Question 1

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 2

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 3

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 4

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 5

If $5\%$ more is gained by selling an article for ₹ 350 than by selling it for ₹ 340, the cost of the article is

Accepted Answer

Concept & Proportional Difference: The difference between two selling prices directly corresponds to the difference in their respective profit percentages. Both percentages share the exact same base, which is the Cost Price (CP). \Delta SP = (\Delta Profit %) × CP Step-by-Step Solution: • Given: • Selling Price 1 (SP_1) = ₹ 340 • Selling Price 2 (SP_2) = ₹ 350 • Additional gain percentage = 5% • Calculate the absolute difference in Selling Price: • Difference in SP = SP_2 - SP_1 • Difference in SP = 350 - 340 = ₹ 10 • This difference in money corresponds strictly to the additional profit percentage gained on the Cost Price (CP). • 5% of CP = ₹ 10 • (5/100) × CP = 10 • Solve for CP: • CP = (10 × 100/5) • CP = 2 × 100 = 200 Exam Strategy & Shortcut: Equate the percentages directly to the currency values in your head. The gap in price is 350 - 340 = 10. The gap in profit is 5%. Therefore, 5% represents 10. To find 100% (the full CP), multiply both sides by 20. 10 × 20 = 200. Common Pitfall: Assuming that the 5% gain applies to the selling prices directly rather than the hidden Cost Price. Always remember that profit percentages are tied to the CP. Final Answer: Therefore, the correct answer is ₹ 200.

Question 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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.

Question 14

A man sells two articles at ₹ 99 each. He gains $10\%$ on one and loses $10\%$ on the other. Then on overall basis he (P.C.S., 2006)

Accepted Answer

Concept & Absolute Net Loss Calculation: Similar to the previous problem, determine the overall percentage loss first, and then translate that percentage into the actual cash amount lost based on the Total Selling Price. Net Loss % = (x^2/100) Step-by-Step Solution: • Total Selling Price (SP) = ₹ 99 + ₹ 99 = ₹ 198. • Since SPs are identical and percentage gain (10%) equals percentage loss (10%), we find the overall loss percentage. • Net Loss % = (10^2/100) = (100/100) = 1% loss. • This means the Total Selling Price represents 99% (100% - 1%) of the Total Cost Price. • Let Total CP be x. Then 99% of x = 198. • 0.99x = 198 \implies x = (198/0.99) = 200. • The Total Cost Price is ₹ 200, and the Total Selling Price is ₹ 198. • Absolute Loss = Total CP - Total SP = 200 - 198 = 2. • The transaction results in a loss of ₹ 2. Exam Strategy & Shortcut: Use the ratio method. The net loss is 1%, which is (1/100). This means if CP is 100 units, Loss is 1 unit, and SP is 99 units. We know Total SP = 198. If 99 units = ₹ 198, then 1 unit = ₹ 2. The loss is exactly 1 unit, which equals ₹ 2. Common Pitfall: Calculating the 1% loss on the Total Selling Price (taking 1% of 198 = 1.98) instead of recognizing that 198 represents 99% of the original base (Cost Price). Final Answer: Therefore, the correct answer is loses ₹ 2.

Question 15

A man sold two steel chairs for ₹ $500$ each. On one, he gains $20\%$ and on the other, he loses $12\%$. How much does he gain or lose in the whole transaction? (M.A.T., 2006)

Accepted Answer

Concept & Formula: To find the overall profit or loss percentage, we must first calculate the total Cost Price (CP) and total Selling Price (SP). The formula for CP when SP and Profit/Loss percentages are known is: CP = (SP × 100/100 \pm Profit/Loss)% The overall gain percentage is computed as \frac{Total SP - Total CP}{Total CP} × 100. Step-by-Step Solution: • Given: Selling price of each chair (SP_1 = SP_2) = ₹ 500. • Total SP: 500 + 500 = ₹ 1000. • CP of first chair (20% gain): CP_1 = 500 × (100/120) = 500 × (5/6) = (2500/6) = (1250/3). • CP of second chair (12% loss): CP_2 = 500 × (100/88) = 500 × (25/22) = (12500/22) = (6250/11). • Total CP: (1250/3) + (6250/11) = (13750 + 18750/33) = (32500/33). • Overall Gain: Total SP - Total CP = 1000 - (32500/33) = (33000 - 32500/33) = (500/33). • Gain Percentage: \frac{Gain}{Total CP} × 100 = (\frac{500/33)}{(32500/33)} × 100 = (500/32500) × 100 = (20/13)% \approx 1.538%. • Rounding to the nearest half percent gives a 1.5% gain. Exam Strategy & Shortcut: Unlike problems where the gain and loss percentages are equal (where the net result is always a loss of (x^2/100)%), different percentages require finding the exact ratio. Using fractions directly instead of decimals prevents rounding errors mid-calculation. Common Pitfall: A frequent mistake is averaging the percentages: (20 - 12/2) = 4% gain, which is entirely incorrect because the cost prices of the two items are different. Final Answer: Therefore, the correct answer is 1.5% gain.

Question 16

Hemant sold $10$ sarees for a total profit of ₹ $460$ and $12$ sarees for a total profit of ₹ $144$. At what profit per saree should he sell the remaining $20$ sarees so that he gets an average profit of ₹ $18$ per saree?

Accepted Answer

Concept & Target Average: To achieve a specific average profit across multiple items, the sum of all profits must equal the target average multiplied by the total number of items. Total Target Profit = Total Quantity × Target Average Profit Step-by-Step Solution: • Total number of sarees: 10 + 12 + 20 = 42 sarees. • Target overall profit: 42 sarees × ₹ 18/saree = ₹ 756. • Profit already earned: Profit from first 10 + Profit from next 12 = 460 + 144 = ₹ 604. • Required profit from remaining 20 sarees: Target Profit - Profit already earned = 756 - 604 = ₹ 152. • Required profit per saree for the remaining batch: \frac{₹ 152}{20 sarees} = ₹ 7.60. Exam Strategy & Shortcut: You can also use the method of deviations from the mean. Target mean = 18. Batch 1 (10 items): average profit = 460/10 = 46 (deviation = +28). Total deviation = +280. Batch 2 (12 items): average profit = 144/12 = 12 (deviation = -6). Total deviation = -72. Net deviation so far = 280 - 72 = +208. To balance to 0, the remaining 20 items must have a deviation of -208. Deviation per item = -208 / 20 = -10.4. Required profit per item = 18 - 10.4 = 7.60. Common Pitfall: Misreading "total profit" as "profit per saree" for the first two batches. Ensure you note whether a value represents an aggregate or a per-unit metric. Final Answer: Therefore, the correct answer is ₹ 7.60.

Question 17

Sanket purchased $20$ dozen notebooks at ₹ $48$ per dozen. He sold $8$ dozen at $10\%$ profit and the remaining $12$ dozen with $20\%$ profit. What is his profit percentage in the transaction?

Accepted Answer

Concept & Weighted Average Percentage: When a total quantity is split into parts and sold at different profit percentages, the overall profit percentage is the weighted average of the individual profit percentages. The base cost price (₹ 48) is uniform across all items, so it cancels out of the percentage calculation. Overall Profit % = (Q_1P_1 + Q_2P_2/Q_1 + Q_2) Step-by-Step Solution: • Total Quantity: 20 dozen. • Quantity 1 (Q_1): 8 dozen. • Profit 1 (P_1): 10%. • Quantity 2 (Q_2): 12 dozen. • Profit 2 (P_2): 20%. • Overall Profit Percentage calculation: (8 × 10% + 12 × 20%/20) = (80 + 240/20) = (320/20) = 16% Exam Strategy & Shortcut: Since the cost price is the same for all items, completely ignore the "₹ 48 per dozen" figure. It is unnecessary data designed to slow you down. The problem simplifies to a pure mixture/weighted average calculation. You can even simplify the ratio of quantities: 8 : 12 = 2 : 3. Weighted average = (2(10) + 3(20)/2 + 3) = (20 + 60/5) = (80/5) = 16%. Common Pitfall: A common error is to calculate the exact cost prices and selling prices for all items (Cost = 960, SP1 = 422.4, SP2 = 691.2, Total SP = 1113.6). This yields the right answer but takes substantially more time. Final Answer: Therefore, the correct answer is 16.

Question 18

If a person makes a profit of $10\%$ on one-fourth of the quantity sold and a loss of $20\%$ on the rest, then what is the average percent profit or loss? (J.M.E.T., 2004)

Accepted Answer

Concept & Fractional Weighted Average: To find the total percentage profit or loss across an entire quantity, apply the given profit and loss percentages to their respective fractions of the whole. Losses are represented as negative values. Net % = (Fraction_1 × Profit_1) - (Fraction_2 × Loss_2) Step-by-Step Solution: • First Part: One-fourth ((1/4)) of the quantity. • Profit on First Part: 10%. • Second Part: The rest, which is 1 - (1/4) = (3/4) of the quantity. • Loss on Second Part: 20%. • Calculating the net effect: Net % = ((1/4) × 10%) + ((3/4) × -20%) Net % = 2.5% - 15% Net % = -12.5% • The negative sign indicates an overall loss. Exam Strategy & Shortcut: Assume the total cost price is 100 units consisting of 4 equal parts costing 25 each. Part 1 (Cost = 25): +10% of 25 = +2.5 profit. Remaining 3 parts (Cost = 75): -20% of 75 = -15 loss. Net effect = +2.5 - 15 = -12.5. Since the base is 100, the net effect is exactly -12.5%, meaning a 12.5% loss. Common Pitfall: A common error is misinterpreting "the rest". If a quarter is used, the remainder is three-quarters ((3/4)), not a half or some other fraction. Final Answer: Therefore, the correct answer is 12.5% loss.

Question 19

I purchased $120$ exercise books at the rate of ₹ $3$ each and sold $\frac{1}{3}$ of them at the rate of ₹ $4$ each, $\frac{1}{2}$ of them at the rate of ₹ $5$ each and the rest at the cost price. My profit percent is (C.P.O., 2006)

Accepted Answer

Concept & Profit Calculation: To find the overall profit percentage, we need to determine the total Cost Price (CP) of all items and the total Selling Price (SP) of all items after dividing them into their respective sales batches. Profit % = \frac{Total SP - Total CP}{Total CP} × 100 Step-by-Step Solution: • Total Cost Price: 120 books × ₹ 3/book = ₹ 360. • Batch 1 (1/3 of total): (1/3) × 120 = 40 books. • Selling Price for Batch 1 = 40 × ₹ 4 = ₹ 160. • Batch 2 (1/2 of total): (1/2) × 120 = 60 books. • Selling Price for Batch 2 = 60 × ₹ 5 = ₹ 300. • Batch 3 (The rest): Total books - (Batch 1 + Batch 2) = 120 - (40 + 60) = 120 - 100 = 20 books. • These are sold at cost price (₹ 3 each). • Selling Price for Batch 3 = 20 × ₹ 3 = ₹ 60. • Total Selling Price: 160 + 300 + 60 = ₹ 520. • Total Profit: Total SP - Total CP = 520 - 360 = ₹ 160. • Profit Percentage: (160/360) × 100 = (16/36) × 100 = (4/9) × 100 = (400/9)% (400/9)% = 44 (4/9)% Exam Strategy & Shortcut: You can solve this using only ratios and per-item profits without calculating the total large values. Assume total items = 6 (LCM of 3 and 2). Cost per item = 3. Total CP for 6 items = 18. - 1/3 of 6 = 2 items sold at 4 (Profit = +1 each, total profit = +2) - 1/2 of 6 = 3 items sold at 5 (Profit = +2 each, total profit = +6) - Remaining = 1 item sold at CP (Profit = 0) Total Profit on 6 items = 8. Profit % = (8/18) × 100 = (4/9) × 100 = 44 (4/9)%. Common Pitfall: A common error is miscalculating the "remaining" fraction by subtracting from 1 incorrectly, or forgetting to multiply the remaining quantity by the original cost price. Final Answer: Therefore, the correct answer is 44 (4/9)%.

Question 20

A departmental store receives a shipment of $1000$ pens, for which it pays ₹ $9000$. The store sells the pens at a price $80$ percent above cost for one month, after which it reduces the price of the pens to $20$ percent above cost. The store sells …

Accepted Answer

Concept & Gross Income Calculation: Gross income is the total revenue generated from sales before any expenses are deducted. To find it, calculate the Cost Price (CP) per unit, apply the respective markups to find the Selling Prices (SP) for different periods, and multiply by the quantities sold during those periods. Gross Income = (Q_1 × SP_1) + (Q_2 × SP_2) Step-by-Step Solution: • Cost Price per pen: The store pays ₹ 9000 for 1000 pens. CP per pen = (9000/1000) = ₹ 9 • First Month Sales (Q_1 & SP_1): • Price is 80% above cost: SP_1 = 9 + (0.80 × 9) = 9 + 7.20 = ₹ 16.20. • Quantity sold (75% of 1000) = 750 pens. • Revenue 1 = 750 × 16.20 = ₹ 12150. • Afterwards Sales (Q_2 & SP_2): • Remaining pens after first month = 1000 - 750 = 250 pens. • Quantity sold (50% of remaining) = 0.50 × 250 = 125 pens. • New Price is 20% above cost: SP_2 = 9 + (0.20 × 9) = 9 + 1.80 = ₹ 10.80. • Revenue 2 = 125 × 10.80 = ₹ 1350. • Total Gross Income: Total Revenue = 12150 + 1350 = ₹ 13500 Exam Strategy & Shortcut: Work with markups fractionally to speed up multiplication. CP = 9. SP1 markup is 80% (4/5). SP1 = 9 × (9/5). Rev 1 = 750 × (81/5) = 150 × 81 = 12150. SP2 markup is 20% (1/5). SP2 = 9 × (6/5). Rev 2 = 125 × (54/5) = 25 × 54 = 1350. Sum = 13500. Common Pitfall: Many students mistakenly assume all remaining 250 pens were sold in the second phase. The problem specifically states "50 percent of the remaining pens", which is only 125 pens. Reading carefully prevents this trap. Final Answer: Therefore, the correct answer is ₹ 13500.

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