Question 1

0.06% of 250% of 1600 is equal to how much?

Accepted Answer

Introduction / Context: This question is about handling nested percentages correctly. It asks the value of 0.06% of 250% of 1600, which involves working with both percentage values and their decimal equivalents. It tests careful handling of small percentages and multiple stages of percentage application. Given Data / Assumptions: We need to compute 250% of 1600 first. Then we must take 0.06% of that result. All percentage operations follow the standard definition: x% = x/100. Concept / Approach: The standard approach is to convert each percentage to its decimal form. 250% becomes 2.5 and 0.06% becomes 0.0006. We first multiply 2.5 by 1600 to get an intermediate value, and then multiply that result by 0.0006. Since multiplication is associative, we can also combine the percentages first and then apply to 1600, but doing it step by step is clearer. Step-by-Step Solution: Step 1: Convert 250% to decimal: 250% = 250/100 = 2.5. Step 2: Compute 250% of 1600: 2.5 * 1600 = 4000. Step 3: Convert 0.06% to decimal: 0.06% = 0.06/100 = 0.0006. Step 4: Compute 0.06% of the previous result: 0.0006 * 4000. Step 5: 0.0006 * 4000 = 2.4. Step 6: Therefore, the required value is 2.4. Verification / Alternative check: We can also combine the operations algebraically. The expression is (0.06/100) * (250/100) * 1600. Compute the combined fraction: (0.06 * 250) / 10000 = 15 / 10000 = 0.0015. Then 0.0015 * 1600 = 2.4. This matches the earlier result and confirms the answer. Why Other Options Are Wrong: 24: This is ten times the correct answer, possibly from treating 0.06% as 0.06 instead of 0.0006. 0.24: This is a factor of ten smaller than 2.4 and could come from one misplaced decimal. 0.024: This is two decimal places off and indicates confusion in handling the percent conversion. 4.8: This may arise from doubling 2.4 by mistake. Common Pitfalls: Students often handle very small percentages like 0.06% incorrectly by converting them to 0.06 instead of 0.0006. Another common issue is not maintaining enough care with decimal places, leading to answers off by factors of ten or hundred. Writing each conversion step explicitly as division by 100, and checking the order of magnitude at the end, helps avoid these errors. Final Answer: The value of 0.06% of 250% of 1600 is 2.4.

Question 2

Two numbers are 90% and 75% less than a certain third number. By what percentage should the smaller number be increased so that it becomes equal to the larger number?

Accepted Answer

Introduction / Context: This question combines relative percentage comparisons with percentage increase. It involves three numbers, where two are described as being 90% and 75% less than a third number. The task is to find how much the smaller number should be increased, in percentage terms, so that it equals the larger number. It is a common pattern in ratio and percentage problems. Given Data / Assumptions: Let the third number be N. First number is 90% less than N. Second number is 75% less than N. We must find the required percentage increase of the smaller number to reach the larger number. Concept / Approach: If a number is 90% less than N, it means it is 10% of N. Similarly, 75% less than N means 25% of N. We convert these verbal descriptions into actual values using N as a base. Then we consider the smaller number as the base value from which we need to reach the larger number, and compute the percentage increase as (difference / smaller) * 100%. The final answer will be independent of N, so we can leave N symbolic. Step-by-Step Solution: Step 1: Let the third number be N. Step 2: First number is 90% less than N, so it is (100% - 90%) of N = 10% of N = 0.10N. Step 3: Second number is 75% less than N, so it is (100% - 75%) of N = 25% of N = 0.25N. Step 4: The smaller number is 0.10N and the larger number is 0.25N. Step 5: Required increase to go from 0.10N to 0.25N is 0.25N - 0.10N = 0.15N. Step 6: Percentage increase = (increase / original smaller number) * 100% = (0.15N / 0.10N) * 100%. Step 7: Simplify the ratio 0.15N / 0.10N = 1.5. Step 8: Therefore, percentage increase = 1.5 * 100% = 150%. Verification / Alternative check: Choose a convenient value for N, say N = 100. Then the smaller number = 10% of 100 = 10, and the larger number = 25% of 100 = 25. To go from 10 to 25, the increase is 15. Percentage increase = (15 / 10) * 100% = 150%. This numerical check confirms the algebraic result. Why Other Options Are Wrong: 250: This would mean the new number is 3.5 times the old one, which is not the case here. 200: This would imply the larger number is three times the smaller, but from 10 to 25 the ratio is 2.5. 100: This would be correct only if the larger number were exactly double the smaller number. 300: This is too high and indicates a confusion between absolute difference and relative difference. Common Pitfalls: Students sometimes confuse “x percent less than N” with “x percent of N” and may treat 90% less as 90% of N. Another common error is to compute percentage difference based on the larger number instead of the smaller one when calculating percentage increase. Always identify clearly which value is being used as the base for the percentage calculation. Final Answer: The smaller number must be increased by 150% to become equal to the larger number.

Question 3

When a number is increased by 216, it becomes 140% of its original value. What is the original number?

Accepted Answer

Introduction / Context: This is another algebraic percentage problem similar to earlier ones, where a number is increased by a fixed amount and described as a certain percent of itself. The goal is to find the original number by converting the verbal description into an equation and solving it. These types of questions commonly appear in aptitude tests. Given Data / Assumptions: Let the original number be N. When 216 is added to N, the result becomes 140% of N. We need to find the value of N. Concept / Approach: If the final value is 140% of the original, then final value = 1.40 * N. At the same time, the final value is given to be N + 216. Setting N + 216 equal to 1.40N gives a linear equation. Solving this equation provides the original number. Careful handling of decimal multipliers is important. Step-by-Step Solution: Step 1: Let the original number be N. Step 2: After increasing it by 216, the new value is N + 216. Step 3: According to the problem, N + 216 = 140% of N. Step 4: Write 140% of N as 1.40N. Step 5: So the equation is N + 216 = 1.40N. Step 6: Rearrange to get 216 = 1.40N - N = 0.40N. Step 7: Therefore N = 216 / 0.40. Step 8: Compute N = 540. Step 9: So the original number is 540. Verification / Alternative check: Substitute N = 540 back into the condition. N + 216 = 540 + 216 = 756. Now compute 140% of 540: 1.40 * 540 = 756. Both values match exactly, confirming that N = 540 is correct. Why Other Options Are Wrong: 756: This is the final value after the increase, not the original number. 450: 140% of 450 is 630, and 450 + 216 = 666, so the condition is not satisfied. 675: 140% of 675 is 945, while 675 + 216 = 891. 600: 140% of 600 is 840, while 600 + 216 = 816. Common Pitfalls: Students may misinterpret 140% as 1.4 plus N instead of 1.4 times N. Others may perform the algebra incorrectly, for example by moving terms incorrectly or dividing by the wrong coefficient. Double checking the derived number by substituting it back into the original statement is a reliable way to catch such mistakes. Final Answer: The original number is 540.

Question 4

A man donates 30% of his wealth to charity. Of the remaining amount, he gives 30% to his wife and 25% to his son. The remaining wealth is divided equally among his three daughters, and each daughter receives Rs 42 lakh. What was the man's total weal…

Accepted Answer

Introduction / Context: This question deals with successive percentage allocations and then an equal division of the remaining amount. It combines percentage reduction and sharing problems, which are very common in aptitude sections that involve distribution of money among family members or partners. The key is to carefully track the remaining wealth after each stage. Given Data / Assumptions: Total wealth of the man is W lakh rupees (unknown). He donates 30% of W to charity. From the remaining amount, he gives 30% to his wife. From the same remaining amount, he gives 25% to his son. The rest is equally divided among three daughters and each daughter gets Rs 42 lakh. Concept / Approach: We first determine what fraction of the original wealth remains after the donation to charity. Then we find how much of that remaining amount goes to the wife and son, and thus what fraction is left for the daughters. Once we have the fraction for the daughters, dividing it equally among three girls gives the share per daughter as a fraction of W. Since each daughter gets 42 lakh, we equate that fraction to 42 and solve for W. Percentages are converted into fractions to simplify the calculations. Step-by-Step Solution: Step 1: Let total wealth be W. Step 2: Donation to charity = 30% of W = 0.30W. Step 3: Wealth left after charity = W - 0.30W = 0.70W. Step 4: From this 0.70W, wife gets 30%: 0.30 * 0.70W = 0.21W. Step 5: From the same 0.70W, son gets 25%: 0.25 * 0.70W = 0.175W. Step 6: Total given to wife and son = 0.21W + 0.175W = 0.385W. Step 7: Wealth left for three daughters = 0.70W - 0.385W = 0.315W. Step 8: This 0.315W is divided equally among three daughters, so each daughter gets 0.315W / 3 = 0.105W. Step 9: Given that each daughter gets 42 lakh, so 0.105W = 42. Step 10: Therefore W = 42 / 0.105 = 400. Step 11: So the man's total wealth is 400 lakh rupees. Verification / Alternative check: Check using W = 400. Charity = 30% of 400 = 120, so remaining = 280. Wife gets 30% of 280 = 84. Son gets 25% of 280 = 70. Remaining for daughters = 280 - 84 - 70 = 126. Divided among three daughters, each gets 126 / 3 = 42. This matches the given share, so W = 400 is correct. Why Other Options Are Wrong: 280: With this wealth, each daughter would get less than 42 lakh. 500: With 500 lakh, each daughter would get more than 42 lakh. 350 and 320: These values also fail to produce the required 42 lakh share when the same percentage distribution is applied. Common Pitfalls: Students sometimes incorrectly apply the percentages for the wife and son directly on the total W instead of on the remaining 0.70W, which changes all subsequent calculations. Another frequent error is to add the percentage amounts instead of properly subtracting them from the remaining wealth. Keeping track of each stage clearly and always noting the current base amount is essential for accuracy. Final Answer: The man's total wealth was 400 lakh rupees.

Question 5

An investor places his savings in the stock market. The value of his investment increases by 12% in the first year and by 9% in the second year. After these two years, the value of his investments becomes Rs 97,664. How much did he originally invest…

Accepted Answer

Introduction / Context: This problem deals with successive percentage increases over two years. It models real life situations where investments grow year by year at different rates. The task is to work backwards from the final amount to find the initial investment. The key idea is to understand compound growth through multiplication by growth factors. Given Data / Assumptions: Initial investment is P (unknown). Increase in first year = 12% of the amount at the beginning of the year. Increase in second year = 9% of the amount at the beginning of the second year. Final value after two years = Rs 97,664. Concept / Approach: A 12% increase means the amount is multiplied by 1.12. A 9% increase means multiplication by 1.09. After two years, the total multiplication factor is 1.12 * 1.09. The final amount equals the initial investment times this combined factor. Thus, we write 97664 = P * 1.12 * 1.09 and solve for P by dividing the final value by the product of the growth factors. Step-by-Step Solution: Step 1: Let the initial investment be P. Step 2: After the first year, amount = P * 1.12. Step 3: After the second year, new amount = (P * 1.12) * 1.09 = P * 1.12 * 1.09. Step 4: Given that this final amount is Rs 97,664. Step 5: So P * 1.12 * 1.09 = 97664. Step 6: Compute the combined factor: 1.12 * 1.09 = 1.2208. Step 7: Therefore P = 97664 / 1.2208. Step 8: Calculate P = 80000. Step 9: So the initial investment was Rs 80,000. Verification / Alternative check: Starting from P = 80000, after first year amount = 80000 * 1.12 = 89600. After second year amount = 89600 * 1.09 = 97664. This matches the given final amount exactly, confirming our solution. Why Other Options Are Wrong: 81000 and 75000: These values, when increased by 12% and then 9%, result in final amounts different from 97664. 72000: This is even smaller and would produce a much lower final value. 90000: Starting with 90000 would end up with a final amount larger than 97664. Common Pitfalls: One frequent mistake is to simply add the percentages and consider the total increase as 21% over two years. That assumes simple rather than compound growth and gives a wrong initial value. Another error is to divide 97664 by only one of the factors instead of the product. Always remember that successive percentage increases multiply, and the correct way to reverse them is to divide by the combined multiplier. Final Answer: The investor originally invested Rs 80,000 in the stock market.

Question 6

One bottle of honey costs Rs 240, but a pack of 4 bottles of the same honey is sold for Rs 768. What is the effective discount (in %) on the pack compared to buying 4 single bottles separately?

Accepted Answer

Introduction / Context: This problem focuses on finding the effective discount when a combo pack is offered cheaper than buying items individually. It connects unit price, total price, and percentage discount, which is a common real life scenario in shopping and a standard concept in profit and loss topics. Given Data / Assumptions: Price of one bottle of honey = Rs 240. If bought separately, 4 bottles would cost 4 * 240 rupees. Pack of 4 bottles is sold for Rs 768. We need the percentage discount on the pack relative to the individual buying cost. Concept / Approach: First, compute the total cost of 4 bottles at the usual unit price. Then, find the difference between that total and the pack price to determine the savings. The effective discount percentage is (savings / original total cost) * 100%. This is the standard formula for discount when we know both original and reduced prices. Step-by-Step Solution: Step 1: Cost of 1 bottle = Rs 240. Step 2: Cost of 4 bottles at single-bottle price = 4 * 240 = Rs 960. Step 3: Pack price for 4 bottles = Rs 768. Step 4: Savings (discount amount) = 960 - 768 = Rs 192. Step 5: Discount percentage = (192 / 960) * 100%. Step 6: Compute 192 / 960 = 0.20. Step 7: Therefore, discount percentage = 0.20 * 100% = 20%. Verification / Alternative check: We can also think in terms of effective price per bottle. In the pack, price per bottle = 768 / 4 = Rs 192. Compare this with the original price 240. The reduction per bottle is 240 - 192 = 48. Percentage discount per bottle = (48 / 240) * 100% = 20%. This matches the earlier result. Why Other Options Are Wrong: 16: This would imply a smaller saving; the actual saving is higher. 25: This is too high and would require the pack price to be 720 instead of 768. 10: This corresponds to a much smaller difference between original and reduced price. 12: Also too low and not supported by the given numbers. Common Pitfalls: Sometimes students calculate the discount on the pack price instead of on the original cost, or they mistakenly divide the savings by the pack price. The discount percentage is always calculated on the original price, which in this case is the total cost of buying 4 bottles individually. Keeping track of which value is the base for the percentage is crucial. Final Answer: The effective discount on the four-bottle honey pack is 20% compared to buying the bottles separately.

Question 7

The cost price of an article is Rs x. The shopkeeper marks it up by 100% and then sells it for Rs 1,200 after allowing a discount of 20% on the marked price. What is the value of x (the cost price in Rs)?

Accepted Answer

Introduction / Context: This question combines markup and discount concepts in one transaction. The shopkeeper increases the price by a certain percentage over the cost price (markup), and then offers a discount to sell the item. The final selling price is given, and we are asked to find the original cost price. It is a classic profit and loss problem that requires stepwise reverse calculation. Given Data / Assumptions: Cost price of the article = Rs x. Marked price is obtained by marking up the cost price by 100%. A discount of 20% is given on this marked price. Final selling price after discount = Rs 1,200. Concept / Approach: A 100% markup means the marked price is double the cost price. A discount of 20% on the marked price means the selling price is 80% of the marked price. We can express the selling price in terms of x by multiplying the cost price by the markup factor and then by the discount factor. Setting this equal to 1200 and solving for x gives the cost price. Step-by-Step Solution: Step 1: Cost price (CP) = Rs x. Step 2: Markup of 100% means marked price (MP) = x + 100% of x = 2x. Step 3: Discount of 20% means selling price (SP) = 80% of MP = 0.80 * MP. Step 4: So SP = 0.80 * 2x = 1.6x. Step 5: Given SP = 1200, so 1.6x = 1200. Step 6: Solve for x: x = 1200 / 1.6. Step 7: Compute x = 750. Step 8: Therefore, the cost price of the article is Rs 750. Verification / Alternative check: Use x = 750 to verify. Marked price = 2 * 750 = Rs 1500. Discount = 20% of 1500 = 300. Selling price after discount = 1500 - 300 = Rs 1200, exactly matching the given selling price. This confirms the correctness of the cost price. Why Other Options Are Wrong: 1500: This would be the marked price, not the cost price, given a 100% markup. 1000: If x were 1000, marked price would be 2000 and 20% discount gives 1600, not 1200. 2000: This cost price is too high and leads to a far larger selling price after discount. 600: With this value, marked price becomes 1200 and 20% discount gives 960, which does not match 1200. Common Pitfalls: A frequent mistake is to confuse markup with profit and assume that a 100% markup means selling price is simply twice the cost price without considering the discount. Another error is to subtract the discount rate from the markup rate directly, which is not valid because they are applied at different stages. Always express each step with precise multiplication factors to avoid these conceptual traps. Final Answer: The cost price of the article is Rs 750.

Question 8

A Rs 1000 box of cookies is offered at 10% discount and a Rs 400 bar of chocolate is offered at 8% discount. If a customer buys 2 boxes of cookies and 3 bars of chocolate in one purchase, what is the overall effective discount percentage on the tota…

Accepted Answer

Introduction / Context: This question tests the concept of effective discount when multiple items with different discount rates are bought together. Instead of looking at each discount separately, we must compare the total amount saved with the total marked price to find a single equivalent discount percentage on the whole purchase. Given Data / Assumptions: Marked price of one box of cookies = Rs 1000 Marked price of one bar of chocolate = Rs 400 Discount on cookies = 10 percent Discount on chocolates = 8 percent Number of cookie boxes bought = 2 Number of chocolate bars bought = 3 We assume there are no extra taxes or additional discounts beyond those mentioned. Concept / Approach: The effective discount is calculated by first finding: Total marked price of all items without discount Total amount actually paid after applying individual discounts Total savings = Total marked price - Total selling price Then, Effective discount percent = (Total savings / Total marked price) * 100. Step-by-Step Solution: Step 1: Total marked price of cookies = 2 * 1000 = Rs 2000. Step 2: Total marked price of chocolates = 3 * 400 = Rs 1200. Step 3: Combined total marked price = 2000 + 1200 = Rs 3200. Step 4: Selling price of one cookie box after 10 percent discount = 1000 * (1 - 10/100) = 1000 * 0.90 = Rs 900. Step 5: Selling price for 2 boxes of cookies = 2 * 900 = Rs 1800. Step 6: Selling price of one chocolate bar after 8 percent discount = 400 * (1 - 8/100) = 400 * 0.92 = Rs 368. Step 7: Selling price for 3 bars of chocolate = 3 * 368 = Rs 1104. Step 8: Total selling price = 1800 + 1104 = Rs 2904. Step 9: Total savings = Total marked price - Total selling price = 3200 - 2904 = Rs 296. Step 10: Effective discount percent = (296 / 3200) * 100 = 9.25 percent. Verification / Alternative check: We can also compute the effective discount by first finding the average discount per rupee. The total discount is Rs 296 on a base of Rs 3200. This ratio is 296 / 3200 = 0.0925, which corresponds to 9.25 percent. This confirms our earlier calculation, so the result is consistent. Why Other Options Are Wrong: 9 percent: This underestimates the savings because it assumes a slightly smaller total discount than actually received. 8.75 percent: This also underestimates the effective discount and does not match the exact ratio of savings to marked price. 8.5 percent: This is even further from the correct value and ignores the precise calculation of selling prices. 9.5 percent: This overestimates the discount and would imply more savings than are actually obtained from the given offers. Common Pitfalls: Students often try to average the discount rates directly, for example by taking the simple average of 10 percent and 8 percent. That method is incorrect because the items have different prices and quantities, so each part contributes differently to the total. Another common mistake is to ignore the total marked price and compute discount only on one item type. Always base the effective discount on the complete bill, not on individual products separately. Final Answer: The overall effective discount on the combined purchase of cookies and chocolates is 9.25 percent.

Question 9

The price of a product after a 20% discount is Rs 3024, and this amount already includes 5% tax on the selling price. What was the original marked price of the product in rupees before discount and tax?

Accepted Answer

Introduction / Context: This problem combines the ideas of discount and tax on the same product. Many entrance and competitive exams test whether a student can correctly unwind both effects to recover the original marked price before any discount or tax is applied. Given Data / Assumptions: Marked price of the product = unknown (let it be M). Discount on marked price = 20 percent. Tax on selling price after discount = 5 percent. Final price paid by the customer after discount and tax = Rs 3024. Tax is calculated on the reduced price after discount, not on the original marked price. Concept / Approach: When a discount and then a tax are applied, the final price can be written as: Final price = Marked price * (1 - discount percent) * (1 + tax percent) We are given the final price and both percentages, so we can treat this as an equation in one variable, solve for the marked price M, and then compare with the options. Step-by-Step Solution: Step 1: Let the marked price be M. Step 2: After a 20 percent discount, selling price before tax = M * (1 - 20/100) = M * 0.80. Step 3: On this selling price, a 5 percent tax is added, so final price = M * 0.80 * (1 + 5/100) = M * 0.80 * 1.05. Step 4: Multiply the factors: 0.80 * 1.05 = 0.84. Step 5: Therefore, final price = M * 0.84. Step 6: Given final price = Rs 3024, so M * 0.84 = 3024. Step 7: Solve for M: M = 3024 / 0.84. Step 8: Divide to get M = Rs 3600. Verification / Alternative check: Take the marked price as Rs 3600 and reapply the operations to confirm. After 20 percent discount: 3600 * 0.80 = Rs 2880. Tax at 5 percent on 2880 is 2880 * 0.05 = Rs 144. Final price = 2880 + 144 = Rs 3024, which matches the given amount. So the marked price of Rs 3600 is correct. Why Other Options Are Wrong: 2742: This value is too low and would not reach 3024 after discount and tax. 2880: This is actually the price after discount but before tax, not the marked price. 3780: This is higher than required and would result in a final price greater than Rs 3024. 3200: This does not satisfy the relation 3024 = 0.84 * M when checked directly. Common Pitfalls: A typical mistake is to subtract or add percentages directly without using multiplication factors. Another error is to treat tax as being applied on the original marked price instead of on the discounted selling price. Students may also forget that a discount reduces the price, while tax increases the price, so both operations must be applied in the correct order using multiplication factors like 0.80 and 1.05. Final Answer: The original marked price of the product is Rs 3600.

Question 10

In an examination, 35% is the minimum percentage required to pass. A student scores 200 marks but still fails by 24 marks. What are the maximum total marks of the examination?

Accepted Answer

Introduction / Context: This question checks understanding of passing percentage and maximum marks. It is a common type of aptitude problem where the candidate must relate the marks obtained, the margin by which the student fails, and the required passing percentage to compute the total marks of the exam. Given Data / Assumptions: Passing percentage = 35 percent. Marks obtained by the student = 200. The student fails by 24 marks. Maximum marks of the examination = unknown (let it be M). Passing marks are calculated as 35 percent of M. Concept / Approach: The key idea is that failing by 24 marks means the pass mark is 24 marks more than the marks obtained. So: Passing marks = 200 + 24 = 224. At the same time, passing marks are 35 percent of maximum marks. Therefore: 0.35 * M = 224. We can solve this equation to find M, the maximum marks. Step-by-Step Solution: Step 1: Interpret "fails by 24 marks". It means passing marks = obtained marks + 24. Step 2: So passing marks = 200 + 24 = 224. Step 3: Passing marks are 35 percent of total marks, so 35/100 * M = 224. Step 4: Write this as 0.35 * M = 224. Step 5: Solve for M: M = 224 / 0.35. Step 6: Divide to get M = 640. Verification / Alternative check: Check with M = 640. Passing marks = 35 percent of 640 = 0.35 * 640 = 224. The student has 200 marks, so shortfall = 224 - 200 = 24 marks, which matches the given failure margin. Thus the calculated maximum marks are consistent with the question statement. Why Other Options Are Wrong: 550: 35 percent of 550 is 192.5, which does not differ from 200 by 24 marks. 680: 35 percent of 680 is 238, which again does not match the failing margin of 24 marks. 820: 35 percent of 820 is 287, which is far from the required pass mark. 600: 35 percent of 600 is 210, and the difference from 200 is 10, not 24. Common Pitfalls: Many students confuse "fails by 24 marks" with "scores 24 percent less than passing marks", which is incorrect. The phrase refers to a simple difference in raw marks. Another common error is to treat 35 percent of 200 as the pass mark, rather than 35 percent of the unknown total. Always express passing marks in terms of the total maximum marks, set up an equation, and then solve for the unknown quantity. Final Answer: The maximum marks of the examination are 640.

Question 11

A man's annual income increases by Rs 1.2 lakhs, but the income tax rate on his income reduces from 12% to 10%. Even after these changes, the amount of tax he pays every year remains exactly the same as before. What is his new increased annual incom…

Accepted Answer

Introduction / Context: This question involves percentage changes applied to income and tax rates. It checks whether you can set up an equation that keeps the tax amount constant while both the income and the tax rate change. Such problems are common in aptitude tests related to percentage applications and income tax scenarios. Given Data / Assumptions: Original annual income = x lakhs (unknown). Original tax rate = 12 percent. New annual income = x + 1.2 lakhs. New tax rate = 10 percent. Tax amount paid before and after the change is the same. We assume only the rate and income change; no extra deductions are involved. Concept / Approach: Tax paid is calculated as: Tax = Income * Tax rate. We equate the old tax and the new tax because the question states that the tax amount remains unchanged. This gives an equation in x. After solving for x, we add 1.2 to get the new income and then compare it with the options given in rupees lakhs. Step-by-Step Solution: Step 1: Let original income be x lakhs. Step 2: Original tax = 12 percent of x = 0.12 * x. Step 3: New income = x + 1.2 lakhs. Step 4: New tax = 10 percent of new income = 0.10 * (x + 1.2). Step 5: Since tax amount is unchanged, set 0.12 * x = 0.10 * (x + 1.2). Step 6: Expand right side: 0.12x = 0.10x + 0.12. Step 7: Subtract 0.10x from both sides to get 0.02x = 0.12. Step 8: Solve for x: x = 0.12 / 0.02 = 6 lakhs. Step 9: New income = x + 1.2 = 6 + 1.2 = 7.2 lakhs. Verification / Alternative check: Original income = 6 lakhs, original tax at 12 percent = 0.12 * 6 = 0.72 lakhs. New income = 7.2 lakhs, new tax at 10 percent = 0.10 * 7.2 = 0.72 lakhs. The tax amount is 0.72 lakhs in both cases, which confirms that the condition of equal tax is satisfied and validates the calculation. Why Other Options Are Wrong: 6: This is the original income rather than the new increased income. 8.4: At this income, tax at 10 percent would be 0.84 lakhs, which is greater than the original tax. 9.6: This would further increase the tax well above the original amount. 10.2: This also leads to a higher tax and does not satisfy the condition of equal tax amounts. Common Pitfalls: Some students subtract percentages directly or confuse percentage change in income with percentage change in tax. Another mistake is to assume that a lower tax rate always means less tax, forgetting that the income may have increased. Always write the explicit expressions for tax before and after the change and then equate them to form a correct equation. Final Answer: The increased annual income of the man is 7.2 lakhs.

Question 12

A daily local train ticket costs Rs 110, while a monthly pass costs Rs 2524.5. If Ram buys the monthly pass and travels on the train for 27 days in a month, what percentage of money does he save compared to buying daily tickets every day?

Accepted Answer

Introduction / Context: This problem compares the cost of daily tickets with a monthly pass and asks for the percentage savings. It is a classic cost comparison question that involves calculating total expenditure under two scenarios and then converting the difference into a percentage saving. Given Data / Assumptions: Cost of one daily local train ticket = Rs 110. Cost of one monthly pass = Rs 2524.5. Number of days Ram travels in the month = 27. We assume one ticket per day and that the monthly pass covers unlimited travel for those days. Concept / Approach: To find the percentage saving, we follow these steps: Compute total cost if Ram buys daily tickets for 27 days. Compute cost using the monthly pass. Find the difference between these two costs. Calculate saving percent = (Saving / Cost with daily tickets) * 100. This gives the effective benefit of choosing the monthly pass over daily tickets. Step-by-Step Solution: Step 1: Total cost with daily tickets = 27 * 110. Step 2: Calculate 27 * 110 = 2970. Step 3: Cost with monthly pass = Rs 2524.5. Step 4: Saving in rupees = 2970 - 2524.5 = 445.5. Step 5: Saving percent = (445.5 / 2970) * 100. Step 6: Compute 445.5 / 2970 = 0.15. Step 7: Multiply by 100 to get 0.15 * 100 = 15 percent. Verification / Alternative check: We can also compare average cost per day. With daily tickets: 2970 / 27 = Rs 110 per day. With the monthly pass: 2524.5 / 27 = Rs 93.5 per day. Saving per day = 110 - 93.5 = Rs 16.5. Ratio = 16.5 / 110 = 0.15, which again gives a 15 percent saving. The two methods agree, so the answer is verified. Why Other Options Are Wrong: 12 percent: Underestimates the saving and does not match the correct fraction 445.5 / 2970. 14 percent: Close but still less than the correct calculated percentage. 18 percent: Overestimates the saving and leads to an incorrect daily cost comparison. 10 percent: This is too low and does not reflect the actual price difference between the two options. Common Pitfalls: Students sometimes divide the saving by the monthly pass cost instead of the daily-ticket cost when calculating the percentage saving. Another mistake is miscalculating the number of days of travel or ignoring the fractional part in 2524.5, which changes the final percent. Always ensure you use the correct base value in the percentage formula and do careful arithmetic when decimals are involved. Final Answer: Ram saves 15 percent by buying the monthly pass instead of daily tickets for 27 days.

Question 13

The selling price of an item is Rs 728 after the shopkeeper offers a 9% discount on its marked price. What was the original marked price of the item in rupees?

Accepted Answer

Introduction / Context: This question focuses on reversing a percentage discount to recover the marked price of an item. Such problems are very common in profit and loss or percentage chapters and test the ability to correctly handle percentage reduction in reverse. Given Data / Assumptions: Selling price after discount = Rs 728. Discount on marked price = 9 percent. Marked price of the item = unknown (let it be M). No taxes or additional discounts are involved beyond the given 9 percent discount. Concept / Approach: If a discount of d percent is given on the marked price, then: Selling price = Marked price * (1 - d/100). Here, d = 9, so factor = 1 - 9/100 = 0.91. We know the selling price and need to find the marked price M using: M = Selling price / 0.91. Step-by-Step Solution: Step 1: Let the marked price be M. Step 2: A 9 percent discount means selling price = M * 0.91. Step 3: Given selling price = Rs 728, so M * 0.91 = 728. Step 4: Solve for M: M = 728 / 0.91. Step 5: Compute the division to get M = 800. Verification / Alternative check: Take M = Rs 800 and apply the discount again to verify. Discount at 9 percent = 800 * 9/100 = Rs 72. Selling price = 800 - 72 = Rs 728. This matches the given selling price exactly, confirming that the marked price is correct. Why Other Options Are Wrong: Rs 667 and Rs 662.48: These values are too low. After a 9 percent discount, they would give a selling price much less than Rs 728. Rs 793.52: This is close but not exact; applying 9 percent discount to it does not give exactly Rs 728. Rs 760: This also leads to a selling price below Rs 728 after a 9 percent reduction. Common Pitfalls: A frequent mistake is to subtract 9 percent of the selling price from 728 instead of reversing the operation on the marked price. Another error is to treat the discount as 728 * (1 + 9/100) which is completely incorrect. Remember that when the discounted price is known, you must divide by (1 - discount percent) rather than subtracting or adding percentages blindly. Final Answer: The original marked price of the item is Rs 800.

Question 14

Two numbers are 30% and 37% less than a third number. By what percentage should the smaller number be increased so that it becomes equal to the larger number?

Accepted Answer

Introduction / Context: This question tests comparative percentage change between two quantities that are both defined relative to a common reference number. It is a typical aptitude problem where one must express each number as a percentage of a base and then compute the required percentage increase from one to the other. Given Data / Assumptions: Let the third number be T. First number is 30 percent less than T. Second number is 37 percent less than T. We are asked by what percent the smaller number must be increased to equal the larger number. We assume T is positive so that all percentages make sense. Concept / Approach: A number that is 30 percent less than T is equal to 70 percent of T. A number that is 37 percent less than T is equal to 63 percent of T. If we call the larger one A and the smaller one B, then: A = 0.70 * T, B = 0.63 * T. Required percentage increase from B to A is: [(A - B) / B] * 100. Step-by-Step Solution: Step 1: Let T be the base number. Step 2: First number A = 70 percent of T = 0.70 * T. Step 3: Second number B = 63 percent of T = 0.63 * T. Step 4: Clearly A > B, so B must be increased. Step 5: Required increase = A - B = (0.70T - 0.63T) = 0.07T. Step 6: Percentage increase relative to B = (0.07T / 0.63T) * 100. Step 7: Simplify 0.07T / 0.63T = 7 / 63 = 1 / 9. Step 8: 1 / 9 in percent form = (1 / 9) * 100 = 11.11 percent approximately. Verification / Alternative check: Choose a convenient value for T, for example T = 100. Then A = 70 and B = 63. Increase needed from 63 to 70 = 7. Percentage increase = 7 / 63 * 100 = 11.11 percent approximately. This numeric example confirms that the formula-based calculation is correct. Why Other Options Are Wrong: 7 percent: Comes from confusing the absolute difference 7 with the percentage difference. 10 percent: A rough guess, but it does not match the true ratio 7 / 63. 18.92 percent: Much larger than the real value, often arising from incorrect denominator choice. 9 percent: Another common guess but not supported by the correct fraction 1 / 9. Common Pitfalls: A frequent mistake is to compute the percentage change using the larger number as the base instead of the smaller one. Another error is to confuse percentage less than with percentage of. Always convert verbal descriptions like "30 percent less" into precise multipliers such as 0.70, then handle the comparison with a proper fraction for percentage change. Final Answer: The smaller number must be increased by 11.11 percent to become equal to the larger number.

Question 15

A website is selling a smartphone at a discounted price of Rs 18,000, while the marked price printed on the phone is Rs 25,000. At what discount percentage on the marked price is the smartphone being sold?

Accepted Answer

Introduction / Context: This question is a straightforward application of the discount formula, where the marked price and the selling price are given. You are required to determine the percentage discount offered on the marked price of the smartphone. Given Data / Assumptions: Marked price of smartphone = Rs 25,000. Selling price after discount = Rs 18,000. Discount percent on marked price = unknown. No additional taxes or extra discounts are mentioned. Concept / Approach: Discount amount is the difference between marked price and selling price: Discount = Marked price - Selling price. Discount percent is calculated with respect to the marked price, using: Discount percent = (Discount / Marked price) * 100. Substituting the given values will directly give the required percentage. Step-by-Step Solution: Step 1: Marked price = Rs 25,000. Step 2: Selling price = Rs 18,000. Step 3: Discount amount = 25,000 - 18,000 = Rs 7,000. Step 4: Discount percent = (7,000 / 25,000) * 100. Step 5: Simplify the fraction 7,000 / 25,000 = 7 / 25. Step 6: Compute (7 / 25) * 100 = 28 percent. Verification / Alternative check: If the discount is 28 percent on Rs 25,000, the discount amount = 25,000 * 28/100 = 25,000 * 0.28 = Rs 7,000. Selling price = 25,000 - 7,000 = Rs 18,000, which matches the given selling price. This confirms that the calculated discount rate is correct. Why Other Options Are Wrong: 22 percent and 20 percent: These are much lower than required; they produce smaller discount amounts that do not reduce the price to Rs 18,000. 25 percent: This would give a discount of 6,250, resulting in a selling price of 18,750, not 18,000. 30 percent: This would give a discount of 7,500 and would reduce the price to 17,500, which is lower than the given selling price. Common Pitfalls: Students sometimes mistakenly compute discount percent based on the selling price instead of the marked price. Another common mistake is to mix up the direction of change and compute percentage increase instead of discount. Always remember that discount percent is calculated on the marked price, because that is the original reference value before any reduction. Final Answer: The smartphone is being sold at a discount of 28 percent on the marked price.

Question 16

On an item with a marked price of Rs 10,000 there is a listed discount of 12%. However, the item is finally sold for Rs 8,360. What additional discount percentage, beyond the original 12% discount, did the customer effectively receive on the already…

Accepted Answer

Introduction / Context: This question examines the idea of successive discounts. An item first receives a standard discount on the marked price, and then the final selling price is even lower than expected. You must calculate the extra discount that has effectively been given beyond the initial advertised discount. Given Data / Assumptions: Marked price of the item = Rs 10,000. Advertised discount on marked price = 12 percent. Expected price after 12 percent discount = unknown. Actual selling price to the customer = Rs 8,360. We assume the extra discount is applied on the already reduced price after the first discount. Concept / Approach: First we compute the price after the original 12 percent discount. Then we compare this discounted price with the final selling price 8,360. The difference between these two prices is treated as an extra discount on the already discounted price. The extra discount percent is then: Extra discount percent = (Extra discount amount / Price after first discount) * 100. Step-by-Step Solution: Step 1: Marked price = Rs 10,000. Step 2: Price after 12 percent discount = 10,000 * (1 - 12/100) = 10,000 * 0.88 = Rs 8,800. Step 3: Actual selling price = Rs 8,360. Step 4: Extra discount amount = 8,800 - 8,360 = Rs 440. Step 5: Extra discount percent = (440 / 8,800) * 100. Step 6: Simplify 440 / 8,800 = 1 / 20. Step 7: (1 / 20) * 100 = 5 percent. Verification / Alternative check: We can check the effective overall discount from the marked price. Overall effective discount = 10,000 - 8,360 = Rs 1,640. Effective discount percent = (1,640 / 10,000) * 100 = 16.4 percent. The sequence 12 percent first and then 5 percent on the reduced price gives an effective discount of: 10,000 * (1 - 0.12) = 8,800, then 8,800 * (1 - 0.05) = 8,360. This matches the final price, confirming that the extra discount is 5 percent. Why Other Options Are Wrong: 4 percent: Gives a smaller reduction and would not bring the price down to Rs 8,360. 6 percent or 7 percent: These would reduce the price too much and lead to a selling price lower than Rs 8,360. 8 percent: This is even larger and clearly inconsistent with the given final price. Common Pitfalls: One frequent mistake is to compute the extra discount as a percentage of the marked price instead of the already discounted price. Another error is to subtract the percentages directly (for example assuming 16.4 percent minus 12 percent equals extra 4.4 percent) without realising that the base changes after the first discount. Always determine the intermediate selling price after the first discount, then calculate any further percentage reductions on that intermediate value. Final Answer: The customer obtained an additional discount of 5 percent on the already discounted price.

Question 17

Find the value of 0.09% of 25% of 1200, that is, first take 25% of 1200 and then find 0.09% of that result.

Accepted Answer

Introduction / Context: This question tests your understanding of nested percentages, where one percentage is taken of another percentage of a base number. It is a straightforward numerical computation but requires careful handling of decimal forms of percentages to avoid calculation errors. Given Data / Assumptions: We need to compute 0.09 percent of 25 percent of 1200. Base number = 1200. First percentage = 25 percent. Second percentage = 0.09 percent. All operations are standard percentage calculations with no rounding specified. Concept / Approach: Percentages can be converted into decimal multipliers. 25 percent = 25 / 100 = 0.25. 0.09 percent = 0.09 / 100 = 0.0009. The required value is: Result = 0.09 percent * (25 percent of 1200). This can be written as: Result = 0.0009 * 0.25 * 1200. Step-by-Step Solution: Step 1: Convert 25 percent to decimal: 25 percent = 0.25. Step 2: Find 25 percent of 1200: 0.25 * 1200 = 300. Step 3: Convert 0.09 percent to decimal: 0.09 percent = 0.0009. Step 4: Compute 0.09 percent of 300: 0.0009 * 300. Step 5: Multiply 0.0009 * 300 = 0.27. Verification / Alternative check: Instead of doing the steps separately, we can combine all factors at once. Result = (0.09 / 100) * (25 / 100) * 1200 = (0.09 * 25 * 1200) / 10,000. Compute the numerator: 0.09 * 25 = 2.25, then 2.25 * 1200 = 2700. Now 2700 / 10,000 = 0.27. This alternate approach gives the same result and confirms the accuracy of the calculation. Why Other Options Are Wrong: 2.7: This is ten times larger than the correct answer and arises if the decimal place is misplaced. 27 and 270: These are huge compared with the fractions of a percent being applied and indicate that percentages were not converted properly. 0.027: This is one tenth of the correct value and may result from incorrectly treating one of the percentages as a plain decimal rather than a percent. Common Pitfalls: Students often forget that a percent itself needs to be divided by 100, and when two percentages are multiplied, the effect is divided by 10,000. Misplacing decimal points is very common in such questions. The safest approach is to convert each percentage step by step into a decimal and multiply carefully, or to carry out the entire calculation as a fraction over 10,000. Final Answer: The value of 0.09 percent of 25 percent of 1200 is 0.27.

Question 18

The price of petrol increases from Rs 60 per litre to Rs 75 per litre. By what percentage should a household reduce its consumption of petrol so that its total expenditure on petrol increases by only 10%?

Accepted Answer

Introduction / Context: This question involves a price increase for petrol and a controlled increase in expenditure. It tests the understanding of how changes in price and quantity interact. The key idea is that to limit the rise in spending, consumption must be reduced appropriately to offset part of the price increase. Given Data / Assumptions: Original price of petrol = Rs 60 per litre. New price of petrol = Rs 75 per litre. Desired increase in total expenditure = 10 percent. Original consumption = q litres (unknown). New consumption after adjustment = q dash litres. We assume no other changes in conditions. Concept / Approach: Original expenditure is: E1 = 60 * q. New expenditure should be 10 percent more, so: E2 = 1.10 * E1 = 1.10 * 60 * q. At the new price of 75 per litre, the new expenditure is also: E2 = 75 * q dash. We equate these two expressions for E2 and solve for q dash in terms of q, then compute the percentage reduction in consumption: Reduction percent = ((q - q dash) / q) * 100. Step-by-Step Solution: Step 1: Original expenditure E1 = 60 * q. Step 2: New expenditure allowed, with 10 percent increase, E2 = 1.10 * 60 * q = 66 * q. Step 3: New expenditure at new price is also E2 = 75 * q dash. Step 4: Equate the two: 75 * q dash = 66 * q. Step 5: Solve for q dash: q dash = (66 / 75) * q. Step 6: Simplify 66 / 75 = 0.88, so q dash = 0.88 * q. Step 7: Reduction in consumption = q - q dash = q - 0.88q = 0.12q. Step 8: Reduction percent = (0.12q / q) * 100 = 12 percent. Verification / Alternative check: Assume an original consumption of 100 litres for simplicity. Original expenditure = 60 * 100 = Rs 6,000. Permitted new expenditure = 10 percent more = 6,000 * 1.10 = Rs 6,600. At Rs 75 per litre, new consumption = 6,600 / 75 = 88 litres. Reduction in consumption = 100 - 88 = 12 litres, which is 12 percent of 100. This numerical example confirms the theoretical result. Why Other Options Are Wrong: 15 percent and 18 percent: These imply a larger reduction than needed, which would lead to an expenditure increase of less than 10 percent. 20 percent: This reduction is too high and would significantly underuse the allowed increased budget. 10 percent: This value is smaller than required and would cause the expenditure to rise by more than 10 percent after the price increase. Common Pitfalls: Candidates often incorrectly compute the percentage reduction by comparing price changes alone, ignoring the controlled expenditure increase. Some also mistakenly treat the change in price from 60 to 75 as 15 percent instead of 25 percent. The correct approach is to equate the target new expenditure with the product of new price and new quantity, then work out the ratio of new quantity to old quantity carefully. Final Answer: The household should reduce petrol consumption by 12 percent to limit the expenditure increase to 10 percent.

Question 19

There are two buckets, one large and one small. The smaller bucket can hold only 3/5 of the water that the larger bucket can hold. If 6,000 buckets of the larger capacity are needed to fill a pond, how many buckets of the smaller capacity would be r…

Accepted Answer

Introduction / Context: This problem involves ratios and proportional reasoning. The capacities of two buckets are related by a simple fraction, and you are asked how many of the smaller buckets are needed when you already know how many larger buckets can fill the same pond. Given Data / Assumptions: Capacity of smaller bucket = 3/5 of capacity of larger bucket. Number of larger buckets needed to fill the pond = 6000. We need to find the number of smaller buckets required. The volume of the pond remains the same in both scenarios. Concept / Approach: If capacity changes, the number of containers needed is inversely proportional to capacity. Since the small bucket holds only 3/5 as much water as the large bucket, more small buckets are required. Let capacity of large bucket = C units. Then capacity of small bucket = (3/5) * C. Total pond volume in terms of large buckets = 6000 * C. Number of small buckets needed = Total volume / small bucket capacity. Step-by-Step Solution: Step 1: Assume capacity of one large bucket is C units of water. Step 2: Then capacity of one small bucket = (3/5) * C. Step 3: Total volume of the pond = 6000 * C (because 6000 large buckets fill it). Step 4: Let N be the number of small buckets needed. Then N * (3/5) * C = 6000 * C. Step 5: Cancel C on both sides to get N * (3/5) = 6000. Step 6: Solve for N: N = 6000 * (5/3). Step 7: Compute 6000 * (5/3) = 6000 * 5 / 3 = 30,000 / 3 = 10,000. Verification / Alternative check: We can also think in terms of a unit volume like 1 pond. Larger bucket capacity = 1/6000 of the pond. Smaller bucket capacity is 3/5 of that = 3 / (5 * 6000) of the pond. To cover the full pond, we need: 1 divided by [3 / (5 * 6000)] = (5 * 6000) / 3 = 10,000 buckets. This matches the result from the algebraic approach. Why Other Options Are Wrong: 8000: This assumes fewer small buckets than large ones, which is impossible since each small bucket holds less water. 12000 and 15000: These numbers are larger than necessary and result from miscalculating the ratio or inverting it incorrectly. 7500: This is again too low and does not satisfy the proportional relationship between capacities and counts. Common Pitfalls: A common mistake is to multiply by 3/5 instead of dividing by 3/5 when going from number of large buckets to number of small buckets. Students may also confuse direct proportion with inverse proportion. Remember that when capacity decreases, the number of buckets needed must increase in inverse proportion to that capacity change. Final Answer: To fill the same pond with the smaller bucket, 10,000 buckets are required.

Question 20

It is given that 90% of X is equal to 40% of Y, and also that Y is equal to a% of X. What is the value of a?

Accepted Answer

Introduction / Context: This question is about relating two variables X and Y using percentage statements and then finding the exact percentage relationship between them. It checks whether you can translate percentage equations into algebraic form and solve for an unknown percentage factor a. Given Data / Assumptions: 90 percent of X is equal to 40 percent of Y. Y is equal to a percent of X. We need to find the value of a. Both X and Y are assumed to be positive real numbers. Concept / Approach: Expressions like 90 percent of X and 40 percent of Y are written as: 90 percent of X = 90/100 * X = 0.90 * X. 40 percent of Y = 40/100 * Y = 0.40 * Y. The equation 90 percent of X = 40 percent of Y becomes: 0.90 * X = 0.40 * Y. From this, we obtain Y in terms of X, and then interpret Y as a percent of X to read off a. Step-by-Step Solution: Step 1: Write the given relationship as an equation: 0.90 * X = 0.40 * Y. Step 2: Solve for Y in terms of X: Y = (0.90 / 0.40) * X. Step 3: Simplify 0.90 / 0.40 = 9/4 = 2.25. Step 4: Hence, Y = 2.25 * X. Step 5: The statement "Y = a percent of X" means Y = (a / 100) * X. Step 6: Compare the two expressions for Y: (a / 100) * X = 2.25 * X. Step 7: Cancel X on both sides to get a / 100 = 2.25. Step 8: Multiply both sides by 100 to find a: a = 2.25 * 100 = 225. Verification / Alternative check: Choose a convenient number for X, such as X = 100. Then 90 percent of X = 90. So 40 percent of Y must be 90, giving Y = 90 / 0.40 = 225. Now Y as a percent of X = 225 / 100 * 100 percent = 225 percent. Thus a = 225, confirming the algebraic solution. Why Other Options Are Wrong: 125: This would correspond to Y being only 1.25 times X, which does not satisfy the given relation 0.90X = 0.40Y. 2.25: This is the multiplier, not the percentage. The question specifically asks for a, which is a percent value, not a plain factor. 1.25: Also a multiplier, and too small in comparison with the correct factor 2.25. 2250: This is ten times larger than the correct percentage and does not match any logical interpretation of the relationship between X and Y. Common Pitfalls: One common error is to forget that a percent is obtained by multiplying the decimal factor by 100, so students may choose 2.25 instead of 225. Another mistake is to mix up which variable is expressed as a percent of the other. Always translate the wording carefully and keep track of which quantity is being described as a percent of which base. Final Answer: The value of a is 225, so Y is 225 percent of X.

Percentage Questions