A man purchases two clocks A and B at a total cost of Rs. 650. He sells A with 20% profit and B at a loss of 25% and gets the same selling price for both the clocks. What are the purchasing prices of A and B respectively ?

1. Foundations: Percentages and Decimal Multipliers (basic)

At its heart, a percentage is simply a way of expressing a number as a fraction of 100. In the UPSC journey, you will encounter this everywhere—from the distribution of soil types like Inceptisols, which cover 39.74% of our land Geography of India, Soils, p.13, to the commodity composition of our exports where manufactured goods hold a 65.7% share Geography of India, Transport, Communications and Trade, p.47. Understanding percentages isn't just about finding a part of a whole; it is about understanding relative change. When we say a price has increased by 5%, we are looking at how a quantity shifts relative to its original state Microeconomics, Theory of Consumer Behaviour, p.35. To master competitive exams, we move beyond basic school-level addition and subtraction and embrace the Decimal Multiplier. Think of your original value (the 'Base') as 100% or simply 1.00. If a value increases by 20%, it becomes 120% of the original, which corresponds to a multiplier of 1.20. Conversely, if a value decreases by 25% (like a loss in trade), you are left with 75% of the original, giving you a multiplier of 0.75. Using these multipliers allows you to calculate final values in a single step (e.g., New Value = Original × Multiplier), which is a vital time-saving skill for the CSAT paper.

To help you visualize how these multipliers work in both profit (increase) and loss (decrease) scenarios, refer to the table below:

Scenario	Percentage Change	Mathematical Logic	Decimal Multiplier
Profit / Increase	+20%	100% + 20% = 120%	1.20
Loss / Decrease	-25%	100% - 25% = 75%	0.75
Marginal Increase	+5%	100% + 5% = 105%	1.05

Sources: Geography of India, Soils, p.13; Geography of India, Transport, Communications and Trade, p.47; Microeconomics, Theory of Consumer Behaviour, p.35

2. Core Definitions: CP, SP, Profit, and Loss (basic)

To master any problem involving business transactions, we must first anchor ourselves in four fundamental terms: Cost Price (CP), Selling Price (SP), Profit, and Loss. Think of these not just as variables, but as the story of a trade. The Cost Price (CP) is the total investment made by a seller to acquire or produce a good. In economic terms, this is often referred to as the Total Cost (TC) Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.56. On the other side of the transaction is the Selling Price (SP), which is the amount the consumer pays. The interaction between what a seller wants and what a buyer is willing to pay eventually settles at this market price through negotiation Exploring Society: India and Beyond, Social Science-Class VII, Understanding Markets, p.252.

The relationship between these two values determines the financial outcome of the trade. If the seller manages to sell the item for more than they paid (SP > CP), they earn a Profit. Conversely, if the market conditions force the seller to accept a price lower than their investment (SP < CP), they incur a Loss. Mathematically, a firm’s profit (denoted by π) is the gap between its total revenue and its total costs Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.56. In competitive markets, a seller will generally refuse to sell at a price that isn't profitable unless absolutely necessary.

Scenario	Condition	Formula
Profit	SP > CP	Profit = SP − CP
Loss	CP > SP	Loss = CP − SP
Break-even	SP = CP	No Profit, No Loss

Crucially, for UPSC and general arithmetic, Profit or Loss percentage is always calculated on the Cost Price unless the question explicitly states otherwise. This is because the CP represents the original base or the 100% value from which we measure growth (profit) or decline (loss). For example, a 20% profit means the SP is 120% of the CP (SP = 1.20 × CP), while a 25% loss means the SP is only 75% of the CP (SP = 0.75 × CP).

Sources: Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.56; Exploring Society: India and Beyond, Social Science-Class VII, Understanding Markets, p.252

3. Ratio and Proportion: Splitting a Total Value (basic)

At its heart, Splitting a Total Value is the art of breaking down a whole into constituent parts based on a specific rule or relationship. Whether you are dividing a budget, calculating a firm's profit by subtracting Total Cost from Total Revenue Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.68, or understanding how the monetary base is split between currency and reserves Indian Economy, Vivek Singh (7th ed. 2023-24), Money and Banking- Part I, p.59, the logic remains the same. If the total is S and you have two parts, $A$ and $B$, you can define $A$ as $x$ and $B$ as $(S - x)$. This simple algebraic substitution ensures that the sum of the parts always equals the original whole.

While simple ratios (like 2:3) are the most common way to split values, competitive exams often hide the ratio behind percentage changes. For instance, if you are told that two parts of a total lead to the same result after different percentage adjustments, you are effectively being given a hidden ratio. In economic terms, this is similar to how a firm must balance its output to ensure that its marginal revenue matches its marginal cost to maximize profit Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.58. When you set up an equation like 1.20x = 0.75(Total - x), you are finding the exact point where these two proportions meet.

Sources: Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.68; Indian Economy, Vivek Singh (7th ed. 2023-24), Money and Banking- Part I, p.59; Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.58

4. Connected Concepts: Markups and Successive Discounts (intermediate)

In the world of commerce, price isn't static; it travels through layers. The first layer is the Markup. A seller rarely sells an item at the Cost Price (CP). Instead, they increase the price by a certain percentage to arrive at the Marked Price (MP). This is essentially a percentage increase. For instance, if you anticipate that the value of goods will rise or if you need to cover overheads, you apply a markup. This logic of adjusting prices to reflect value is similar to how we use a 'deflator' to find the real value of the GDP by discounting the inflationary impact from the nominal figures, as seen in Indian Economy, Nitin Singhania (ed 2nd 2021-22), National Income, p.7.

The second layer involves Successive Discounts. Sellers often lure customers with 'cascading' discounts, such as '10% + 10% off'. A crucial rule to remember: discounts are successive, not additive. A 20% discount followed by a 10% discount is not a 30% total reduction. The first discount is applied to the Marked Price, and the second is applied to the resulting reduced price. This relationship between price changes and the final value mirrors the way total expenditure shifts when both price and quantity change, where the net effect depends on the percentage change of each component Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.32.

To calculate the net effect of a markup and a discount, we treat them as successive percentage changes. If a price is increased by a% (markup) and then decreased by b% (discount), the effective change is given by the formula: Net % Change = a - b - (ab/100). If the result is positive, it’s a profit; if negative, it’s a loss.

Process	Direction	Base Value
Markup	Upward (+)	Calculated on Cost Price (CP)
Discount	Downward (-)	Calculated on Marked Price (MP)
Profit/Loss	Net Result	Comparison of Final SP vs Original CP

Sources: Indian Economy, Nitin Singhania (ed 2nd 2021-22), National Income, p.7; Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.32

5. Connected Concepts: Simple Interest and Growth Rates (intermediate)

When we move from basic percentages to financial concepts, Simple Interest (SI) is the most direct application. Think of interest as the "rent" or the price paid for using someone else's money. In economic terms, it is the opportunity cost of holding money rather than investing it Macroeconomics (NCERT class XII 2025 ed.), Money and Banking, p.46. At its core, Simple Interest is just a fixed percentage of the principal amount applied over a specific duration.

However, in the real world and for your UPSC preparation, understanding the Growth Rate requires looking beyond the "nominal" number. This is where we distinguish between the Nominal Interest Rate (the rate the bank tells you) and the Real Interest Rate (your actual growth in purchasing power). If your money grows at 8% but the price of goods (inflation) also grows by 6%, your actual wealth has only increased by 2% Indian Economy, Vivek Singh (7th ed. 2023-24), Money and Banking- Part I, p.111. This relationship is crucial for understanding how the economy behaves during recessions or periods of high inflation.

Term	Definition	Impact on Wealth
Nominal Rate	The stated interest rate on a deposit or loan.	The numerical increase in your bank balance.
Real Rate	Nominal Rate minus the Inflation Rate.	The actual increase in what you can buy (purchasing power).

Finally, remember that the "market interest rate" is a general term. It could refer to the deposit rate (what you earn) or the lending rate (what you pay the bank) Indian Economy, Vivek Singh (7th ed. 2023-24), Money and Banking- Part I, p.92. An important rule in macroeconomics is the inverse relationship between interest rates and bond prices: when the market interest rate rises, the value of existing bonds falls because new bonds are now being issued at higher, more attractive rates Macroeconomics (NCERT class XII 2025 ed.), Money and Banking, p.45.

Sources: Indian Economy, Vivek Singh (7th ed. 2023-24), Money and Banking- Part I, p.111; Macroeconomics (NCERT class XII 2025 ed.), Money and Banking, p.45-46; Indian Economy, Vivek Singh (7th ed. 2023-24), Money and Banking- Part I, p.92

6. The Alligation Method for Weighted Averages (exam-level)

In our previous hops, we looked at individual profit and loss calculations. However, in the UPSC CSAT and even in Economics, we often deal with scenarios where two different groups are combined. The Alligation Method is a powerful shortcut used to find the ratio in which two components are mixed to achieve a specific weighted average. Instead of using complex algebraic equations, we use a visual 'cross' method to determine how much of 'Part A' and 'Part B' contribute to the 'Whole.'

Think of it this way: if you have two quantities with different values (like profit percentages or interest rates), the final combined value will always lie somewhere between the two. The closer the final average is to one value, the higher the 'weight' or quantity of that component. This concept of weighting is fundamental in economics. For instance, when calculating the Index of Industrial Production (IIP), the Manufacturing sector is given a much higher weight (77.6%) compared to Electricity (8%), meaning changes in manufacturing have a far greater impact on the total index than electricity does Indian Economy, Nitin Singhania, Indian Industry, p.385.

To apply the Alligation rule, follow these steps:

• Place the lower value (L) on the top left and the higher value (H) on the top right.
• Place the mean/average value (M) in the center.
• Calculate the differences diagonally: (H − M) and (M − L).
• The ratio of the quantities is (H − M) : (M − L).

This method is incredibly versatile. Whether a firm is calculating its total revenue across different product lines to maximize profit Microeconomics, NCERT class XII, The Theory of the Firm under Perfect Competition, p.56, or a trader is mixing two types of pulses, the underlying principle of balance remains the same.

Sources: Indian Economy, Nitin Singhania, Indian Industry, p.385; Microeconomics, NCERT class XII, The Theory of the Firm under Perfect Competition, p.56

7. Special Case: Equal Selling Price Scenario (exam-level)

In competitive aptitude, the Equal Selling Price scenario is a classic challenge. This occurs when two different items are sold for the exact same amount of money, despite having different initial costs and different profit or loss outcomes. While the Selling Price (SP) is identical, their Cost Prices (CP) must differ to account for their respective percentage changes. In financial terms, we can think of these as two different 'assets' with varying yields that ultimately liquidate for the same value. To solve these problems from first principles, we set up an equation of equality. If an item is sold at a 20% profit, its SP is 120% (or 1.20) of its CP. If another is sold at a 25% loss, its SP is only 75% (or 0.75) of its CP. By equating these two expressions—$1.20 imes CP_1 = 0.75 imes CP_2$—we can derive the ratio of their costs. This inverse relationship is fundamental: the item with the higher profit percentage must have a lower initial Cost Price to reach the same final selling amount. Interestingly, the abbreviation CP is also common in the Indian money market, where it stands for Commercial Paper—a short-term, unsecured promissory note used by companies to raise funds Indian Economy, Nitin Singhania, p.261. Similarly, understanding the ratios between these values is as vital as understanding the Statutory Liquidity Ratio (SLR) in banking, which dictates how a bank must allocate its deposits before lending Macroeconomics, Money and Banking, p.40. Just as banks balance their reserves, a student must balance the ratio of Cost Prices to solve for the total amount.

Condition	Multiplier for SP	Cost Price (CP) Logic
Profit of x%	(100 + x) / 100	CP is relatively Lower
Loss of y%	(100 - y) / 100	CP is relatively Higher

Sources: Indian Economy, Agriculture, p.261; Macroeconomics, Money and Banking, p.40

8. Solving the Original PYQ (exam-level)

Now that you have mastered the building blocks of Profit and Loss and Ratio Proportions, this UPSC PYQ serves as the perfect synthesis of those concepts. The core challenge here is to translate the phrase "gets the same selling price" into a mathematical equation. In your previous lessons, you learned that a 20% profit is equivalent to a multiplier of 1.20 and a 25% loss is equivalent to a multiplier of 0.75. By equating the two selling prices (1.20 × CP_A = 0.75 × CP_B), you are essentially applying the concept of inverse proportionality: the ratio of the Cost Prices must be the inverse of the ratio of their price change factors.

To arrive at the answer efficiently, look at the ratio of the Cost Prices: CP_A / CP_B = 0.75 / 1.20. Simplifying this fraction gives you a ratio of 5:8. Since the total cost is Rs. 650, you simply need to divide 650 into 13 equal parts (5 + 8). One part equals Rs. 50, making CP_A = 5 × 50 = Rs. 250 and CP_B = 8 × 50 = Rs. 400. This logical flow from percentage multipliers to simplified ratios is a hallmark of successful CSAT problem-solving, allowing you to bypass tedious long division.

UPSC designed this question with specific traps in mind. Notice that in all four options, the sum of the two prices is exactly Rs. 650. This prevents you from using simple elimination based on the total cost. Options like (D) Rs. 300; Rs. 350 are common distractors for students who try to "split the difference" or guess based on proximity. However, only (B) Rs. 250; Rs. 400 maintains the exact mathematical balance where a smaller profit on a smaller base equals a larger loss on a larger base, satisfying the same selling price condition of Rs. 300.