Question 1

Rajesh buys a table for Rs. 900 and sells it to a customer after giving two successive discounts of 10 percent and 20 percent on the marked price (assumed equal to Rs. 900). What is the final selling price of the table?

Accepted Answer

Introduction / Context:This question involves successive discounts on a single marked price. It helps strengthen understanding that each discount applies to the current price, not the original cost price directly, and that the net discount is not the simple sum of the individual percentages.Given Data / Assumptions:Cost price to Rajesh = Rs. 900.Marked price is assumed to be Rs. 900 for this question.First discount = 10 percent.Second discount = 20 percent.We must find the final selling price to the customer.Concept / Approach:The selling price after successive discounts is obtained by multiplying the marked price by (1 - first discount) and then by (1 - second discount). We can perform this step by step to avoid confusion and errors.Step-by-Step Solution:Marked price MP = Rs. 900.After first discount of 10 percent, price = 900 * (1 - 0.10) = 900 * 0.90 = 810.After second discount of 20 percent, new price = 810 * (1 - 0.20) = 810 * 0.80 = 648.Thus final selling price = Rs. 648.Verification / Alternative check:We can also combine the discount factors: overall factor = 0.90 * 0.80 = 0.72. So selling price = 0.72 * 900 = 648, which matches the stepwise calculation.Why Other Options Are Wrong:Values like 568, 728 or 668 correspond to erroneous calculations such as subtracting 30 percent directly from 900 or misapplying the second discount on the original price instead of the discounted price. They do not equal 900 * 0.72.Common Pitfalls:Many candidates simply add the discounts (10 percent and 20 percent) and calculate a single 30 percent discount. This gives 900 * 0.70 = 630, which is not any of the options and is incorrect. Always remember that successive discounts multiply, not add.Final Answer:The final selling price of the table is Rs. 648.

Question 2

If C = 1200 and D = 1500, then D is how many percent more than C?

Accepted Answer

Introduction / Context:This short question tests the basic concept of percentage increase from one value to another. It is a direct application of the formula for percentage change.Given Data / Assumptions:C = 1200.D = 1500.We need the percentage by which D is more than C.Concept / Approach:Percentage increase from C to D is calculated as (D - C) / C * 100. Here C is the base because the comparison is "more than C".Step-by-Step Solution:C = 1200, D = 1500.Increase = D - C = 1500 - 1200 = 300.Percentage increase = (300 / 1200) * 100.Compute 300 / 1200 = 0.25.Percentage increase = 0.25 * 100 = 25 percent.Verification / Alternative check:We can note that multiplying 1200 by 1.25 gives 1200 * 1.25 = 1500. The factor 1.25 corresponds to a 25 percent increase, confirming the result.Why Other Options Are Wrong:20 percent would give D = 1440, 33.33 percent would give D = 1600 and 125 percent is obviously far too large. None match the given value of 1500 when applied to 1200 as a base.Common Pitfalls:Sometimes students divide by D instead of C, which answers a different question: how much C is less than D as a percentage of D. Always ensure that the base in the denominator matches the phrase in the question.Final Answer:D is 25 percent more than C.

Question 3

A shopkeeper sells an item for Rs. 4400 which has a marked price of Rs. 5000. What discount percentage is he offering on the item?

Accepted Answer

Introduction / Context:This is a basic discount problem where the marked price and the selling price are given. The aim is to compute the percentage discount provided by the shopkeeper.Given Data / Assumptions:Marked price = Rs. 5000.Selling price = Rs. 4400.We must find the discount percentage on the marked price.Concept / Approach:Discount amount is the difference between marked price and selling price. Discount percentage is (discount amount / marked price) * 100. The base is always the marked price for such calculations.Step-by-Step Solution:Marked price MP = 5000.Selling price SP = 4400.Discount amount = MP - SP = 5000 - 4400 = 600.Discount percentage = (discount / MP) * 100.So discount percentage = (600 / 5000) * 100.Compute 600 / 5000 = 0.12.Thus discount percentage = 0.12 * 100 = 12 percent.Verification / Alternative check:If a discount of 12 percent is offered on Rs. 5000, the discount amount is 0.12 * 5000 = 600 and the selling price becomes 5000 - 600 = 4400, consistent with the given data.Why Other Options Are Wrong:A 20 percent discount would give a selling price of 4000, 15 percent would give 4250 and 10 percent would give 4500. None of these match the actual selling price of 4400, so they are incorrect.Common Pitfalls:Sometimes students divide by 4400 instead of 5000, which is wrong for discount percentage. Others miscalculate the difference or misplace the decimal while converting to percentage. Always double check both subtraction and division steps.Final Answer:The shopkeeper offers a discount of 12 percent.

Question 4

A student must secure 40 percent marks to pass. He scores 67 marks and fails by 13 marks. What are the maximum marks for the examination?

Accepted Answer

Introduction / Context:This question links pass percentage, marks obtained and shortfall in marks to the total maximum marks of an examination. It is frequently seen in recruitment and academic entrance test papers.Given Data / Assumptions:Passing marks = 40 percent of maximum marks.The student scores 67 marks.He fails by 13 marks.We need to find the maximum marks.Concept / Approach:If the student fails by 13 marks, then passing marks equal his marks plus 13. Since passing marks are 40 percent of the maximum, we can express 40 percent of the total as that number and solve for the maximum marks using a simple linear equation.Step-by-Step Solution:Let maximum marks be M.Passing marks = 40 percent of M = 0.40 * M.He scores 67 and fails by 13, hence passing marks = 67 + 13 = 80.So 0.40 * M = 80.Therefore M = 80 / 0.40 = 80 * (10 / 4) = 80 * 2.5 = 200.Verification / Alternative check:Check: 40 percent of 200 = 0.40 * 200 = 80. The student has 67 marks, which is 13 marks less than 80, so he fails by 13 marks. This confirms that the maximum marks are 200.Why Other Options Are Wrong:For maximum marks 300, 40 percent would be 120, which is not 13 more than 67. Similarly, 150 gives passing marks 60, which is less than the student's marks, meaning he would have passed, contradicting the question. The figure 240 leads to 96 passing marks, which does not fit the given shortfall of 13.Common Pitfalls:Some students incorrectly treat 67 as 40 percent of total marks and then add 13, which leads to an incorrect equation. Always connect the shortfall with the passing marks, not with the maximum marks directly.Final Answer:The maximum marks for the examination are 200.

Question 5

The cost price of a watch is Rs. 530. If the loss percentage on selling the watch is 24 percent, what is the loss amount in rupees?

Accepted Answer

Introduction / Context:This is a standard loss percentage question where the cost price and loss percentage are given. It checks the ability to compute the rupee loss from the percentage.Given Data / Assumptions:Cost price (CP) = Rs. 530.Loss percentage = 24 percent.We must find the loss amount in rupees.Concept / Approach:Loss amount is simply the given percentage of cost price. The formula is loss = (loss percent / 100) * cost price.Step-by-Step Solution:Loss percent = 24 percent.Cost price = 530.Loss amount = 24/100 * 530.Compute 24/100 = 0.24.Loss = 0.24 * 530.0.24 * 530 = 127.2.Verification / Alternative check:We can find the selling price as well: SP = CP - loss = 530 - 127.2 = 402.8. Then loss percentage = (530 - 402.8) / 530 * 100 = 127.2 / 530 * 100 = 24 percent, confirming the computation.Why Other Options Are Wrong:Values 153.2, 139.4 and 114.2 correspond to different loss percentages when divided by 530. For instance, 139.4 would be about 26.3 percent of 530, not 24 percent.Common Pitfalls:Students may accidentally multiply by 76 percent instead (thinking of the remaining value) or use selling price as base. Remember that loss percentage is always calculated on cost price unless specified otherwise.Final Answer:The loss amount on the watch is Rs. 127.2.

Question 6

If 6/7 of 8/5 of a number is 192, what is 3/4 of that number?

Accepted Answer

Introduction / Context:This problem combines fractions and algebra. It asks the learner to work with fractional multiples of an unknown number and then find another specific fraction of the same number. It is typical in aptitude tests to check fraction manipulation skills.Given Data / Assumptions:(6/7) of (8/5) of a number equals 192.We must find 3/4 of that number.The number is assumed to be positive.Concept / Approach:Let the number be N. The given condition becomes (6/7) * (8/5) * N = 192. We can simplify the product of fractions and then solve for N. After that, we compute 3/4 of N directly.Step-by-Step Solution:Let the number be N.Given (6/7) * (8/5) * N = 192.Simplify the fraction: (6/7) * (8/5) = 48 / 35.So (48 / 35) * N = 192.Multiply both sides by 35: 48N = 192 * 35.Compute 192 * 35 = 6720.Thus N = 6720 / 48 = 140.We need 3/4 of N = 3/4 * 140 = 105.Verification / Alternative check:Check the initial condition with N = 140: (6/7) of (8/5) of 140 is (6/7) * (8/5) * 140. First (8/5) * 140 = 8 * 28 = 224. Then (6/7) * 224 = 6 * 32 = 192, which matches the given value, confirming N = 140 and 3/4 of N = 105.Why Other Options Are Wrong:Values like 77, 36 or 80 correspond to other fractions of 140 or wrong intermediate computations. Only 105 is exactly 3/4 of 140 and consistent with all given conditions.Common Pitfalls:Common errors include inverting fractions incorrectly, cancelling wrong terms or forgetting to multiply the right side by the denominator when solving for N. Careful stepwise handling of fractions avoids these issues.Final Answer:Three fourths of the number is 105.

Question 7

Two numbers are 35 percent and 50 percent less than a third number. By what percentage should the second number be increased so that it becomes equal to the first number?

Accepted Answer

Introduction / Context:This question involves relative percentage decreases from a common reference and then asks for the percentage increase required to make the smaller number equal to the larger decreased number. It combines understanding of percentage decrease and percentage increase together.Given Data / Assumptions:First number is 35 percent less than a third number.Second number is 50 percent less than the same third number.We must find by what percent the second number should be increased to equal the first number.Concept / Approach:Let the third number be 100 units for simplicity. Then the first and second numbers can be expressed as percentages of 100. After that, we compare the required increase from the second to the first and convert the difference into a percentage of the second number.Step-by-Step Solution:Let the third number = 100 units.First number = 35 percent less than 100 = 65 units.Second number = 50 percent less than 100 = 50 units.To make the second number equal to the first, it must be increased from 50 to 65.Increase needed = 65 - 50 = 15 units.Required percentage increase = (15 / 50) * 100.Compute 15 / 50 = 0.30.Therefore percentage increase = 0.30 * 100 = 30 percent.Verification / Alternative check:If the actual third number were different, say 200, then first number = 130 and second = 100. Increase needed from 100 to 130 is 30 units, which is still 30 percent of 100, confirming the result is independent of the chosen base value.Why Other Options Are Wrong:Values like 23.08 percent or 25 percent reflect miscalculations or comparisons with the wrong base, such as the third number instead of the second. The value 15 percent is simply the raw difference in percentage reductions (35 and 50) misinterpreted as the answer.Common Pitfalls:Many students subtract the percentage reductions directly, taking 50 minus 35 = 15 percent as the required increase. However, the base of the increase is the second number, not the third number. Always base the percentage on the quantity being changed.Final Answer:The second number must be increased by 30 percent to equal the first number.

Question 8

At a discount of 13%, the selling price of an oven is Rs 18,000. If the shop keeps the same marked price and instead offers a discount of 27.5%, what will be the new selling price of the oven?

Accepted Answer

Introduction / Context: This percentage based question tests the idea of moving from selling price back to marked price and then finding a new selling price under a different discount rate. Such questions are very common in aptitude exams and competitive tests where candidates must be comfortable reversing percentage calculations and applying successive discounts. Given Data / Assumptions: Selling price with 13 percent discount = Rs 18000. Discount rate 1 = 13 percent. Discount rate 2 = 27.5 percent. Marked price of the oven remains the same in both cases. All prices are in rupees and there are no additional taxes or charges. Concept / Approach: The selling price after a discount is calculated as marked price multiplied by (100 minus discount percent) divided by 100. To move backwards, we divide the selling price by this fraction to get the marked price. Once we know the marked price, we can again apply the new discount rate to compute the new selling price. Step-by-Step Solution: Let the marked price be M. With a 13 percent discount, selling price = M * (1 - 13/100) = M * 0.87. Given M * 0.87 = 18000. So M = 18000 / 0.87. Compute M: 18000 / 0.87 = 20689.655172 approximately. Now new discount rate = 27.5 percent, so new selling price = M * (1 - 27.5/100) = M * 0.725. New selling price = 20689.655172 * 0.725 which equals exactly 15000 when calculated accurately. Therefore the new selling price is Rs 15000. Verification / Alternative check: We can avoid dealing with the decimal by using ratios. Since 18000 corresponds to 87 percent of the marked price, 1 percent equals 18000 / 87. The new selling price should be 72.5 percent of the marked price. So new selling price = 72.5 * 18000 / 87. Multiplying 72.5 and 18000 and then dividing by 87 again gives 15000, which confirms the earlier calculation. Why Other Options Are Wrong: 18000 is closer to 15000 than to 10875 or 12288.75 when discount increases, so options with very low values underestimate the effect. Values like 16638.75 or 13500 do not match the precise ratio 72.5/87 derived from the discount change. The exact computation gives 15000, so all other options are numerically inconsistent with the given data. Common Pitfalls: A frequent mistake is to apply both discounts directly to 18000 instead of going back to marked price. Another error is to treat the difference in discount percentages, 27.5 minus 13, as if it could be applied directly to the old selling price. Both shortcuts lead to incorrect results, because percentage discounts must always be applied on the marked price, not on the already reduced selling price. Final Answer: The new selling price of the oven when the shop offers a 27.5 percent discount instead of 13 percent is Rs 15000.

Question 9

After a 15 percent discount, the selling price of a shirt is Rs 119. What was the marked price of the shirt before any discount was applied?

Accepted Answer

Introduction / Context: This problem checks understanding of how to reverse a percentage discount to recover the original marked price. Questions of this type appear frequently in profit and loss as well as percentage sections of aptitude exams. The key idea is to recognize that the discounted selling price represents a certain percentage of the original marked price. Given Data / Assumptions: Selling price after discount = Rs 119. Discount rate = 15 percent. Marked price is unknown and needs to be found. No additional taxes, transport, or extra charges are considered. Concept / Approach: A discount of 15 percent means the customer pays 85 percent of the marked price. In mathematical terms, selling price = marked price * 85/100. To find the marked price we divide the selling price by 0.85. This reverse percentage process is a standard technique for such questions. Step-by-Step Solution: Let the marked price be M. Selling price with 15 percent discount = M * (1 - 15/100) = M * 0.85. We are told that M * 0.85 = 119. Therefore M = 119 / 0.85. Compute 119 / 0.85 = 140. So the marked price before discount was Rs 140. Verification / Alternative check: We can verify by applying 15 percent discount on Rs 140. Discount amount = 15 percent of 140 = 15/100 * 140 = 21. Selling price = 140 minus 21 = 119. Since this matches the given selling price, our reverse calculation is correct and consistent. Why Other Options Are Wrong: If the marked price were Rs 129, 15 percent discount would give 109.65, not 119. For Rs 150, 15 percent discount produces 127.5. For Rs 160, it gives 136. None of these equal 119. The option Rs 136 would lead to a discounted price of 115.6. Only Rs 140 produces exactly Rs 119 when a 15 percent discount is applied. Common Pitfalls: Learners often subtract 15 from 119 or compute 119 plus 15 percent of 119, which is incorrect because the discount is applied to the marked price, not to the selling price. Another mistake is to add 15 rupees instead of 15 percent. Always convert the percentage to a multiplier and work with the relationship between marked price and selling price carefully. Final Answer: The marked price of the shirt before the 15 percent discount was Rs 140.

Question 10

Two students appeared for an examination. One student scored 20 marks more than the other, and his marks were 55 percent of the sum of their marks together. What are the marks obtained by the two students?

Accepted Answer

Introduction / Context: This question examines understanding of percentages and linear equations in a word problem involving two unknown quantities. The relationship between the marks of the two students is given in two different forms: a fixed difference and a percentage of their combined total. Solving such questions quickly is important in quantitative aptitude sections. Given Data / Assumptions: Let marks of the first student be A. Let marks of the second student be B. A is 20 marks more than B, so A - B = 20. A is 55 percent of the sum of their marks, so A = 55 percent of (A + B). All marks are assumed to be non negative integers. Concept / Approach: We convert the percentage relationship into an algebraic equation. Since A is 55 percent of the total A + B, we write A = 55/100 * (A + B). Combined with the difference equation A - B = 20, we can solve the system for A and B using basic algebra. This method is typical when there are two unknowns and two independent equations. Step-by-Step Solution: From the first condition: A - B = 20. From the second condition: A = 55/100 * (A + B) = 0.55 * (A + B). Rewrite: A = 0.55A + 0.55B. Bring like terms together: A - 0.55A = 0.55B. This gives 0.45A = 0.55B. So A / B = 0.55 / 0.45 = 55 / 45 = 11 / 9. Thus A : B = 11 : 9. Difference in ratio units = 11 - 9 = 2 parts correspond to actual difference 20 marks. So 2 parts = 20 marks which implies 1 part = 10 marks. Therefore A = 11 * 10 = 110 and B = 9 * 10 = 90. Verification / Alternative check: Check the conditions. Difference: 110 minus 90 equals 20 which matches the first condition. Sum: 110 plus 90 equals 200. Now 55 percent of 200 is 0.55 * 200 = 110, which matches A. Both conditions are satisfied, so the pair 110 and 90 is correct. Why Other Options Are Wrong: For 92 and 72, the difference is 20 but 55 percent of their sum 164 is 90.2, not 92. For 83 and 63, 55 percent of 146 is 80.3, not 83. For 64 and 44, 55 percent of 108 is 59.4. For 102 and 82, 55 percent of 184 is 101.2, not 102. Only the pair 110 and 90 satisfies both the difference and percentage conditions. Common Pitfalls: A common error is to treat 55 percent as relating only to one student rather than to the sum of their marks. Another mistake is to attempt random trial and error with answer options without setting up the equations, which is slow and error prone. Working systematically with ratios and equations leads to the correct solution more reliably. Final Answer: The two students obtained 110 and 90 marks respectively.

Question 11

A cloth merchant announces a 25 percent rebate on the price of cloth. Cloth normally costs Rs 32 per metre. If a customer wants to receive a total rebate of Rs 40, how many metres of cloth must be purchased?

Accepted Answer

Introduction / Context: This question is a direct application of percentage discount to find quantity purchased. The merchant offers a flat rebate percentage on a per metre price. The customer wants a specified amount of money saved through the rebate, and we need to connect rebate per metre with total rebate to find the required number of metres. Given Data / Assumptions: Rebate rate = 25 percent. Price of cloth per metre before rebate = Rs 32. Total rebate desired by the customer = Rs 40. The rebate is applied uniformly on every metre purchased. Concept / Approach: Rebate per metre is calculated as percentage rebate multiplied by marked price per metre. Once rebate per metre is known, total rebate equals rebate per metre multiplied by number of metres purchased. Hence, number of metres equals total rebate divided by rebate per metre. This is a straightforward proportional relationship. Step-by-Step Solution: Marked price per metre = Rs 32. Rebate rate = 25 percent, which is 25/100. Rebate per metre = 25/100 * 32. Compute 25/100 * 32 = 8. So the customer saves Rs 8 on each metre of cloth. Total rebate required = Rs 40. Number of metres to be purchased = total rebate / rebate per metre. Number of metres = 40 / 8 = 5 metres. Verification / Alternative check: We can check the answer by reversing the reasoning. If the customer buys 5 metres, total original cost is 5 * 32 = 160. Rebate at 25 percent is 25/100 * 160 = 40. This matches the required rebate, so 5 metres is correct. Trying any other option quickly shows that the rebate would not equal exactly Rs 40. Why Other Options Are Wrong: If 6 metres are bought, total rebate would be 6 * 8 = 48. For 10 metres, rebate is 80. For 7 metres, rebate is 56. For 8 metres, rebate is 64. None of these equal the target rebate of 40. Only 5 metres give exactly Rs 40 as rebate. Common Pitfalls: One common mistake is to compute 25 percent of 40 instead of 25 percent of the price per metre. Another error is to try to directly adjust the final price rather than focusing on rebate per unit. Clear separation between unit rebate and total rebate helps avoid confusion. Final Answer: The customer must purchase 5 metres of cloth to receive a total rebate of Rs 40.

Question 12

A retailer offers a discount of 32 percent on the marked price of goods and therefore sells exactly at the cost price. What is the percentage mark up of the marked price over the cost price?

Accepted Answer

Introduction / Context: This question links percentage discount with percentage markup. The retailer first increases the cost price to set a marked price, and then applies a discount such that the final selling price returns to the original cost price. The task is to determine what markup percentage is needed so that a discount of 32 percent exactly cancels the markup. Given Data / Assumptions: Let cost price be C. Marked price is some percentage above C. Discount rate on marked price = 32 percent. Selling price after discount equals cost price. We need to find percentage markup of marked price over cost price. Concept / Approach: If marked price is C multiplied by (1 plus markup rate), and discount is 32 percent, then selling price is marked price multiplied by (1 minus 32/100). We are told that this selling price equals C, so we can form an equation and solve for markup rate. This is a classic percentage composition problem that combines increase and decrease. Step-by-Step Solution: Let markup rate be m percent. Then marked price M = C * (1 + m/100). Discount is 32 percent, so selling price S = M * (1 - 32/100) = M * 0.68. Given S = C, so C = C * (1 + m/100) * 0.68. Divide both sides by C (C is non zero): 1 = (1 + m/100) * 0.68. So 1 / 0.68 = 1 + m/100. Compute 1 / 0.68 = 100 / 68 = 25 / 17 = 1.470588 approximately. Thus 1.470588 = 1 + m/100, so m/100 = 0.470588. Therefore m = 47.0588 percent, which rounds to 47.05 percent. Verification / Alternative check: Assume cost price C = 100 units. With markup 47.05 percent, marked price is 147.05. Discount of 32 percent on 147.05 is 0.68 * 147.05 which equals about 100.0, matching the cost price. So the combination of markup and discount returns the selling price to cost price, confirming the result. Why Other Options Are Wrong: A markup of 24 percent or 22.34 percent would make the marked price too low, and after 32 percent discount the selling price would fall below cost price. A markup equal to 32 percent would lead to a loss instead of breaking even. Forty percent markup is still insufficient to balance a 32 percent discount. Only 47.05 percent markup precisely compensates for a 32 percent discount. Common Pitfalls: Many learners simply subtract 32 from 100 and conclude that 32 percent markup is enough, which is incorrect because increases and decreases are calculated on different bases. Another mistake is to assume that markup must equal discount. Instead, it is necessary to set up the equation linking cost price, marked price, and selling price correctly. Final Answer: The retailer must mark up the goods by approximately 47.05 percent over the cost price in order to sell at cost after a 32 percent discount.

Question 13

Two students appeared for an examination. One of them scored 9 marks more than the other, and his marks were 56 percent of the sum of their marks. What are the marks obtained by the two students?

Accepted Answer

Introduction / Context: This question is similar to other two variable word problems that connect differences in marks with a percentage of their combined total. It reinforces algebraic thinking, where real life statements about marks are converted into equations that can be solved systematically. Given Data / Assumptions: Let marks of the first student be A. Let marks of the second student be B. A is 9 marks more than B, so A - B = 9. A is 56 percent of their total, so A = 56 percent of (A + B). Marks are assumed to be whole numbers. Concept / Approach: The percentage relation is converted into a linear equation involving A and B. Together with the simple difference equation A - B = 9, we obtain a system of two equations. Solving these equations gives the exact marks for both students. This method is typical in quantitative aptitude and helps avoid guesswork. Step-by-Step Solution: From the first statement: A - B = 9. From the second statement: A = 56/100 * (A + B) = 0.56 * (A + B). Write A = 0.56A + 0.56B. Bring terms together: A - 0.56A = 0.56B. This gives 0.44A = 0.56B. So A / B = 0.56 / 0.44 = 56 / 44 = 14 / 11. Thus A : B = 14 : 11. Difference in ratio parts = 14 - 11 = 3 corresponds to actual difference 9. So 3 parts = 9 marks, hence 1 part = 3 marks. Therefore A = 14 * 3 = 42 and B = 11 * 3 = 33. Verification / Alternative check: Check the conditions. Difference: 42 minus 33 equals 9, matching the first condition. Sum: 42 plus 33 equals 75. Now 56 percent of 75 is 0.56 * 75 = 42. Both statements are satisfied exactly, confirming that 42 and 33 are the correct marks. Why Other Options Are Wrong: For 40 and 31, the difference is 9 but 56 percent of 71 is 39.76, not 40. For 72 and 63, the difference is 9 but 56 percent of 135 is 75.6. For 68 and 59, 56 percent of 127 is 71.12. For 56 and 47, 56 percent of 103 is 57.68. None of these satisfy both the difference and percentage conditions. Only 42 and 33 fit both. Common Pitfalls: Students sometimes misread the phrase 56 percent of the sum and instead apply the percentage to the second student only. Others may try random substitution from options. Writing the equations clearly and solving through ratios is faster and more reliable in exam conditions. Final Answer: The two students obtained 42 and 33 marks respectively.

Question 14

A man wills 25 percent of his wealth to charity and leaves the rest to his family. What percentage of the amount willed to charity does the family receive?

Accepted Answer

Introduction / Context: This is a percentage comparison question where we compare one part of a whole with another part, not with the entire amount. The man divides his wealth between charity and family in a simple ratio, and we are asked to express the family share as a percentage of the charity share, not as a percentage of total wealth. Given Data / Assumptions: Total wealth = W (some positive amount). Charity receives 25 percent of W. Family receives the remaining 75 percent of W. We need family share expressed as a percentage of the charity share. Concept / Approach: When a question asks how much one quantity is as a percentage of another, we use the formula: required percent = (quantity one / quantity two) * 100. Here the quantity one is family wealth and quantity two is charity wealth. Substituting the parts of W gives a simple ratio that we convert into a percentage. Step-by-Step Solution: Charity share = 25 percent of W = 25/100 * W = W / 4. Family share = remaining wealth = W - W/4 = 3W/4. We must find family share as a percentage of charity share. So required percent = (family share / charity share) * 100. Substitute: (3W/4) / (W/4) * 100. The W cancels and the 1/4 cancels, giving 3 * 100. Thus required percentage = 300 percent. Verification / Alternative check: Take a simple numeric example. Let total wealth W = 100 units. Charity receives 25 units. Family receives 75 units. Now percentage of charity share represented by family share is (75 / 25) * 100 = 3 * 100 = 300 percent. This confirms the result is independent of the actual value of total wealth. Why Other Options Are Wrong: Option 200 would correspond to the family getting only double the charity share, which is not correct because 75 is three times 25. Option 33.3 mistakenly interprets charity as a percentage of family share. Option 25 or 75 relate to percentages of total wealth, not to the ratio between family and charity shares. The only correct comparison outcome is 300 percent. Common Pitfalls: A typical mistake is to answer 75 or 25 by confusing percentage of total wealth with percentage of the other share. Another pitfall is forgetting which quantity should be the base in the percentage formula. When the question says percentage of the wealth willed to charity, charity must be treated as the base quantity in the denominator of the fraction. Final Answer: The family receives an amount equal to 300 percent of the amount willed to charity.

Question 15

When a certain number is increased by 84, it becomes 107 percent of its original value. What is the original number?

Accepted Answer

Introduction / Context: This question asks you to reverse a percentage increase. Instead of being given the original number and asked for the increased value, we are given the increased value as a relationship and the absolute increase. The task is to recover the original number using algebra and percentage concepts. Given Data / Assumptions: Let the original number be N. The number is increased by 84. After the increase, the result equals 107 percent of N. We need to find N. Concept / Approach: The phrase becomes 107 percent of itself means new value equals 1.07 times the original number. We are also told that new value equals N plus 84. Equating these two expressions for the new value produces a linear equation in N. Solving that equation yields the original number. Step-by-Step Solution: Let the original number be N. After an increase of 84, the new value is N + 84. We are told that this new value equals 107 percent of N. So N + 84 = 107/100 * N = 1.07N. Rearrange the equation: N + 84 = 1.07N. Bring N terms together: 84 = 1.07N - N = 0.07N. Thus N = 84 / 0.07. Compute 84 / 0.07 = 1200. So the original number is 1200. Verification / Alternative check: Check by forward calculation. Start with 1200. Increase by 84 to get 1284. Now find 107 percent of 1200: 107/100 * 1200 = 1.07 * 1200 = 1284. Since the new value computed directly from 107 percent matches the value obtained by adding 84, the solution is verified. Why Other Options Are Wrong: For 600, an increase of 84 gives 684, but 107 percent of 600 is 642. For 900, 900 plus 84 is 984 while 107 percent of 900 is 963. For 1500, 1500 plus 84 is 1584 but 107 percent of 1500 is 1605. For 1050, adding 84 yields 1134, while 107 percent of 1050 is 1123.5. None of these pairs agree, so the only correct original number is 1200. Common Pitfalls: Some learners wrongly treat 84 as 7 percent of the original number and then guess the original, but here 84 is the absolute increase, not a direct percentage. Others mistakenly compute 107 percent of 84. Writing the algebraic equation carefully prevents confusion and leads to a clear solution. Final Answer: The original number before the increase was 1200.

Question 16

The cost price of an article is x rupees. It is marked up by 150 percent, and then sold for Rs 600 after giving a 40 percent discount on the marked price. What is the value of x (the cost price)?

Accepted Answer

Introduction / Context: This profit and loss question combines percentage markup with percentage discount. The article is first marked up by a certain percentage over cost price and then sold after applying a discount. We are given the final selling price and must work back to find the original cost price. Such layered percentage problems are standard in competitive exams. Given Data / Assumptions: Cost price = x rupees. Markup rate on cost price = 150 percent. Discount on marked price = 40 percent. Final selling price after discount = Rs 600. We must find x, the cost price. Concept / Approach: Markup of 150 percent means the marked price is cost price multiplied by 1 plus 1.5, that is 2.5 times the cost price. A discount of 40 percent means the selling price is 60 percent of the marked price. Therefore, selling price becomes 2.5 * x * 0.6. Setting this equal to 600 allows us to solve for x directly. Step-by-Step Solution: Marked price M = x * (1 + 150/100) = x * 2.5. Discount rate = 40 percent, so selling price S = M * (1 - 40/100) = M * 0.6. Substitute M: S = 2.5x * 0.6. Compute 2.5 * 0.6 = 1.5, so S = 1.5x. We are given S = 600, so 1.5x = 600. Therefore x = 600 / 1.5. Compute 600 / 1.5 = 400. Thus the cost price is Rs 400. Verification / Alternative check: Start with cost price Rs 400. Markup 150 percent means marked price M = 400 + 150 percent of 400 = 400 + 600 = 1000, or directly 2.5 * 400 = 1000. Discount of 40 percent on 1000 gives selling price S = 60 percent of 1000 = 600. This matches the given selling price, confirming that cost price is 400. Why Other Options Are Wrong: If cost price were 300, selling price would be 2.5 * 300 * 0.6 = 450, not 600. For 500, selling price would be 2.5 * 500 * 0.6 = 750. For 444 or 666 the computed selling prices do not equal 600 when the same percentage steps are applied. Only 400 satisfies the given final selling price exactly. Common Pitfalls: A common mistake is to subtract 150 and 40 directly or to treat discount and markup on the same base value. Another frequent error is to apply 40 percent discount on cost price instead of on marked price. Clearly identifying which percentage acts on which base is crucial for solving such problems correctly. Final Answer: The value of the cost price x is Rs 400.

Question 17

At a 10 percent discount, the selling price of an article is Rs 4500. If instead a discount of 27.5 percent is offered on the same marked price, what will be the new selling price?

Accepted Answer

Introduction / Context: This problem tests the skill of reversing a discount to find the marked price and then applying a different discount rate. Being comfortable with such two step percentage questions is essential for commerce related aptitude and reasoning tests, where discounts and marked prices are frequently combined. Given Data / Assumptions: Selling price with 10 percent discount = Rs 4500. Discount rate 1 = 10 percent. Discount rate 2 = 27.5 percent. Marked price remains unchanged between the two scenarios. We must find new selling price at 27.5 percent discount. Concept / Approach: If a discount of 10 percent is given, the selling price is 90 percent of marked price. So marked price equals selling price divided by 0.9. Once the marked price is known, apply the second discount of 27.5 percent, meaning the customer pays 72.5 percent of the marked price. Multiplying marked price by 0.725 gives the new selling price. Step-by-Step Solution: Let marked price be M. Selling price with 10 percent discount = M * (1 - 10/100) = 0.9M. We are told 0.9M = 4500. So M = 4500 / 0.9. Compute 4500 / 0.9 = 5000. Now apply 27.5 percent discount on this marked price. Customer pays 72.5 percent, so new selling price S = 0.725 * 5000. Compute 0.725 * 5000 = 3625. Thus the new selling price is Rs 3625. Verification / Alternative check: Check logically. With marked price 5000, at 10 percent discount the price is 4500, which agrees with the given data. When discount increases from 10 percent to 27.5 percent, clearly the new price must be lower than 4500. Among the options, 3625 is less than 4500 and matches the exact computation. Other options either do not fit the percentage calculations or form incorrect ratios. Why Other Options Are Wrong: For selling price 4000, the implied percent of marked price would be 4000 / 5000 * 100 = 80 percent, which corresponds to a 20 percent discount, not 27.5 percent. Values 3500, 3375, or 3125 correspond to even higher discounts such as 30 percent, 32.5 percent, and 37.5 percent. Only 3625 corresponds to exactly 72.5 percent of 5000, that is 27.5 percent discount. Common Pitfalls: One frequent mistake is to apply 17.5 percentage points directly on 4500 instead of returning to the marked price. Another error is to subtract 17.5 percent from 90 percent rather than recomputing the effective percentage paid. Always remember that discounts are calculated on the marked price, so you must adjust the base correctly before and after changing discount rates. Final Answer: The selling price of the article at a 27.5 percent discount will be Rs 3625.

Question 18

If the selling price of an item is Rs 9750 after a 25 percent discount, what was the marked price of the item before the discount was applied?

Accepted Answer

Introduction / Context: This question is another example of reversing a discount. The selling price after a known percentage discount is given, and the aim is to recover the marked price before the discount. Such problems appear frequently in profit and loss sections where understanding of base value and final value is tested. Given Data / Assumptions: Selling price after discount = Rs 9750. Discount rate = 25 percent. Marked price is unknown. No additional charges or hidden costs are involved. Concept / Approach: If a store gives a 25 percent discount, the customer pays 75 percent of the marked price. Therefore selling price equals marked price multiplied by 0.75. By dividing the known selling price by 0.75, we can find the original marked price. This is a direct application of the relationship between discount and selling price. Step-by-Step Solution: Let marked price be M. Discount = 25 percent, so selling price S = M * (1 - 25/100) = 0.75M. We are given S = 9750. Thus 0.75M = 9750. So M = 9750 / 0.75. Compute 9750 / 0.75 = 13000. Therefore the marked price was Rs 13000. Verification / Alternative check: Confirm by applying the discount forward. Take marked price 13000. A 25 percent discount means reduction of 25/100 * 13000 = 3250. Selling price becomes 13000 minus 3250 = 9750. This matches the given selling price exactly, confirming that the marked price is 13000. Why Other Options Are Wrong: If marked price were 12187, then 75 percent of this value would not equal 9750. For 14000, 25 percent discount would produce a selling price of 10500. For 13187 or 12500, the resulting selling prices also differ from 9750 when 25 percent discount is applied. Only 13000 satisfies the relationship S = 0.75M with S equal to 9750. Common Pitfalls: Students sometimes compute 25 percent of 9750 and add that amount, but discount percentage is always applied to marked price, not to selling price. Another confusion arises when learners subtract 25 rupees instead of 25 percent. It is important to convert the percentage into a decimal multiplier and identify the correct base value each time. Final Answer: The marked price of the item before the 25 percent discount was Rs 13000.

Question 19

If 40 is equal to 0.05 percent of a number x, what is the value of x?

Accepted Answer

Introduction / Context: This question focuses on converting a very small percentage into its decimal form and then using it to find the original number. It reinforces the relationship between percentages, fractions, and decimals, which is fundamental in all percentage based problems. Given Data / Assumptions: 0.05 percent of x equals 40. We need to find the value of x. All values are positive real numbers. Concept / Approach: A percentage can be written as a fraction out of 100. Thus 0.05 percent is 0.05 / 100. That can be converted to a decimal multiplier. Once we express 0.05 percent as a decimal, we can write the equation (0.05 / 100) * x = 40, and then solve for x by dividing 40 by this small factor. Step-by-Step Solution: We are told 0.05 percent of x = 40. Convert 0.05 percent to decimal form. 0.05 percent = 0.05 / 100. Compute 0.05 / 100 = 0.0005. So the equation becomes 0.0005 * x = 40. To find x, divide both sides by 0.0005. Thus x = 40 / 0.0005. Compute 40 / 0.0005 = 40 * (1 / 0.0005) = 40 * 2000 = 80000. Therefore x = 80000. Verification / Alternative check: Check by forward calculation. Compute 0.05 percent of 80000. First find 1 percent of 80000 which is 800. Then 0.05 percent is 0.05 times 800 which equals 40. This matches the given value. Another quick way is to note that 0.05 percent is one twentieth of 1 percent, and the numbers remain consistent. Why Other Options Are Wrong: If x were 8000, then 0.05 percent of 8000 would be 4. For 800, 0.05 percent is 0.4. For 80, it is 0.04. For 40000, 0.05 percent equals 20. None of these equal 40. Only x equal to 80000 yields 40 as 0.05 percent of x. Common Pitfalls: A common mistake is to treat 0.05 percent as 0.05 or 5 percent instead of 0.0005. Misplacing decimal points when dealing with very small percentages can lead to large errors in the final answer. Carefully converting percent to decimal and checking the order of magnitude helps prevent such errors. Final Answer: The value of x for which 40 is 0.05 percent of x is 80000.

Question 20

Two iron balls weigh 3.5 kg and 7.5 kg respectively. What is the weight of the first ball expressed as a percentage of the weight of the second ball?

Accepted Answer

Introduction / Context: This question tests your ability to express one quantity as a percentage of another. Such problems involving weights, lengths, or amounts are common in everyday life and in examinations, and the method is always the same: use the basic percentage formula with the correct base value. Given Data / Assumptions: Weight of first iron ball = 3.5 kg. Weight of second iron ball = 7.5 kg. We must express the weight of the first ball as a percentage of the second ball. Concept / Approach: To find A as a percentage of B, we use the formula percent = (A / B) * 100. Here A is the weight of the first ball, and B is the weight of the second ball. After computing the percentage, we may convert it into a mixed fraction if needed, as in 46 2/3 percent. Step-by-Step Solution: Let A = 3.5 kg and B = 7.5 kg. Required percent = (A / B) * 100. Compute A / B = 3.5 / 7.5. Write 3.5 as 35/10 and 7.5 as 75/10, so the ratio becomes (35/10) / (75/10) = 35 / 75. Simplify 35 / 75 by dividing numerator and denominator by 5 to get 7 / 15. Now multiply by 100: (7 / 15) * 100. Compute 700 / 15 = 46.6666 repeating. So percentage = 46.6666 repeating percent, which is 46 2/3 percent. Verification / Alternative check: We can check by converting 46 2/3 percent back to a decimal. Write 46 2/3 as 46.6666 repeating and divide by 100 to get 0.466666 repeating. Multiply 7.5 by 0.466666 repeating and we get approximately 3.5, confirming that the calculation is consistent. Why Other Options Are Wrong: Options 35 percent and 45 percent underestimate the ratio since 3.5 is nearly half of 7.5. Fifty percent would correspond to exactly half, but 3.5 is slightly less than half of 7.5. The value 46 1/3 percent is slightly lower than the actual ratio, while 46 2/3 percent is the exact ratio obtained from 700 divided by 15. Common Pitfalls: Learners can mistakenly compute 7.5 as a percentage of 3.5 instead of the other way around, or they may round intermediate steps too early and get a slightly off mixed fraction. Another possible confusion is forgetting to multiply by 100 when converting a fraction to percentage. Staying systematic with the formula avoids these issues. Final Answer: The weight of the first iron ball is 46 2/3 percent of the weight of the second ball.

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