ratios are equal. In the context of governance, proportions are vital for ensuring fairness; for example, the weight of a legislator's vote in Presidential elections is determined by a specific ratio of population to elected members to maintain parity across states (Indian Polity, President, p.187).

In quantitative aptitude, we often use algebraic manipulation to find these ratios. When you are given a linear equation like X + 2Y = 2X + Y, your goal is to group like terms to reveal the relationship between the variables. By subtracting X and Y from both sides, you simplify the expression to X = Y. This tells us that the ratio of X to Y is 1:1. Once you establish this fundamental relationship, you can solve for more complex forms. For example, if X = Y, then any power of their ratio, such as X²/Y², will simply be 1²/1², which equals 1.

Understanding ratios also helps us grasp the concept of proportional justice. In social systems, we often argue that rewards should be distributed in proportion to the effort or quality of work provided (Political Theory, Social Justice, p.56). This logic follows a functional relation where one variable (reward) depends on another (effort), a concept frequently visualized in economic graphs where independent and dependent variables are compared (Microeconomics, Theory of Consumer Behaviour, p.22).

Sources: Contemporary World Politics, Contemporary South Asia, p.40; Indian Polity, President, p.187; Political Theory, Social Justice, p.56; Microeconomics, Theory of Consumer Behaviour, p.22

4. Algebraic Identities and Powers (intermediate)

In quantitative aptitude, mastering the manipulation of linear equations is a fundamental skill that allows us to uncover hidden relationships between variables. An equation is essentially a scale in balance; whatever operation you perform on one side, you must perform on the other to maintain that balance. When we encounter an equation like X + 2Y = 2X + Y, our first instinct should be to group "like terms" together. By subtracting X and Y from both sides, we simplify the expression to reveal that X = Y. This logical deduction is the cornerstone of algebraic problem-solving, turning a complex-looking expression into a simple identity.

Once a relationship between variables is established, we can evaluate more complex exponential expressions or ratios. For instance, if we know X = Y, then any power of X divided by the same power of Y (such as X²/Y² or X³/Y³) will inevitably result in 1, provided the variables are non-zero. This is because the numerator and denominator become identical. Understanding these algebraic identities is not just for abstract math; as seen in economic modeling, expressing variables algebraically helps us find points of equilibrium where opposing forces (like demand and supply) cancel each other out Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74.

When dealing with powers in these ratios, remember that the laws of exponents apply. If X = Y, then Xⁿ / Yⁿ = (X/Y)ⁿ. Since X/Y = 1, and 1 raised to any power remains 1, the ratio simplifies instantly. This technique of substitution and simplification is a high-yield strategy in competitive exams, saving precious time by avoiding cumbersome calculations and focusing on the underlying logical relationship instead.

Sources: Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74

5. Solving Simultaneous Linear Equations (intermediate)

At its core, Simultaneous Linear Equations are a set of equations that we deal with at the same time to find a common solution for the variables involved. In a standard two-variable system, each equation represents a straight line on a graph. The point where these lines intersect is the unique solution (X, Y) that satisfies both equations simultaneously. As noted in the study of economic relations, a linear relation between two variables like X and Y can be expressed in an "intercept form," where the relationship is defined by a constant slope and a starting point on the axis Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58.

To solve these equations, we generally rely on two primary techniques: Substitution and Elimination. The Elimination Method involves adding or subtracting the equations to cancel out one variable, making it easier to solve for the remaining one Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Pressure Systems and Wind System, p.324. For instance, if you have X + Y = 10 and X - Y = 2, adding them together eliminates Y, giving you 2X = 12, and thus X = 6. The Substitution Method, on the other hand, involves expressing one variable in terms of the other (e.g., Y = 10 - X) and plugging that into the second equation.

Sometimes, equations are presented in a way that requires rearrangement or simplification before they can be solved. If you encounter an expression like X + 2Y = 2X + Y, your first step is to group the "like terms" on opposite sides of the equals sign. By subtracting X from both sides and Y from both sides, you simplify the relation to Y = X. This tells us that in any valid solution for this specific system, the value of X and Y must be identical. Understanding how one variable depends on another is a fundamental concept in functional mathematics, where one is the independent variable and the other is the dependent variable Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.21.

Method	Core Strategy	Best Used When...
Elimination	Add or subtract equations to remove one variable.	Coefficients of one variable are the same or easily made the same.
Substitution	Replace one variable with an expression derived from the other equation.	One variable is already isolated or has a coefficient of 1.

Sources: Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Pressure Systems and Wind System, p.324; Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.21

6. Determining Ratio from a Single Equation (exam-level)

In quantitative aptitude, we often encounter scenarios where we are given a single linear equation involving two variables, such as X and Y. While a single equation is usually insufficient to find the exact numerical values of both variables, it is perfectly sufficient to determine the ratio between them. This is because a linear equation establishes a fixed relative relationship, or a "bridge," between the two quantities.

To find the ratio, your primary goal is rearrangement and isolation. You must group all terms containing one variable on the Left-Hand Side (LHS) and all terms containing the other variable on the Right-Hand Side (RHS). For example, if you have an equation like X + 2Y = 2X + Y, you subtract X from both sides and Y from both sides to arrive at Y = X. Just as we ensure a chemical equation is balanced by having equal atoms on both sides Science, class X (NCERT 2025 ed.), Chemical Reactions and Equations, p.5, we maintain the algebraic balance to reveal that the ratio of X to Y is 1:1.

This concept is deeply linked to the intercept form of a linear equation, which is expressed as Y = a + bX. In this form, the constant b represents the slope or the rate at which Y changes relative to X Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58. If the intercept a is zero, the ratio Y/X is simply the constant b. Once you have this fundamental ratio (e.g., X/Y = k), you can solve for more complex expressions like X²/Y² or (X+Y)/(X-Y) by substituting one variable in terms of the other.

Sources: Science, class X (NCERT 2025 ed.), Chemical Reactions and Equations, p.5; Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58

7. Solving the Original PYQ (exam-level)

Congratulations on completing the conceptual modules! This specific PYQ is a classic example of how UPSC tests your ability to apply algebraic manipulation to simplify a relationship before jumping into a calculation. The building blocks you've learned—specifically isolating variables and simplifying ratios—are the exact tools needed here. Instead of searching for specific numerical values for X and Y, the goal is to establish the proportional relationship between them, transforming a complex-looking fraction into a simple identity.

Walking through the logic, we start with the equation X + 2Y = 2X + Y. By subtracting X and Y from both sides, we use standard operations on equations to arrive at the conclusion that X = Y. As emphasized in NASA Stargaze: Algebra Review, this identity means that for any non-zero value, X and Y are identical. Therefore, when we substitute this into the target expression, X²/Y² becomes X²/X², which mathematically simplifies to (B) 1. The beauty of this problem is that the actual values of the variables don't matter; only their 1:1 ratio does.

UPSC often includes distractor options to catch students who rush. Option (D) 4 is a common trap designed for those who might square the coefficients (2²) without first simplifying the equation. Similarly, Option (A) 0 targets students who might confuse division with subtraction when terms cancel out. By sticking to the systematic rearrangement you practiced in the concept phase, you ensure that you don't fall for these arithmetic decoys and consistently find the correct answer (B).