Q: A bought a radio set and spent ₹ $110$ on its repairs. He then sold it to B at $20\%$ profit, B sold it to C at a loss of $10\%$ and C sold it for ₹ $1188$ at a profit of $10\%$. What is the amount for which A bought the radio set?

Concept & Reverse Supply Chain Tracking: When the final selling price is given at the end of a chain of transactions, we calculate backwards to find the initial cost price. Additionally, any overhead expenses (like repairs) must be factored into the original purchaser's total cost price before applying profit percentages. Step-by-Step Solution: Let's track the prices backwards from C to A. • C's Transaction: C sold it for ₹ 1188 at a 10% profit. • Let C's Cost Price be CP_c. • CP_c × (1 + 0.10) = 1188 • 1.10 × CP_c = 1188 \implies CP_c = (1188/1.1) = ₹ 1080. • B's Transaction: B sold it to C at a 10% loss. B's Selling Price is C's Cost Price (₹ 1080). • Let B's Cost Price be CP_b. • CP_b × (1 - 0.10) = 1080 • 0.90 × CP_b = 1080 \implies CP_b = (1080/0.9) = ₹ 1200. • A's Transaction: A sold it to B at a 20% profit. A's Selling Price is B's Cost Price (₹ 1200). • Let A's *Total* Cost Price be CP_a. • CP_a × (1 + 0.20) = 1200 • 1.20 × CP_a = 1200 \implies CP_a = (1200/1.2) = ₹ 1000. • A's total cost includes the purchase price plus the repair costs. • Purchase Price + Repairs = 1000 • Purchase Price + 110 = 1000 • Purchase Price = 1000 - 110 = ₹ 890. Exam Strategy & Shortcut: Chain the multipliers in a single equation. Let the initial purchase price be x. Total initial cost = (x + 110). Chain: (x + 110) × 1.2 × 0.9 × 1.1 = 1188 (x + 110) × 1.188 = 1188 (x + 110) = 1000 x = 890. Common Pitfall: A frequent mistake is applying the repair cost of ₹ 110 to the end of the calculation rather than incorporating it into A's baseline cost price before calculating A's 20% profit. Final Answer: Therefore, the correct answer is ₹ 890.

Q: A car worth ₹ $1,50,000$ was sold by X to Y at $5\%$ profit. Y sold the car back to X at $2\%$ loss. In the entire transaction (C.P.O., 2007)

Concept & Net Gain in Circular Transactions: When an asset is sold and then bought back by the original owner, the net gain or loss is determined by tracking the net cash flow. The actual cash extracted from the other party defines the net financial profit, given that the asset ends up back where it started. Step-by-Step Solution: • Initial State: X has a car worth ₹ 1,50,000 and ₹ 0 in cash. • First Transaction (X sells to Y): • X sells at a 5% profit. • Selling Price for X (Cost Price for Y) = 1,50,000 + (5% of 1,50,000) = 1,50,000 + 7,500 = ₹ 1,57,500. • Now, X has ₹ 1,57,500 in cash and Y has the car. • Second Transaction (Y sells back to X): • Y sells at a 2% loss on his cost price. • Loss for Y = 2% of 1,57,500 = (2/100) × 157500 = ₹ 3,150. • Selling Price for Y (Purchase price for X) = 1,57,500 - 3,150 = ₹ 1,54,350. • X pays ₹ 1,54,350 to get the car back. • Net Result for X: • X gets his original car back. • Cash remaining with X = Cash received - Cash paid = 1,57,500 - 1,54,350 = ₹ 3,150. • Since X started with the car and ₹ 0 cash, and ended with the same car and ₹ 3,150 cash, X has made a net gain of ₹ 3,150. Exam Strategy & Shortcut: Identify the net transfer of cash. Y paid X ₹ 1,57,500. X paid Y ₹ 1,54,350. Net cash X extracted from Y = 1,57,500 - 1,54,350 = 3150. Since X retained the asset in the end, this positive cash flow is pure profit for X. Common Pitfall: A very common mistake is stating X gained ₹ 7500 in the first step and lost ₹ 4350 (difference between 1,54,350 and intrinsic value of 1,50,000) in the second. Profit in this closed loop is simply the delta in cash reserves after the asset is retrieved. Final Answer: Therefore, the correct answer is X gained ₹ 3150.

Q: A manufacturer sold a machine to a wholesale dealer at a profit of $10\%$. The wholesale dealer sold it to a retailer at a profit of $20\%$. While transporting some defect occurred in the machine and hence the retailer sold it at a loss of $5\%$. Th…

Concept & Chain Rule in Profit and Loss: When an item passes through multiple hands with successive profits and losses, we apply the chain rule of percentages. Final Price = Initial Price × (1 + Profit_1%) × (1 + Profit_2%) × (1 - Loss_3%) SP = CP × ( (100 \pm P/L_1/100) ) × ( (100 \pm P/L_2/100) ) \dots Step-by-Step Solution: • Given: • Profit by manufacturer = 10% • Profit by wholesale dealer = 20% • Loss by retailer = 5% • Final amount paid by customer = ₹ 627 • Let the original cost of the machine for the manufacturer be x. • Apply the successive change multipliers: • x × (1 + (10/100)) × (1 + (20/100)) × (1 - (5/100)) = 627 • x × (110/100) × (120/100) × (95/100) = 627 • x × (11/10) × (6/5) × (19/20) = 627 • x × (1254/1000) = 627 • Solving for x: • x = 627 × (1000/1254) • x = (1000/2) = 500 Exam Strategy & Shortcut: Work with percentage multipliers directly as fractions for faster cancellation: 10% profit = (11/10), 20% profit = (6/5), 5% loss = (19/20). Set up the equation x × (11 × 6 × 19/10 × 5 × 20) = 627. Notice that 11 × 19 = 209, and 209 × 3 = 627. So, x × (3 × 2/1000) = 3 \Rightarrow x = 500. Common Pitfall: A common mistake is adding the percentages together (10 + 20 - 5 = 25%) and calculating the profit on the base price. This ignores the compounding nature of successive transactions. Final Answer: Therefore, the correct answer is ₹ 500.

Q: Goods pass successively through the hands of three traders and each of them sells his goods at a profit of $25\%$ of his cost price. If the last trader sold the goods for ₹ 250, then how much did the first trader pay for them? (S.S.C., 2005)

Concept & Successive Profit Calculation: When an article is sold sequentially at a profit, the effective selling price is a product of successive multipliers. Final SP = Initial CP × (1 + (P_1/100)) × (1 + (P_2/100)) × (1 + (P_3/100)) Step-by-Step Solution: • Given: • Three traders each make a profit of 25%. • Final selling price (SP) = ₹ 250. • Let the original cost price for the first trader be x. • The multiplier for a 25% profit is 1 + (25/100) = (125/100) = (5/4). • Formulate the equation: • x × ((5/4)) × ((5/4)) × ((5/4)) = 250 • x × (125/64) = 250 • Solve for x: • x = 250 × (64/125) • x = 2 × 64 = 128 Exam Strategy & Shortcut: Memorize fraction equivalents of common percentages. 25% = (1/4). A profit of (1/4) means the multiplying factor is (5/4). Since the profit occurs three times, cube the multiplier: ((5/4))^3 = (125/64). (125/64) of CP = 250. It's easy to see 125 goes into 250 exactly twice, so CP = 64 × 2 = 128. Common Pitfall: A common error is treating the total profit as 25% × 3 = 75% and finding the cost price using 175% of CP = 250, which completely ignores compounding. Final Answer: Therefore, the correct answer is ₹ 128.

Q: By selling an article, a man makes a profit of $25\%$ of its selling price. His profit percent is (S.S.C., 2010)

Concept & Converting Profit Base: When profit is given as a percentage of the Selling Price (SP), you need to find the Cost Price (CP) to calculate the actual (real) profit percentage, which is always based on CP. Actual Profit % = ( \frac{Profit}{CP} ) × 100 Step-by-Step Solution: • Given: • Profit = 25% of SP. • Let the Selling Price (SP) be 100. • Calculate Profit: • Profit = 25% of 100 = 25. • Determine Cost Price (CP): • CP = SP - Profit • CP = 100 - 25 = 75. • Calculate the actual profit percent (on CP): • Profit % = ((25/75)) × 100 • Profit % = ((1/3)) × 100 = 33.33% = 33(1/3)% Exam Strategy & Shortcut: Use fractional equivalents. 25% = (1/4). If profit is (1/4) of SP, it means for every ₹ 4 of SP, profit is ₹ 1. So, CP = 4 - 1 = 3. Actual profit percent on CP is (1/3) × 100 = 33(1/3)%. Common Pitfall: Assuming the given 25% profit on selling price is the same as the actual profit percentage. Always verify the base of the percentage. Final Answer: Therefore, the correct answer is 33(1/3).

Q: If there is a profit of $20\%$ on the cost price of an article, the percentage of profit calculated on its selling price will be (S.S.C., 2010)

Concept & Profit Margin Conversion: Standard profit percentage is calculated on the Cost Price (CP). To find the profit percentage based on the Selling Price (SP), known as the profit margin, we use the formula: Profit on SP % = ( \frac{Profit}{SP} ) × 100 Step-by-Step Solution: • Given: • Actual Profit = 20% on Cost Price. • Let the Cost Price (CP) be 100. • Calculate Profit and Selling Price: • Profit = 20% of 100 = 20. • SP = CP + Profit = 100 + 20 = 120. • Calculate the profit percent calculated on SP: • New Profit % = (\frac{Profit}{SP}) × 100 • New Profit % = ((20/120)) × 100 • New Profit % = ((1/6)) × 100 = 16.66% = 16(2/3)% Exam Strategy & Shortcut: Use fractions. A profit of 20% on CP means a fraction of (1/5). This means Profit = 1 unit, CP = 5 units. SP = CP + Profit = 5 + 1 = 6 units. Profit on SP is \frac{Profit}{SP} = (1/6). The percentage equivalent of (1/6) is universally known as 16(2/3)%. Common Pitfall: A common mistake is assuming profit on SP is always higher or identical to profit on CP. Since SP is generally larger than CP in a profitable transaction, the denominator increases, making the profit percentage on SP mathematically smaller. Final Answer: Therefore, the correct answer is 16(2/3).

Q: By selling an article at $\frac{2}{3}$ of the marked price, there is a loss of $10\%$. The profit percent, when the article is sold at the marked price, is (C.P.O., 2006)

Concept & Marked Price and Discount: The relationship between Cost Price (CP), Selling Price (SP), and Marked Price (MP) can be established through profit/loss percentages and fractional sales conditions. SP = MP × Fraction SP = CP × (1 - \frac{Loss %}{100}) Step-by-Step Solution: • Given: • New SP = (2/3) × MP • Loss at this SP = 10% • Let the Marked Price (MP) be x. • Current SP = (2x/3) • Express CP in terms of x using the loss condition: • With a 10% loss, SP = 90% of CP = 0.9 × CP. • 0.9 × CP = (2x/3) • CP = (2x/3 × 0.9) = (2x/2.7) = (20x/27) • Calculate profit when sold at Marked Price: • New SP = MP = x • Profit = SP - CP = x - (20x/27) = (7x/27) • Calculate profit percentage: • Profit % = (\frac{Profit}{CP}) × 100 • Profit % = (\frac{7x/27)}{(20x/27)} × 100 • Profit % = (7/20) × 100 = 7 × 5 = 35% Exam Strategy & Shortcut: Assume a convenient number for MP that is divisible by 3. Let MP = 300. SP = (2/3) × 300 = 200. Loss is 10%, so 200 = 90% of CP \Rightarrow CP = (200/0.9) = (2000/9). If sold at MP (300), Profit = 300 - (2000/9) = (2700 - 2000/9) = (700/9). Profit % = (\frac{700/9)}{(2000/9)} × 100 = (700/2000) × 100 = 35%. Common Pitfall: A frequent error is calculating the final profit percentage based on the Marked Price instead of the Cost Price. Profit percentage is fundamentally always calculated over the CP unless stated otherwise. Final Answer: Therefore, the correct answer is 35%.

Question 1

A shopkeeper advertises for selling cloth at $4\%$ loss. However, by using a false metre scale he actually gains $20\%$. What is the actual length of the scale?  (R.R.B., 2006)

Accepted Answer

Concept & Reverse Percentage Calculation: When the overall gain percentage and the professed price percentage are given, we can trace backward to find the actual physical quantity delivered by equating the final selling price to the marked up actual cost. Step-by-Step Solution: • Let the true cost price of 100 cm (1 metre) of cloth be Rs. 100. • The shopkeeper advertises a 4% loss. • Selling Price (SP) for a promised 100 cm = 100 - 4 = Rs. 96. • Let the actual length of the false scale be x cm. • The actual Cost Price (CP) for the cloth given is Rs. x. • The problem states he makes an actual gain of 20% on his actual CP. • Therefore, SP = Actual CP + 20% of Actual CP. • 96 = x + 0.20x • 96 = 1.2x • x = (96/1.2) = 80 • The actual length of the scale used is 80 cm. Exam Strategy & Shortcut: Let the false scale length be x. He charges for 96 cm (due to 4% loss on 100 cm). His profit is 20%, meaning the amount he charges represents 120% of what he actually gives. 120% of x = 96 1.2x = 96 \implies x = 80. Common Pitfall: A frequent error is taking 20% of 96 and subtracting it, which incorrectly assumes the 20% margin is calculated on the selling price instead of the cost price. Profit is always based on the CP. Final Answer: Therefore, the correct answer is 80 cm.

Question 2

A trader professes to sell his goods at a nominal gain percentage but actually earns $37 \frac{1}{2}\%$ profit by using false weight. If for a kg he uses a weight of $800$ gm, what is the nominal gain percentage at which he claims to be selling his …

Accepted Answer

Concept & Nominal vs. Actual Profit: A trader's nominal (professed) profit is based on the declared selling price and the assumed standard weight. The actual profit depends on the actual money received compared to the cost of the actual physical weight delivered. Step-by-Step Solution: • Let the cost price (CP) of 1 gm of goods be Re. 1. • Actual Transaction: The trader uses an 800 gm weight for a kilogram (1000 gm). • The actual CP incurred by the trader is for the goods he physically gives away: CP of 800 gm = Rs. 800. • He earns an actual profit of 37 (1/2)%. Convert to fraction: 37.5% = (3/8). • Actual Profit = (3/8) × Actual CP = (3/8) × 800 = Rs. 300. • Actual Selling Price (SP) = Actual CP + Actual Profit = 800 + 300 = Rs. 1100. • Nominal (Professed) Claim: He claims to be selling 1000 gm (which costs Rs. 1000) for the SP of Rs. 1100. • Nominal Profit = SP - Claimed CP = 1100 - 1000 = Rs. 100. • Nominal Gain Percentage = ( \frac{Nominal Profit}{Claimed CP} ) × 100 • Nominal Gain % = (100/1000) × 100 = 10% Exam Strategy & Shortcut: Let professed profit factor be x. The false weight factor is (1000/800) = (5/4) (he gets paid for 5 parts but gives 4). Total profit factor = 1 + 37.5% = 1 + (3/8) = (11/8). Equation: x × (5/4) = (11/8) \implies x = (11/8) × (4/5) = (11/10) = 1.1. A factor of 1.1 means a 10% nominal gain. Common Pitfall: It is easy to confuse the base of the percentages. Ensure that actual profit is calculated on the actual weight delivered (800g), while nominal profit is calculated on the standard weight (1000g). Final Answer: Therefore, the correct answer is 10%.

Question 3

A stockist wants to make some profit by selling sugar. He contemplates about various methods. Which of the following would maximize his profit?

Accepted Answer

Concept & Comparative Profit Scenarios: To compare different profit-making strategies, calculate the profit percentage for each method by strictly identifying the actual Cost Price (value of goods actually given) and the Selling Price (money received). Step-by-Step Solution: Let the standard CP of 1000g (1 kg) of sugar be Rs. 1000 (Re. 1 per gram). • Method (a): Sell sugar at 10% profit • SP = Rs. 1100, CP = Rs. 1000. • Profit = 10%. • Method (b): Use 900 g of weight instead of 1 kg • He gives 900g (Actual CP = Rs. 900) and charges for 1000g (SP = Rs. 1000). • Profit = 1000 - 900 = 100. • Profit % = (100/900) × 100 = (100/9)% \approx 11.11%. • Method (c): Mix 10% impurities in sugar and sell at cost price • He has 1000g sugar and adds 100g free impurities, making an 1100g mixture. • He sells the 1100g at the CP rate of pure sugar (Re. 1/gram). Total SP = Rs. 1100. • His actual cost is only for the 1000g pure sugar. Actual CP = Rs. 1000. • Profit = 1100 - 1000 = 100. • Profit % = (100/1000) × 100 = 10%. • Method (d): Increase the price by 5% and reduce the weight by 5% • He charges 5% more for 1 kg. SP = 1000 + 5% = Rs. 1050. • He gives 5% less weight. Actual weight = 950g. Actual CP = Rs. 950. • Profit = 1050 - 950 = 100. • Profit % = (100/950) × 100 = (1000/95) = (200/19)% \approx 10.52%. Comparing the percentages: 10%, 11.11%, 10%, and 10.52%. The highest is 11.11%. Exam Strategy & Shortcut: You can evaluate these rapidly using fractions: (a) Profit fraction = 1/10 (b) Profit fraction = (1000-900/900) = 1/9 (c) Profit fraction = 100/1000 = 1/10 (d) Profit fraction = (105 - 95/95) = 10/95 = 2/19 Comparing 1/10, 1/9, and 2/19: 1/9 (0.111) is larger than 1/10 (0.10) and 2/19 (\approx 0.105). Common Pitfall: A common misjudgment in option (c) is calculating profit over 1100g instead of the 1000g cost base, or assuming mixing 10% impurities yields more than 10% profit. Always ground the denominator in the actual cost out of pocket. Final Answer: Therefore, the correct answer is Use 900 g of weight instead of 1 kg.

Question 4

A fair price shopkeeper takes $10\%$ profit on his goods. He lost $20\%$ goods during theft. His loss percent is

Accepted Answer

Concept & Effective Profit and Loss: When a portion of goods is lost or destroyed, the profit must be calculated based on the revenue generated solely from the remaining goods compared to the total initial investment. Step-by-Step Solution: • Let the initial number of goods be 100. • Let the cost price (CP) of 1 good be Re. 1. • Total initial Cost Price (CP) for all goods = Rs. 100. • The shopkeeper sets his price to take a 10% profit on individual goods. • Expected Selling Price (SP) of 1 good = Re. 1 + 10% of 1 = Rs. 1.10. • A theft occurs, and he loses 20% of his goods. • Remaining goods available for sale = 100 - 20% of 100 = 80 goods. • Total Selling Price (SP) realized from the remaining goods = 80 × 1.10 = Rs. 88. • Now, compare the actual total SP to the total initial CP. • Total CP = 100, Total SP = 88. • Since SP < CP, he incurs a loss. • Loss amount = 100 - 88 = Rs. 12. • Loss percentage = \frac{Loss}{Total CP} × 100 = (12/100) × 100 = 12%. Exam Strategy & Shortcut: Use the multiplier method for successive changes. Volume multiplier (due to theft) = 0.80 (since only 80% is sold). Price multiplier (due to profit margin) = 1.10. Overall multiplier = 0.80 × 1.10 = 0.88. A final multiplier of 0.88 represents a 12% drop from the original value of 1.00. Thus, a 12% loss. Common Pitfall: A common mistake is simply subtracting the two percentages (-20% + 10% = -10%). This fails because the price markup and the physical volume loss are multiplicative factors affecting total revenue, not additive ones. Final Answer: Therefore, the correct answer is 12.

Question 5

A bought a radio set and spent ₹ $110$ on its repairs. He then sold it to B at $20\%$ profit, B sold it to C at a loss of $10\%$ and C sold it for ₹ $1188$ at a profit of $10\%$. What is the amount for which A bought the radio set?

Accepted Answer

Concept & Reverse Supply Chain Tracking: When the final selling price is given at the end of a chain of transactions, we calculate backwards to find the initial cost price. Additionally, any overhead expenses (like repairs) must be factored into the original purchaser's total cost price before applying profit percentages. Step-by-Step Solution: Let's track the prices backwards from C to A. • C's Transaction: C sold it for ₹ 1188 at a 10% profit. • Let C's Cost Price be CP_c. • CP_c × (1 + 0.10) = 1188 • 1.10 × CP_c = 1188 \implies CP_c = (1188/1.1) = ₹ 1080. • B's Transaction: B sold it to C at a 10% loss. B's Selling Price is C's Cost Price (₹ 1080). • Let B's Cost Price be CP_b. • CP_b × (1 - 0.10) = 1080 • 0.90 × CP_b = 1080 \implies CP_b = (1080/0.9) = ₹ 1200. • A's Transaction: A sold it to B at a 20% profit. A's Selling Price is B's Cost Price (₹ 1200). • Let A's *Total* Cost Price be CP_a. • CP_a × (1 + 0.20) = 1200 • 1.20 × CP_a = 1200 \implies CP_a = (1200/1.2) = ₹ 1000. • A's total cost includes the purchase price plus the repair costs. • Purchase Price + Repairs = 1000 • Purchase Price + 110 = 1000 • Purchase Price = 1000 - 110 = ₹ 890. Exam Strategy & Shortcut: Chain the multipliers in a single equation. Let the initial purchase price be x. Total initial cost = (x + 110). Chain: (x + 110) × 1.2 × 0.9 × 1.1 = 1188 (x + 110) × 1.188 = 1188 (x + 110) = 1000 x = 890. Common Pitfall: A frequent mistake is applying the repair cost of ₹ 110 to the end of the calculation rather than incorporating it into A's baseline cost price before calculating A's 20% profit. Final Answer: Therefore, the correct answer is ₹ 890.

Question 6

A car worth ₹ $1,50,000$ was sold by X to Y at $5\%$ profit. Y sold the car back to X at $2\%$ loss. In the entire transaction  (C.P.O., 2007)

Accepted Answer

Concept & Net Gain in Circular Transactions: When an asset is sold and then bought back by the original owner, the net gain or loss is determined by tracking the net cash flow. The actual cash extracted from the other party defines the net financial profit, given that the asset ends up back where it started. Step-by-Step Solution: • Initial State: X has a car worth ₹ 1,50,000 and ₹ 0 in cash. • First Transaction (X sells to Y): • X sells at a 5% profit. • Selling Price for X (Cost Price for Y) = 1,50,000 + (5% of 1,50,000) = 1,50,000 + 7,500 = ₹ 1,57,500. • Now, X has ₹ 1,57,500 in cash and Y has the car. • Second Transaction (Y sells back to X): • Y sells at a 2% loss on his cost price. • Loss for Y = 2% of 1,57,500 = (2/100) × 157500 = ₹ 3,150. • Selling Price for Y (Purchase price for X) = 1,57,500 - 3,150 = ₹ 1,54,350. • X pays ₹ 1,54,350 to get the car back. • Net Result for X: • X gets his original car back. • Cash remaining with X = Cash received - Cash paid = 1,57,500 - 1,54,350 = ₹ 3,150. • Since X started with the car and ₹ 0 cash, and ended with the same car and ₹ 3,150 cash, X has made a net gain of ₹ 3,150. Exam Strategy & Shortcut: Identify the net transfer of cash. Y paid X ₹ 1,57,500. X paid Y ₹ 1,54,350. Net cash X extracted from Y = 1,57,500 - 1,54,350 = 3150. Since X retained the asset in the end, this positive cash flow is pure profit for X. Common Pitfall: A very common mistake is stating X gained ₹ 7500 in the first step and lost ₹ 4350 (difference between 1,54,350 and intrinsic value of 1,50,000) in the second. Profit in this closed loop is simply the delta in cash reserves after the asset is retrieved. Final Answer: Therefore, the correct answer is X gained ₹ 3150.

Question 7

A manufacturer sold a machine to a wholesale dealer at a profit of $10\%$. The wholesale dealer sold it to a retailer at a profit of $20\%$. While transporting some defect occurred in the machine and hence the retailer sold it at a loss of $5\%$. Th…

Accepted Answer

Concept & Chain Rule in Profit and Loss: When an item passes through multiple hands with successive profits and losses, we apply the chain rule of percentages. Final Price = Initial Price × (1 + Profit_1%) × (1 + Profit_2%) × (1 - Loss_3%) SP = CP × ( (100 \pm P/L_1/100) ) × ( (100 \pm P/L_2/100) ) \dots Step-by-Step Solution: • Given: • Profit by manufacturer = 10% • Profit by wholesale dealer = 20% • Loss by retailer = 5% • Final amount paid by customer = ₹ 627 • Let the original cost of the machine for the manufacturer be x. • Apply the successive change multipliers: • x × (1 + (10/100)) × (1 + (20/100)) × (1 - (5/100)) = 627 • x × (110/100) × (120/100) × (95/100) = 627 • x × (11/10) × (6/5) × (19/20) = 627 • x × (1254/1000) = 627 • Solving for x: • x = 627 × (1000/1254) • x = (1000/2) = 500 Exam Strategy & Shortcut: Work with percentage multipliers directly as fractions for faster cancellation: 10% profit = (11/10), 20% profit = (6/5), 5% loss = (19/20). Set up the equation x × (11 × 6 × 19/10 × 5 × 20) = 627. Notice that 11 × 19 = 209, and 209 × 3 = 627. So, x × (3 × 2/1000) = 3 \Rightarrow x = 500. Common Pitfall: A common mistake is adding the percentages together (10 + 20 - 5 = 25%) and calculating the profit on the base price. This ignores the compounding nature of successive transactions. Final Answer: Therefore, the correct answer is ₹ 500.

Question 8

Goods pass successively through the hands of three traders and each of them sells his goods at a profit of $25\%$ of his cost price. If the last trader sold the goods for ₹ 250, then how much did the first trader pay for them? (S.S.C., 2005)

Accepted Answer

Concept & Successive Profit Calculation: When an article is sold sequentially at a profit, the effective selling price is a product of successive multipliers. Final SP = Initial CP × (1 + (P_1/100)) × (1 + (P_2/100)) × (1 + (P_3/100)) Step-by-Step Solution: • Given: • Three traders each make a profit of 25%. • Final selling price (SP) = ₹ 250. • Let the original cost price for the first trader be x. • The multiplier for a 25% profit is 1 + (25/100) = (125/100) = (5/4). • Formulate the equation: • x × ((5/4)) × ((5/4)) × ((5/4)) = 250 • x × (125/64) = 250 • Solve for x: • x = 250 × (64/125) • x = 2 × 64 = 128 Exam Strategy & Shortcut: Memorize fraction equivalents of common percentages. 25% = (1/4). A profit of (1/4) means the multiplying factor is (5/4). Since the profit occurs three times, cube the multiplier: ((5/4))^3 = (125/64). (125/64) of CP = 250. It's easy to see 125 goes into 250 exactly twice, so CP = 64 × 2 = 128. Common Pitfall: A common error is treating the total profit as 25% × 3 = 75% and finding the cost price using 175% of CP = 250, which completely ignores compounding. Final Answer: Therefore, the correct answer is ₹ 128.

Question 9

By selling an article, a man makes a profit of $25\%$ of its selling price. His profit percent is (S.S.C., 2010)

Accepted Answer

Concept & Converting Profit Base: When profit is given as a percentage of the Selling Price (SP), you need to find the Cost Price (CP) to calculate the actual (real) profit percentage, which is always based on CP. Actual Profit % = ( \frac{Profit}{CP} ) × 100 Step-by-Step Solution: • Given: • Profit = 25% of SP. • Let the Selling Price (SP) be 100. • Calculate Profit: • Profit = 25% of 100 = 25. • Determine Cost Price (CP): • CP = SP - Profit • CP = 100 - 25 = 75. • Calculate the actual profit percent (on CP): • Profit % = ((25/75)) × 100 • Profit % = ((1/3)) × 100 = 33.33% = 33(1/3)% Exam Strategy & Shortcut: Use fractional equivalents. 25% = (1/4). If profit is (1/4) of SP, it means for every ₹ 4 of SP, profit is ₹ 1. So, CP = 4 - 1 = 3. Actual profit percent on CP is (1/3) × 100 = 33(1/3)%. Common Pitfall: Assuming the given 25% profit on selling price is the same as the actual profit percentage. Always verify the base of the percentage. Final Answer: Therefore, the correct answer is 33(1/3).

Question 10

If there is a profit of $20\%$ on the cost price of an article, the percentage of profit calculated on its selling price will be (S.S.C., 2010)

Accepted Answer

Concept & Profit Margin Conversion: Standard profit percentage is calculated on the Cost Price (CP). To find the profit percentage based on the Selling Price (SP), known as the profit margin, we use the formula: Profit on SP % = ( \frac{Profit}{SP} ) × 100 Step-by-Step Solution: • Given: • Actual Profit = 20% on Cost Price. • Let the Cost Price (CP) be 100. • Calculate Profit and Selling Price: • Profit = 20% of 100 = 20. • SP = CP + Profit = 100 + 20 = 120. • Calculate the profit percent calculated on SP: • New Profit % = (\frac{Profit}{SP}) × 100 • New Profit % = ((20/120)) × 100 • New Profit % = ((1/6)) × 100 = 16.66% = 16(2/3)% Exam Strategy & Shortcut: Use fractions. A profit of 20% on CP means a fraction of (1/5). This means Profit = 1 unit, CP = 5 units. SP = CP + Profit = 5 + 1 = 6 units. Profit on SP is \frac{Profit}{SP} = (1/6). The percentage equivalent of (1/6) is universally known as 16(2/3)%. Common Pitfall: A common mistake is assuming profit on SP is always higher or identical to profit on CP. Since SP is generally larger than CP in a profitable transaction, the denominator increases, making the profit percentage on SP mathematically smaller. Final Answer: Therefore, the correct answer is 16(2/3).

Question 11

By selling an article at $\frac{2}{3}$ of the marked price, there is a loss of $10\%$. The profit percent, when the article is sold at the marked price, is (C.P.O., 2006)

Accepted Answer

Concept & Marked Price and Discount: The relationship between Cost Price (CP), Selling Price (SP), and Marked Price (MP) can be established through profit/loss percentages and fractional sales conditions. SP = MP × Fraction SP = CP × (1 - \frac{Loss %}{100}) Step-by-Step Solution: • Given: • New SP = (2/3) × MP • Loss at this SP = 10% • Let the Marked Price (MP) be x. • Current SP = (2x/3) • Express CP in terms of x using the loss condition: • With a 10% loss, SP = 90% of CP = 0.9 × CP. • 0.9 × CP = (2x/3) • CP = (2x/3 × 0.9) = (2x/2.7) = (20x/27) • Calculate profit when sold at Marked Price: • New SP = MP = x • Profit = SP - CP = x - (20x/27) = (7x/27) • Calculate profit percentage: • Profit % = (\frac{Profit}{CP}) × 100 • Profit % = (\frac{7x/27)}{(20x/27)} × 100 • Profit % = (7/20) × 100 = 7 × 5 = 35% Exam Strategy & Shortcut: Assume a convenient number for MP that is divisible by 3. Let MP = 300. SP = (2/3) × 300 = 200. Loss is 10%, so 200 = 90% of CP \Rightarrow CP = (200/0.9) = (2000/9). If sold at MP (300), Profit = 300 - (2000/9) = (2700 - 2000/9) = (700/9). Profit % = (\frac{700/9)}{(2000/9)} × 100 = (700/2000) × 100 = 35%. Common Pitfall: A frequent error is calculating the final profit percentage based on the Marked Price instead of the Cost Price. Profit percentage is fundamentally always calculated over the CP unless stated otherwise. Final Answer: Therefore, the correct answer is 35%.

Question 12

A man buys an article for $10\%$ less than its value and sells it for $10\%$ more than its value. His gain or loss percent is

Accepted Answer

Concept & Value vs Cost Price: When an article is bought at a discount relative to its "value" and sold at a premium relative to the same "value", the profit percentage must be calculated on the discounted Cost Price, not the baseline value. Step-by-Step Solution: • Given: • Cost Price (CP) = Value - 10% of Value • Selling Price (SP) = Value + 10% of Value • Let the intrinsic Value of the article be 100. • Calculate Cost Price (CP): • CP = 100 - (10% of 100) = 100 - 10 = 90 • Calculate Selling Price (SP): • SP = 100 + (10% of 100) = 100 + 10 = 110 • Calculate actual Gain: • Gain = SP - CP = 110 - 90 = 20 • Calculate Gain Percentage: • Gain % = (\frac{Gain}{CP}) × 100 • Gain % = ((20/90)) × 100 • Gain % = (200/9)% = 22.22% • Evaluate against options: • 22.22% is strictly more than 20%. Exam Strategy & Shortcut: If you buy at a 10% discount and sell at a 10% markup on a base of 100, your CP is 90 and SP is 110. Your absolute profit is 20. If your CP was 100, your profit would be exactly 20%. Since your CP (90) is *less* than 100, dividing the same profit margin (20) by a smaller base mathematically results in a percentage *greater* than 20%. No exact calculation is needed. Common Pitfall: A classic trap is to assume that a 10% drop on purchase and a 10% increase on sale relative to a base value cancel out, or that they sum exactly to a 20% profit. Always normalize back to the actual Cost Price. Final Answer: Therefore, the correct answer is more than 20% profit.

Question 13

If a man reduces the selling price of a fan from ₹ 400 to ₹ 380, his loss increases by $2\%$. The cost price of the fan is

Accepted Answer

Concept & Proportional Difference in Loss: Just like profit, any change in loss percentage directly correlates to the absolute change in the Selling Price (SP). Both refer to the same baseline Cost Price (CP). \Delta SP = (\Delta Loss %) × CP Step-by-Step Solution: • Given: • Original Selling Price (SP_1) = ₹ 400 • Reduced Selling Price (SP_2) = ₹ 380 • Increase in loss percentage = 2% • Find the reduction in Selling Price: • Difference in SP = 400 - 380 = ₹ 20 • This ₹ 20 drop directly caused the loss to increase by 2%. Therefore, 2% of the Cost Price is exactly equal to ₹ 20. • 2% of CP = 20 • (2/100) × CP = 20 • Solve for CP: • CP = (20 × 100/2) • CP = 10 × 100 = 1000 • The actual cost price of the fan is ₹ 1000. Let's check the options provided. • Options given: (a) ₹ 480, (b) ₹ 500, (c) ₹ 600, (d) None of these. • Since ₹ 1000 is not present in options (a), (b), or (c), the correct choice is "None of these". Exam Strategy & Shortcut: Mental math is the fastest route here. The price dropped by 20. This drop equates to a 2% loss. If 2% \rightarrow 20, then 1% \rightarrow 10. The full Cost Price is 100%, so 100% \rightarrow 100 × 10 = 1000. Common Pitfall: A frequent error is trying to formulate complex equations involving initial loss percentages (e.g., L%). Since the *change* in loss directly links to the *change* in price, the initial loss percentage is completely irrelevant and unnecessary to find. Final Answer: Therefore, the correct answer is None of these.

Question 14

A shopkeeper sells an article at $12\frac{1}{2}\%$ loss. If he sells it for ₹ 92.50 more then he gains $6\%$. What is the cost price of the article? (MA.T., 2008)

Accepted Answer

Concept & Total Percentage Distance: When adjusting the Selling Price (SP) transforms a loss into a profit, the increase in price equals the absolute sum of the initial loss percentage and the final profit percentage, calculated strictly on the Cost Price (CP). Price Increase = (Loss % + Profit %) of CP Step-by-Step Solution: • Given: • Initial condition: Loss of 12(1/2)% = 12.5% • Final condition: Gain of 6% • Price increment (\Delta SP) = ₹ 92.50 • Determine the total percentage gap bridging the loss and the gain: • Total % difference = 12.5% (loss) + 6% (gain) = 18.5% • Equate this total percentage to the absolute price difference: • 18.5% of CP = 92.50 • (18.5/100) × CP = 92.50 • Solve for CP: • CP = (92.50 × 100/18.5) • CP = (9250/18.5) • To divide easily, multiply numerator and denominator by 10 to remove decimals: • CP = (92500/185) • Note that 185 × 5 = 925. • Therefore, CP = 500. Exam Strategy & Shortcut: Instead of decimals, you can use fractions. 12.5% = (1/8), and 6% = (6/100) = (3/50). Total fraction = (1/8) + (3/50) = (25/200) + (12/200) = (37/200). So, (37/200) of CP = 92.50. CP = (92.50 × 200/37) = (18500/37) = 500. Fractions often prevent decimal arithmetic errors during exams. Common Pitfall: A typical mistake is treating the 12.5% and 6% as sequential changes and multiplying them, rather than realizing they represent two independent endpoints relative to the same constant Cost Price base. Final Answer: Therefore, the correct answer is ₹ 500.

Question 15

A man sells a book at a profit of $20\%$. If he had bought it at $20\%$ less and sold it for ₹ 18 less, he would have gained $25\%$. The cost price of the book is (M.A.T., 2009)

Accepted Answer

Concept & Successive Profit Scenarios: When both Cost Price and Selling Price change, formulate equations for the initial and final states based on CP and SP, and apply the new profit percentage to the new CP. New SP = New CP × (1 + \frac{New Profit %}{100}) Step-by-Step Solution: • Let the original Cost Price be CP = x. • Initial profit is 20%, so original Selling Price SP = 1.2x. • The man bought it at 20% less: New CP = x - 0.2x = 0.8x. • He sold it for ₹ 18 less: New SP = 1.2x - 18. • On this transaction, he gains 25%. So, New SP = New CP × 1.25. • Substitute the expressions: 1.2x - 18 = 0.8x × 1.25 • 1.2x - 18 = 1.0x • 1.2x - x = 18 • 0.2x = 18 • x = (18/0.2) = 90 Exam Strategy & Shortcut: Use a base of 100 units for CP. Let CP = 100u, SP = 120u. New CP = 80u. New Profit = 25% on 80u = 20u. New SP = 80u + 20u = 100u. The decrease in SP is 120u - 100u = 20u. We are given this decrease is ₹ 18. So, 20u = 18. We need CP, which is 100u. 100u = 18 × 5 = 90. Common Pitfall: Applying the new profit percentage (25%) to the original Cost Price instead of the updated, reduced Cost Price. Final Answer: Therefore, the correct answer is ₹ 90.

Question 16

A watch is sold at a profit of $20\%$. If both the cost price and the selling price of the watch are decreased by ₹ 100, the profit would be $5\%$ more. Original cost price of the watch is (A.A.O. Exam, 2010)

Accepted Answer

Concept & Absolute Value Changes in Profit and Loss: When a fixed absolute amount is deducted from both CP and SP, the profit percentage changes. Set up the foundational equation linking the updated CP and SP. Step-by-Step Solution: • Let original CP = x. • Original Profit = 20%, so Original SP = 1.2x. • Both CP and SP decrease by ₹ 100: • New CP = x - 100 • New SP = 1.2x - 100 • The new profit is 5% more, meaning it is 20% + 5% = 25%. • Set up the equation: New SP = New CP × 1.25 • 1.2x - 100 = (x - 100) × 1.25 • 1.2x - 100 = 1.25x - 125 • 1.25x - 1.2x = 125 - 100 • 0.05x = 25 • x = (25/0.05) = (2500/5) = 500 Exam Strategy & Shortcut: Use the ratio method. Initial CP : SP = 100 : 120 = 5 : 6. New Profit = 25%, so New CP : SP = 100 : 125 = 4 : 5. Both decreased by 100. Initial ratio: 5x, 6x. (5x - 100/6x - 100) = (4/5) Cross multiply: 25x - 500 = 24x - 400 \implies x = 100. Original CP = 5x = 5(100) = 500. This avoids decimals entirely. Common Pitfall: Assuming "profit would be 5% more" means the new profit is 5% of CP, rather than interpreting it as the old percentage plus 5% (i.e., 25%). Final Answer: Therefore, the correct answer is ₹ 500.

Question 17

The cash difference between the selling prices of an article at a profit of $4\%$ and $6\%$ is ₹ 3. The ratio of the two selling prices is (C.B.I., 2003)

Accepted Answer

Concept & Ratio of Selling Prices: When asked for the ratio of two selling prices derived from the same cost price, the absolute cash difference is irrelevant. The ratio depends solely on the profit percentages. Ratio = (100 + P_1%/100 + P_2%) Step-by-Step Solution: • Let the Cost Price (CP) of the article be x. • The first selling price (SP_1) is at a 4% profit: SP_1 = x × (1 + (4/100)) = 1.04x • The second selling price (SP_2) is at a 6% profit: SP_2 = x × (1 + (6/100)) = 1.06x • We need to find the ratio SP_1 : SP_2. • Ratio = (1.04x/1.06x) = (1.04/1.06) • Multiply numerator and denominator by 100 to remove decimals: (104/106) • Simplify the fraction by dividing both by 2: (52/53) • The ratio is 52 : 53. Exam Strategy & Shortcut: Ignore the ₹ 3 completely; it's a distractor intended to make you calculate the actual CP. Immediately write down the ratio of the percentages representing the selling prices: 104 : 106. Simplify it to 52 : 53. Common Pitfall: Wasting time calculating the actual Cost Price using the ₹ 3 difference (2% of CP = 3 \rightarrow CP = 150) before calculating the selling prices and their ratio. Final Answer: Therefore, the correct answer is 52 : 53.

Question 18

Shailja earns 15 percent on an investment but loses 10 percent on another investment. If the ratio of the two investments is $3 : 5$, then the combined loss percent is (S.S.C., 2006)

Accepted Answer

Concept & Weighted Average Profit/Loss: When different investments yield different returns (profits or losses), the overall return percentage is the weighted average of the individual returns, using the investment amounts as weights. Combined Return % = \frac{Total Profit/Loss}{Total Investment} × 100 Step-by-Step Solution: • The ratio of the two investments is 3 : 5. • Let the two investments be 3x and 5x. • Total Investment = 3x + 5x = 8x. • Return on the first investment (15% profit): Profit = 15% of 3x = 0.15 × 3x = +0.45x • Return on the second investment (10% loss): Loss = 10% of 5x = 0.10 × 5x = -0.50x • Net Return (Combined Profit/Loss): Net = +0.45x - 0.50x = -0.05x • Since the net result is negative, it is an overall loss. • Combined Loss Percent = \frac{Net Loss}{Total Investment} × 100 • Loss Percent = ((0.05x/8x)) × 100 = (0.05/8) × 100 = (5/8)% Exam Strategy & Shortcut: Use base numbers for ratios. Let the investments be 300 and 500. Total = 800. 15% of 300 = +45. 10% of 500 = -50. Net = -5. Loss percent = (5/800) × 100 = (5/8)%. This avoids decimals entirely. Common Pitfall: Simply averaging the percentages (15% - 10% = 5%) without factoring in the weight (ratio) of the respective investments. Final Answer: Therefore, the correct answer is (5/8).

Question 19

A man sells two articles for ₹ 240 each. On one he gains $20\%$ and on the other he loses $20\%$. What is the gain or loss percent in the entire transaction? (S.S.C., 2005)

Accepted Answer

Concept & Identical Selling Price Theorem: When two articles are sold for the exact same selling price, and one is sold at a profit of x% while the other is sold at a loss of x%, the overall transaction always results in a net loss. Net Loss % = (x^2/100) Step-by-Step Solution: • We are given that both articles are sold at the same price: SP_1 = SP_2 = 240. • The gain on the first article is x = 20%. • The loss on the second article is x = 20%. • Because the selling prices are equal and the percentage gain and loss are equal in magnitude, we can apply the standard theorem. • The transaction yields an overall loss. • Loss % = ((20)^2/100) • Loss % = (400/100) = 4% Exam Strategy & Shortcut: Commit the formula (x^2/100)% loss to memory. The actual selling price (₹ 240) is completely unnecessary information. As soon as you see "sells two articles for the same price" and "gains 20% and loses 20%", immediately calculate 20 × (20/100) = 4% loss. Common Pitfall: Assuming that a 20% gain and a 20% loss cancel each other out, resulting in a 0% net change (no profit, no loss). This happens if the Cost Prices were equal, not the Selling Prices. Final Answer: Therefore, the correct answer is 4% loss.

Question 20

A house and a shop were sold for ₹ 1 lakh each. In this transaction, the house sale resulted into $20\%$ loss whereas the shop sale resulted into $20\%$ profit. The entire transaction resulted in

Accepted Answer

Concept & Absolute Net Loss Calculation: When asked for the *absolute cash amount* of the loss (rather than just the percentage) in a "same SP, same percentage gain/loss" scenario, you must calculate the exact aggregate Cost Price and subtract the aggregate Selling Price. CP = \frac{SP}{1 \pm \frac{Profit/Loss %}{100}} Step-by-Step Solution: • Total Selling Price (SP) = ₹ 1 lakh + ₹ 1 lakh = ₹ 2 lakhs. • Calculate Cost Price of the house (CP_h): Loss = 20%, so SP_h = 0.8 × CP_h 1 = (4/5) × CP_h \implies CP_h = (5/4) lakhs (or 1.25 lakhs) • Calculate Cost Price of the shop (CP_s): Profit = 20%, so SP_s = 1.2 × CP_s 1 = (6/5) × CP_s \implies CP_s = (5/6) lakhs • Total Cost Price (CP) = CP_h + CP_s Total CP = (5/4) + (5/6) = (15/12) + (10/12) = (25/12) lakhs. • Since Total CP ((25/12) \approx 2.083) is greater than Total SP (2), there is a loss. • Net Loss = Total CP - Total SP Net Loss = (25/12) - 2 = (25/12) - (24/12) = (1/12) lakh. Exam Strategy & Shortcut: First, determine the net loss percentage using the theorem: (20^2/100) = 4% loss. This means Total SP is 96% of Total CP. Total SP = 2 lakhs. 96% of Total CP = 2 lakhs. We want to find the absolute loss, which is the remaining 4% of Total CP. If 96% = 2, then 4% = (2/96) × 4 = (2/24) = (1/12) lakh. This skips finding individual CPs! Common Pitfall: Automatically selecting "no loss, no gain" due to the identical percentages, or calculating the 4% net loss but failing to convert that percentage back into the absolute monetary value requested by the options. Final Answer: Therefore, the correct answer is loss of ₹ (1/12) lakh.

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