Question 1

Thirty eight percent of a certain number is equal to 133. Using this information, determine the value of the original number.

Accepted Answer

Introduction / Context: This is another reverse percentage problem. Instead of finding a percentage of a known quantity, you are given the percentage result and asked to recover the original number. Questions like this are common in topics such as profit and loss, taxation, and data analysis. Given Data / Assumptions: • 38 percent of a number N equals 133. • Formally, (38 / 100) * N = 133. • Your goal is to solve for N. Concept / Approach: The general relation is p percent of N = V, which means (p / 100) * N = V. Rearranging gives N = (V * 100) / p. Here, p = 38 and V = 133. After substituting, we simplify the fraction to find the exact original number. Step-by-Step Solution: Step 1: Write the equation: (38 / 100) * N = 133. Step 2: Solve for N: N = (133 * 100) / 38. Step 3: Compute 133 * 100 = 13,300. Step 4: Divide 13,300 by 38. Since 38 * 350 = 13,300, we have N = 350. Verification / Alternative check: Check the answer by recalculating 38 percent of 350: (38 / 100) * 350 = 0.38 * 350 = 133. This matches the given value in the problem, so N = 350 is confirmed as correct. Why Other Options Are Wrong: • 340 gives 38 percent equal to 129.2, not 133. • 360 gives 38 percent equal to 136.8, which is larger than 133. • 355 gives 38 percent equal to 134.9, which does not match 133 exactly. Common Pitfalls: Learners sometimes mistakenly compute 38 percent of 133 instead of using 133 as the percentage result. Others may round the division incorrectly. Writing the algebraic equation clearly and checking with multiplication ensures accuracy. Final Answer: The original number is 350.

Question 2

In a class test, a student scores 22 marks out of a maximum of 25 marks. Express the student's score as a percentage of the total marks.

Accepted Answer

Introduction / Context: This question asks you to convert a simple fraction of marks obtained into a percentage score. Such conversions are extremely common in exams, grading systems, and everyday performance evaluations, so it is important to be comfortable with them. Given Data / Assumptions: • Marks obtained by the student = 22. • Maximum possible marks = 25. • We need to express 22 out of 25 as a percentage. Concept / Approach: Percentage is defined as (part / whole) * 100. Here, the part is the marks obtained and the whole is the maximum marks. Therefore, percentage score = (marks obtained / maximum marks) * 100. We simply substitute the given numbers and simplify the fraction. Step-by-Step Solution: Step 1: Use the percentage formula: percentage = (22 / 25) * 100. Step 2: Simplify the fraction 22 / 25. Step 3: Multiply numerator and denominator by 4 to get (88 / 100). Step 4: Now (88 / 100) * 100 = 88. Step 5: Therefore the student's score is 88 percent. Verification / Alternative check: We know 25 percent corresponds to 1 out of 4. Here 22 out of 25 is slightly less than full marks. Since 22 is 3 marks below 25, the score is 3/25 below 100 percent. The fraction 3/25 equals 12 percent, so the score is 100 − 12 = 88 percent, which confirms our calculation. Why Other Options Are Wrong: • 90 percent would correspond to 22.5 marks out of 25, not 22 marks. • 80 percent would correspond to 20 marks out of 25. • 75 percent would correspond to 18.75 marks, which is quite far from 22. Common Pitfalls: Mistakes often happen when handling the fraction or when multiplying by 100. Some learners may incorrectly divide 25 by 22 instead of 22 by 25. Always keep the formula in mind: percentage = (obtained / total) * 100, with the total in the denominator. Final Answer: The student’s score is 88% of the total marks.

Question 3

The weights of two friends A and B are in the ratio 7 : 8. If the weight of A increases by 10 percent and the total combined weight of A and B increases by 20 percent, what is the percentage increase in the weight of B?

Accepted Answer

Introduction / Context: This problem links ratio and percentage change. It gives the initial ratio of the weights of two people, then describes different percentage increases for their weights such that the total weight rises by a known percentage. You have to determine the unknown percentage change for one person, which requires setting up and solving an equation. Given Data / Assumptions: • Initial weight ratio A : B = 7 : 8. • Let initial weights be 7x for A and 8x for B. • Weight of A increases by 10 percent. • Combined weight of A and B increases by 20 percent overall. • We need the percentage increase in the weight of B. Concept / Approach: Use variables to represent the initial weights according to the ratio. Apply the 10 percent increase for A and an unknown percentage p for B. Then equate the new total weight to 120 percent (1.2 times) of the original total weight. Solving the resulting equation in p will give the required percentage increase for B. Step-by-Step Solution: Step 1: Let initial weights be A = 7x and B = 8x. Step 2: Initial total weight = 7x + 8x = 15x. Step 3: After 10 percent increase, A becomes 7x * 1.10 = 7.7x. Step 4: Let B increase by p percent, so new B weight = 8x * (1 + p). Step 5: New total weight must be 120 percent of original total, so new total = 1.20 * 15x = 18x. Step 6: Form the equation: 7.7x + 8x * (1 + p) = 18x. Step 7: Simplify: 7.7x + 8x + 8x * p = 18x ⇒ 15.7x + 8x * p = 18x. Step 8: Subtract 15.7x: 8x * p = 2.3x ⇒ p = 2.3 / 8 = 0.2875. Step 9: Convert to percentage: p = 0.2875 * 100 = 28.75 percent. Verification / Alternative check: We can plug back the percentage to verify. New B weight = 8x * 1.2875 = 10.3x. New total = 7.7x + 10.3x = 18x, which equals 120 percent of the original 15x. Hence the calculation is consistent. Why Other Options Are Wrong: • 11.55, 32.85, and 24.25 do not satisfy the equation when substituted as percentage increases for B and give incorrect total weights. Common Pitfalls: Many learners forget to apply the total increase to the combined weight, or they incorrectly treat the ratio as fixed after the percentage changes. It is important to handle each percentage change separately and then apply the total change condition systematically through algebra. Final Answer: The weight of B increases by 28.75 percent.

Question 4

If a person walks at 7/6 of the usual speed, then the person reaches the office 15 minutes earlier than usual. What is the usual time in minutes taken to reach the office at the original speed?

Accepted Answer

Introduction / Context: This problem combines ratios with the inverse relationship between speed and time for a fixed distance. When speed increases, time decreases in inverse proportion. The question gives the factor by which speed changes and the change in time, and asks you to find the original time taken for the journey. Given Data / Assumptions: • Usual time taken to reach the office = T minutes (unknown). • New speed = (7 / 6) times the usual speed. • For a fixed distance, time is inversely proportional to speed. • With higher speed, the person reaches 15 minutes earlier than usual. Concept / Approach: Since distance is fixed, Speed * Time is constant. If speed becomes (7 / 6) times, then time becomes (6 / 7) times the original. Let original time be T, then new time is (6 / 7)T. The difference between original and new times equals 15 minutes. This leads to a simple linear equation in T. Step-by-Step Solution: Step 1: Let original time = T minutes. Step 2: New speed = (7 / 6) * original speed, so new time = (6 / 7) * T. Step 3: The person reaches 15 minutes earlier, so time saved = T − (6 / 7)T = 15. Step 4: Simplify the difference: T − (6 / 7)T = (1 / 7)T. Step 5: Set (1 / 7)T = 15. Step 6: Solve for T: T = 15 * 7 = 105 minutes. Verification / Alternative check: Compute the new time using T = 105 minutes. New time = (6 / 7) * 105 = 90 minutes. The time saved is 105 − 90 = 15 minutes, which matches the information given, so the solution is verified. Why Other Options Are Wrong: • 60, 75, and 90 minutes would give smaller time differences than 15 minutes when adjusted by the factor 6 / 7 and therefore do not satisfy the condition. Common Pitfalls: A frequent error is to assume the time changes by the same factor as the speed instead of the inverse factor. Another mistake is to set up the equation with reversed ratios. Always remember that for the same distance, speed and time are inversely proportional, so if speed is multiplied by a factor, time is divided by that factor. Final Answer: The usual time taken to reach the office is 105 minutes.

Question 5

After a discount of 23 percent on its marked price, an article is sold for Rs. 1,848. What is the marked price of the article before the discount is applied?

Accepted Answer

Introduction / Context: This question is a typical discount and marked price problem found in profit and loss topics. It gives the final selling price after a known percentage discount and asks you to find the original marked price before the discount was applied. Given Data / Assumptions: • Selling price after discount = Rs. 1,848. • Discount rate on marked price = 23 percent. • Marked price is the original price before discount. • No additional taxes or charges are mentioned. Concept / Approach: A discount of 23 percent means the customer pays 77 percent of the marked price. If M is the marked price, then selling price S equals (100 − discount percent) / 100 * M, that is S = (77 / 100) * M. We know S and need to solve for M by rearranging the equation to M = S * 100 / 77. Step-by-Step Solution: Step 1: Let the marked price be M. Step 2: After a 23 percent discount, selling price S = (100 − 23) percent of M = 77 percent of M. Step 3: Write S in equation form: S = (77 / 100) * M. Step 4: Substitute S = 1,848: 1,848 = (77 / 100) * M. Step 5: Rearrange to find M: M = (1,848 * 100) / 77. Step 6: Compute 1,848 / 77 = 24, so M = 24 * 100 = 2,400. Verification / Alternative check: Check by applying the discount to the marked price: 23 percent of 2,400 is (23 / 100) * 2,400 = 552. Subtracting this from 2,400 gives 2,400 − 552 = 1,848, which matches the given selling price. This confirms that the marked price is correct. Why Other Options Are Wrong: • 2,150, 2,275, and 2,350, when reduced by 23 percent, do not produce a selling price of 1,848. Their corresponding discounted prices are all different from 1,848. Common Pitfalls: Many students compute the discount on the selling price instead of the marked price or they subtract 23 percent incorrectly. Always remember that the given discount rate is applied to the marked price. Using the equation S = (100 − discount percent) / 100 * M and then solving for M is the safest approach. Final Answer: The marked price of the article is Rs. 2,400.

Question 6

Two successive discounts of 15 percent and 35 percent are given on the same marked price of an article. What is the single equivalent net discount percentage that would give the same final price?

Accepted Answer

Introduction / Context: This problem deals with successive discounts, which do not simply add up. When two discounts are applied one after another, the net effect is less than the sum of the individual discounts. The question asks for the single equivalent discount that produces the same final price as the two given discounts applied successively. Given Data / Assumptions: • First discount = 15 percent. • Second discount = 35 percent. • Both are applied successively on the same marked price. • We are required to find one net discount that yields the same final result. Concept / Approach: If the marked price is M, after a discount of d1 percent, the price becomes M * (1 − d1 / 100). After a second discount d2 percent on this reduced price, the final price is M * (1 − d1 / 100) * (1 − d2 / 100). The equivalent single discount d satisfies final price = M * (1 − d / 100). We compute the combined factor and then convert it back to a net discount percentage. Step-by-Step Solution: Step 1: After 15 percent discount, price factor = 1 − 15 / 100 = 0.85. Step 2: After 35 percent discount on the reduced price, factor = 1 − 35 / 100 = 0.65. Step 3: Final price factor after both discounts = 0.85 * 0.65. Step 4: Compute 0.85 * 0.65 = 0.5525. Step 5: If d is the equivalent discount, then final factor = 1 − d / 100 = 0.5525. Step 6: So d / 100 = 1 − 0.5525 = 0.4475, and d = 44.75 percent. Verification / Alternative check: To check, assume a marked price of 100 units. Successive discounts give: first discount leaves 85 units, then second leaves 85 * 0.65 = 55.25 units. This final price equals 55.25 percent of 100, meaning the total discount is 100 − 55.25 = 44.75 percent, matching our calculation. Why Other Options Are Wrong: • 51.25, 55.25, and 56.25 all correspond to larger net discounts that would produce a lower final price than the correct combined effect of 15 percent and 35 percent. Common Pitfalls: A common mistake is to simply add the two discounts, 15 percent + 35 percent = 50 percent, which is not correct because the second discount is applied to an already reduced price. Always multiply the remaining fractions and then convert back to a single discount to avoid errors. Final Answer: The net equivalent discount is 44.75 percent.

Question 7

Due to an increase of 33.33 percent in the price of apples, a customer can now buy 4 fewer apples for Rs. 16 than before. What was the original price of one apple in paise before the price increase?

Accepted Answer

Introduction / Context: This question combines unit price, percentage increase, and change in quantity within a fixed budget. The customer spends the same total amount before and after a price change but can buy fewer apples after the price increases. You need to translate the verbal description into algebra and solve for the original price. Given Data / Assumptions: • Original price of one apple = p rupees (unknown). • Price increases by 33.33 percent, which is effectively one third. • New price per apple = p * (4 / 3). • Customer spends Rs. 16 both before and after the price increase. • After the increase, the customer can buy 4 apples fewer. Concept / Approach: The number of apples that can be bought is total money divided by price per unit. Using the original and increased prices, we can express the old and new quantities purchased and then use the fact that the difference between them is 4. This leads to an equation in p that we can solve. Finally we convert the answer into paise by multiplying rupees by 100. Step-by-Step Solution: Step 1: Let original price = p rupees per apple. Step 2: After 33.33 percent (one third) increase, new price = (4 / 3) * p. Step 3: Number of apples originally bought = 16 / p. Step 4: Number of apples after the increase = 16 / ((4 / 3) * p) = 16 * (3 / 4p) = 12 / p. Step 5: The customer now buys 4 fewer apples, so (16 / p) − (12 / p) = 4. Step 6: Simplify: (4 / p) = 4, hence p = 1 rupee. Step 7: Convert the original price into paise: 1 rupee = 100 paise. Verification / Alternative check: Original price = Rs. 1; with Rs. 16 the customer could buy 16 apples. New price after one third increase = Rs. 4 / 3 per apple. Now the number of apples affordable is 16 * (3 / 4) = 12 apples. The difference is 16 − 12 = 4 apples fewer, exactly as stated in the question, confirming that the computation is correct. Why Other Options Are Wrong: • 125, 150, and 400/3 paise correspond to very different original prices and do not produce a difference of 4 apples when the price increases by 33.33 percent. Common Pitfalls: A common mistake is to treat 33.33 percent as exactly 1/3 without adjusting carefully or to set up the quantity difference equation incorrectly. Another error is forgetting to convert rupees to paise at the end. Always keep track of units and carefully form the relationship between old and new quantities. Final Answer: The original price of one apple was 100 paise (that is, Rs. 1).

Question 8

A dealer sells a machine whose marked price is Rs. 3,840 at a discount of 20 percent on the marked price. What is the selling price of the machine after this discount?

Accepted Answer

Introduction / Context: This is a direct application of percentage discount on a marked price. Such questions are basic but very common in the profit and loss section of aptitude tests and also occur frequently in everyday shopping situations. Given Data / Assumptions: • Marked price of the machine = Rs. 3,840. • Discount rate = 20 percent on the marked price. • Selling price equals marked price minus discount amount. Concept / Approach: A discount of 20 percent means the buyer pays 80 percent of the marked price. Instead of first calculating the discount and then subtracting, it is often quicker to directly compute 80 percent of the marked price. Mathematically, selling price S = (100 − discount percent) / 100 * marked price. Step-by-Step Solution: Step 1: Discount percent = 20 percent, so payable percent = 100 − 20 = 80 percent. Step 2: Selling price S = (80 / 100) * 3,840. Step 3: Simplify 80 / 100 to 4 / 5. Step 4: Compute S = (4 / 5) * 3,840. Step 5: First find 3,840 / 5 = 768, then multiply by 4 to get 3,072. Step 6: Thus, the selling price is Rs. 3,072. Verification / Alternative check: We can also compute the discount directly. Discount amount = 20 percent of 3,840 = (20 / 100) * 3,840 = 768. Subtract this from the marked price: 3,840 − 768 = 3,072. This matches the previous calculation and confirms the answer. Why Other Options Are Wrong: • 3,500 implies a much smaller discount than 20 percent. • 4,608 would actually be greater than the marked price, which is not possible after giving a discount. • 3,240 corresponds to a different discount percentage, not exactly 20 percent. Common Pitfalls: Some learners mistakenly add the discount to the marked price or confuse discount percentage with the percentage to be paid. Always remember that a discount reduces the price and that the selling price is the remaining fraction of the marked price after the discount is subtracted. Final Answer: The selling price of the machine is Rs. 3,072.

Question 9

After deducting 12 percent from a certain sum and then deducting 25 percent from the remainder, Rs. 2,508 is left. What was the initial sum before any deductions were made?

Accepted Answer

Introduction / Context: This question involves successive percentage deductions from an unknown original sum. It tests your understanding of how consecutive percentage changes combine and how to reverse them to recover the original amount from the final remaining value. Given Data / Assumptions: • Original sum = S (unknown). • First deduction = 12 percent of S. • Second deduction = 25 percent of the remaining amount after the first deduction. • Final amount left after both deductions = Rs. 2,508. Concept / Approach: A deduction of 12 percent means 88 percent of the amount is left after the first step. A deduction of 25 percent from the remainder means only 75 percent of that remainder is left. Therefore, the final amount is S * 0.88 * 0.75. We equate this product to 2,508 and solve for S using basic algebra. Step-by-Step Solution: Step 1: After the first deduction of 12 percent, amount left = S * (100 − 12) / 100 = S * 0.88. Step 2: From this remainder, 25 percent is deducted, leaving 75 percent of it. Step 3: Amount left after second deduction = S * 0.88 * 0.75. Step 4: Compute 0.88 * 0.75 = 0.66. Step 5: So final amount = S * 0.66 = 2,508. Step 6: Solve for S: S = 2,508 / 0.66. Step 7: Compute 2,508 / 0.66 = 3,800. Verification / Alternative check: Check stepwise: starting from 3,800, first deduct 12 percent. Twelve percent of 3,800 is (12 / 100) * 3,800 = 456. Remainder = 3,800 − 456 = 3,344. Now deduct 25 percent of 3,344, which is (25 / 100) * 3,344 = 836. Final remainder = 3,344 − 836 = 2,508, exactly matching the given amount. This confirms S = 3,800 is correct. Why Other Options Are Wrong: • 3,500, 3,450, and 3,970, when processed through the same successive deductions, do not result in 2,508; their final values are different. Common Pitfalls: One common mistake is to add the percentages 12 percent and 25 percent and treat the total deduction as 37 percent once, which is incorrect because the second deduction is taken on a reduced base. Another error is reversing the operations incorrectly when solving for the original sum. Using the product of the remaining percentages (0.88 and 0.75) and then solving the simple equation is the cleanest method. Final Answer: The initial sum before any deductions were made is Rs. 3,800.

Question 10

In a basic percentage problem, three numbers x, y and z are such that x is 30% of z and y is 40% of z. If x is equal to p% of y, what is the value of p?

Accepted Answer

Introduction / Context: This question tests the concept of percentages as ratios between related quantities. It focuses on how one number can be expressed as a percentage of another when both are defined in terms of a third base number. Such questions are very common in quantitative aptitude tests and competitive exams, because they check conceptual clarity rather than heavy calculation. Given Data / Assumptions: - x is 30 percent of z. - y is 40 percent of z. - We need to find a value p such that x is p percent of y. - All numbers are real and positive in the usual aptitude context. Concept / Approach: A percentage is simply a fraction multiplied by 100. If one quantity is a certain percent of another, we can write it as (percentage / 100) times the reference quantity. Here, x and y are both written as percentages of z, so we first express them in terms of z. Then we compare x and y to find what percent x is of y by using the ratio x / y and converting it into a percentage. Step-by-Step Solution: Let z be any convenient base value (for example, z itself as a symbol). Given: x is 30% of z, so x = (30 / 100) * z = 0.30z. Similarly, y is 40% of z, so y = (40 / 100) * z = 0.40z. We are told that x is p% of y. This means x = (p / 100) * y. Substitute x and y in terms of z: 0.30z = (p / 100) * 0.40z. Cancel z from both sides: 0.30 = (p / 100) * 0.40. Rearrange for p: p / 100 = 0.30 / 0.40 = 3 / 4. So p = (3 / 4) * 100 = 75. Verification / Alternative check: Take an easy numerical value for z, such as z = 100. Then x = 30% of 100 = 30, and y = 40% of 100 = 40. Now find what percent 30 is of 40. The fraction is 30 / 40 = 3 / 4. Converting to a percentage, (3 / 4) * 100 = 75%. This confirms our algebraic result and shows that p must be 75. Why Other Options Are Wrong: 45: This would imply x is less than half of y, but 30 compared with 40 is more than half, so 45% is too small. 55: 55% of 40 is 22, which is less than 30, so this does not match x. 65: 65% of 40 is 26, still below 30, so it is also incorrect. 60: 60% of 40 is 24, which again does not equal x = 30. Common Pitfalls: A common mistake is to subtract percentages directly, for example thinking 40% minus 30% is 10%, and somehow linking that to p. Another error is to confuse the base quantity, using z instead of y when computing p. Always remember that when we say “x is p percent of y”, the base is y, not z. It is also important not to mix fraction and percent without converting correctly. Final Answer: Therefore, the required percentage p such that x is p percent of y is 75 percent.

Question 11

In a hostel, the rent per room is increased by 20% and at the same time the number of rooms is also increased by 20%. If the hostel is always full, what is the percentage change in the total daily collection at the cash counter?

Accepted Answer

Introduction / Context: This problem checks understanding of percentage change when two related quantities change at the same time. Here, the total collection in a hostel depends on both the rent per room and the number of rooms occupied. When both factors are changed, we must multiply the corresponding change factors instead of simply adding percentages. This type of question is standard in percentage and profit related topics in aptitude exams. Given Data / Assumptions: - Rent per room increases by 20 percent. - Number of rooms increases by 20 percent. - The hostel is assumed to be fully occupied both before and after the change. - We are asked to find the net percentage change in total collection at the cash counter. Concept / Approach: Total collection for one day is given by: Total collection = Number of rooms * Rent per room. When both the rent and the number of rooms increase by certain percentages, we convert each percentage change into a multiplicative factor. An increase of 20% corresponds to a factor of 1.20. The new total collection is obtained by multiplying both factors with the original total. The overall percentage change is then derived from the ratio of new total collection to old total collection. Step-by-Step Solution: Let the original number of rooms be R. Let the original rent per room be P. Original total collection = R * P. After a 20% increase in rent, new rent per room = P * 1.20. After a 20% increase in rooms, new number of rooms = R * 1.20. New total collection = (R * 1.20) * (P * 1.20) = R * P * 1.20 * 1.20. So, new total collection = R * P * 1.44. Factor of change = New total / Old total = (R * P * 1.44) / (R * P) = 1.44. Net percentage increase = (1.44 - 1) * 100% = 0.44 * 100% = 44%. Verification / Alternative check: We can choose simple numbers to verify. Suppose there were originally 100 rooms at a rent of Rs. 100 each. Then original collection is 100 * 100 = Rs. 10,000. After a 20% increase, rooms become 120 and rent becomes Rs. 120. New collection is 120 * 120 = Rs. 14,400. The increase is 14,400 - 10,000 = Rs. 4,400, which is 44% of 10,000. This confirms that the correct net increase is 44%. Why Other Options Are Wrong: 30% and 40%: These values might come from simply adding or partially adding the two 20% increases, but adding is not correct when both factors multiply the total. 48%: This could result from an incorrect assumption or arithmetic error when dealing with the factors. 36%: This value does not correspond to the correct multiplicative effect of 1.20 * 1.20. Common Pitfalls: Many learners add percentages directly, for example 20% + 20% = 40%, and think the answer is 40%. This is wrong because total collection depends on the product of two quantities, and both are changing. When two changes occur simultaneously and independently, we must convert each change into a factor and then multiply those factors. Forgetting this rule leads to wrong answers in many percentage change problems. Final Answer: The total collection at the hostel cash counter increases by 44 percent.

Question 12

The price of an article is first decreased by 10%. By what percentage must the new reduced price be increased so that the price is restored to its original value?

Accepted Answer

Introduction / Context: This problem is a classic example of successive percentage changes where a decrease followed by an increase does not cancel out using the same percentage. It tests understanding of how percentage changes are applied on different bases. Many candidates wrongly assume that a 10% increase will undo a 10% decrease, which is not correct. Given Data / Assumptions: - The original price of an article is some amount, say P. - The price is decreased by 10 percent, giving a new reduced price. - We must find the percentage by which this reduced price should be increased to get back to P. Concept / Approach: A percentage decrease or increase is always calculated on the current value, not on the original value, unless explicitly stated. When the price is decreased by 10%, the new price is 90% of the original. To return to the original price, we need to increase this reduced price by a certain percentage. This required percentage can be found using ratios or by setting up a simple equation, then converting the ratio back into a percentage. Step-by-Step Solution: Let the original price be P. After a 10% decrease, the new price is P * (1 - 10 / 100) = P * 0.90. Call this new price P_new = 0.90P. We now increase P_new by some unknown percentage r to get back to P. So, P = P_new * (1 + r / 100) = 0.90P * (1 + r / 100). Divide both sides by P: 1 = 0.90 * (1 + r / 100). Therefore, 1 / 0.90 = 1 + r / 100. Compute 1 / 0.90 = 10 / 9. So, 10 / 9 = 1 + r / 100. Hence, r / 100 = 10 / 9 - 1 = 1 / 9. Thus, r = 100 * (1 / 9) = 100 / 9 = 11 1/9 percent. Verification / Alternative check: Take a simple number for the original price, such as P = 100. After 10% decrease, the new price becomes 90. To know the increase required to go from 90 back to 100, compute the difference 100 - 90 = 10. Now express this as a percentage of 90: (10 / 90) * 100 = 100 / 9 = 11 1/9 percent. This numerical approach confirms the algebraic result. Why Other Options Are Wrong: 9 1/11%: This is a common distractor but does not satisfy the equation needed to return to the original price. 10%: This assumes that a 10% increase cancels a 10% decrease, which is incorrect because the bases are different. 11%: Close to the true answer but slightly less; it would not fully restore the original price. 12%: This overshoots and would make the price higher than the original value. Common Pitfalls: The main mistake is thinking that percentage increases and decreases are symmetric. Decreasing by 10% and then increasing by 10% does not bring you back to the starting point. Another error is forgetting that the second percentage is applied on the reduced amount, not on the original. Learning this idea is essential for dealing with successive discount, profit, and loss questions. Final Answer: To restore the original price after a 10% decrease, the new price must be increased by 11 1/9 percent.

Question 13

An item is sold with a 25% discount on its marked price, and an additional promo code gives 4% cashback on the amount paid after discount. What is the effective overall discount received by the customer?

Accepted Answer

Introduction / Context: This question combines the ideas of discount and cashback, both expressed as percentages. It tests whether the learner can correctly interpret how successive reductions and returns are computed relative to the marked price. Many shoppers meet similar offers in real life, so understanding the effective discount helps in practical decision making as well as examinations. Given Data / Assumptions: - An item has a marked price, say M rupees. - A flat discount of 25 percent is first applied to the marked price. - On the reduced price (the bill amount), a cashback of 4 percent is given to the customer. - We are asked to find the effective discount as a single percentage on the original marked price M. Concept / Approach: The effective discount is calculated by comparing the net amount actually paid by the customer (after discount and cashback) with the original marked price. The 25 percent discount clearly reduces the bill, while the 4 percent cashback returns some money from the amount paid after discount. We treat the cashback as a further reduction to the net cost to the customer, but it is computed as a percentage of the already discounted price, not of the original marked price. Step-by-Step Solution: Let the marked price be M. First discount is 25 percent, so price after discount = M * (1 - 25 / 100) = M * 0.75. This discounted price is the basis for cashback. Cashback is 4 percent of 0.75M, so cashback amount = 0.04 * 0.75M = 0.03M. Net amount actually paid by the customer = discounted price - cashback. So net payment = 0.75M - 0.03M = 0.72M. This means the customer finally pays 72 percent of the marked price. Effective discount = (Marked price - Net payment) / Marked price * 100. Effective discount = (M - 0.72M) / M * 100 = 0.28 * 100 = 28 percent. Verification / Alternative check: Choose a simple value for M, for example M = 1000 rupees. After 25 percent discount, the price becomes 750 rupees. Cashback is 4 percent of 750 = 30 rupees. So the customer effectively pays 750 - 30 = 720 rupees. The effective discount is 1000 - 720 = 280 rupees. As a percentage, 280 / 1000 * 100 = 28 percent. This numerical check confirms our algebraic calculation. Why Other Options Are Wrong: 29.12 percent and 29 percent: These values arise if one confuses successive percentage formulae or incorrectly treats cashback as an extra discount applied directly on the marked price. 5 percent: This ignores the first large discount and wrongly considers only the cashback effect. 25 percent: This takes into account only the first discount and ignores cashback, which provides additional savings. Common Pitfalls: Learners sometimes add 25 and 4 directly to get 29 percent, which is misleading because the 4 percent cashback is not on the original price but on the reduced amount. Another mistake is to think cashback does not change effective discount. In reality, any money returned to the customer reduces the net amount paid and therefore increases the effective discount. Always specify clearly what base each percentage is applied to and then compare the final net payment with the original marked price. Final Answer: The effective overall discount received by the customer is 28 percent of the marked price.

Question 14

At what percentage above the cost price must an article be marked so that, after allowing a customer a discount of 5%, the shopkeeper still makes a profit of 33%?

Accepted Answer

Introduction / Context: This question involves finding the required marked price when both profit and discount percentages are given. It is a common type in profit and loss problems where the shopkeeper wants a certain profit even after giving a discount to attract customers. The key idea is to link cost price, marked price and selling price through percentages. Given Data / Assumptions: - Let the cost price (CP) of the article be some amount, say 100 units for convenience. - The shopkeeper wants a profit of 33 percent on the cost price. - A discount of 5 percent is allowed on the marked price (MP). - We must find how many percent above CP the article should be marked (that is, MP relative to CP). Concept / Approach: The profit percentage is calculated on the cost price, while the discount percentage is calculated on the marked price. The selling price is the link between them. So the steps are: compute the desired selling price using the profit requirement, then relate selling price to marked price using the discount, and finally express marked price as a percentage of cost price. Using CP = 100 makes calculations simpler and more intuitive. Step-by-Step Solution: Assume CP = 100. Desired profit = 33% of CP, so desired selling price SP = 100 + 33 = 133. Let the marked price be MP. A 5% discount is allowed, so SP = MP * (1 - 5 / 100) = MP * 0.95. We know SP should be 133, so MP * 0.95 = 133. Therefore, MP = 133 / 0.95. Compute MP = 140. So MP as a percentage of CP is (140 / 100) * 100% = 140% of cost price. Mark up above cost price = 140% - 100% = 40%. Verification / Alternative check: Using CP = 100 and the computed MP = 140, let us check: discount 5% on 140 gives SP = 140 * 0.95 = 133. Profit over CP = 133 - 100 = 33, which is 33% of 100. This confirms that our marked price indeed allows a 33 percent profit after giving 5 percent discount. Why Other Options Are Wrong: 45% and 47%: These would give a marked price higher than needed, and the resulting profit percentage after discount would be more than 33%. 35% and 30%: These mark ups are too low, resulting in a selling price that fails to provide the desired 33% profit on cost price. Common Pitfalls: A common mistake is to directly add or subtract percentages without checking which base each percentage refers to. Students may also treat profit and discount as if both directly change cost price, which is incorrect. Always remember that profit is on cost price and discount is on marked price. Using assumed values like CP = 100 makes the relationships clear and prevents arithmetic confusion. Final Answer: The article must be marked at 40 percent above its cost price to earn a 33 percent profit after allowing a 5 percent discount.

Question 15

When a liquid freezes into its solid state, its volume increases by 4%. By what percentage does the volume decrease when this solid melts back into the liquid state?

Accepted Answer

Introduction / Context: This question checks the understanding of reverse percentage change. An increase in one direction does not mean the same percentage decrease will bring you back to the original value. Here, the physical context is a liquid expanding slightly when it freezes and then contracting when it melts. The mathematics is the same as the standard successive percentage problems in aptitude tests. Given Data / Assumptions: - When the liquid freezes, its volume increases by 4 percent. - When the solid melts back to liquid, the volume decreases from the larger value back to the original value. - We are asked for the percentage decrease based on the larger (solid) volume. Concept / Approach: If a quantity increases by a certain percentage, say a%, the new value is original * (1 + a / 100). To reverse the change, we need to find a percentage decrease b% such that new value * (1 - b / 100) returns to the original. In general, these percentages a and b are not equal because the bases are different. The same idea appears in price rise and reduction problems. Step-by-Step Solution: Let the original volume of the liquid be V units. After freezing, there is a 4% increase, so solid volume = V * (1 + 4 / 100) = 1.04V. Now, when the solid melts, its volume goes from 1.04V back to V. Let the required percentage decrease be d%. Then V = 1.04V * (1 - d / 100). Divide both sides by V: 1 = 1.04 * (1 - d / 100). So, 1 / 1.04 = 1 - d / 100. Compute 1 / 1.04 = 100 / 104 = 25 / 26. Thus, 25 / 26 = 1 - d / 100. So, d / 100 = 1 - 25 / 26 = 1 / 26. Therefore, d = 100 / 26 = 50 / 13 = 3 11/13 percent (approximately 3.846 percent). Verification / Alternative check: Assume an original volume of 100 units. After a 4% increase, the volume becomes 104 units. To find the decrease needed to return to 100, compute the difference: 104 - 100 = 4 units. The percentage decrease relative to 104 is (4 / 104) * 100 = 400 / 104 = 100 / 26 = 3 11/13 percent. This numerical check confirms the algebraic result. Why Other Options Are Wrong: 3 3/13% and 4 1/13%: These values are close but do not match the exact calculation of 100 / 26 percent. 4%: This assumes symmetry between increase and decrease, which is not correct when the bases differ. 5%: This is too large and would bring the volume below the original value. Common Pitfalls: Many learners think that a 4% increase followed by a 4% decrease restores the original quantity, but this is incorrect because the second percentage is taken on a different base (the increased value). Others may try to compute the decrease as 4 / 1.04 incorrectly or approximate too early. Always set up the equation with the new and original values and solve systematically to find the exact percentage change. Final Answer: When the solid melts back into liquid, the volume decreases by 3 11/13 percent of the solid volume.

Question 16

A retailer offers a discount of 28% on the marked price of goods and thereby ends up selling exactly at cost price. What was the percentage markup of the marked price over the cost price?

Accepted Answer

Introduction / Context: This problem combines the ideas of markup and discount. The retailer first increases the price above cost (markup) and then reduces it by giving a discount, finally selling at cost. We must work backward from the information that after discount the selling price equals the cost price. This type of question appears frequently in profit and loss sections of competitive exams. Given Data / Assumptions: - Cost price (CP) of an item is some amount, say CP units. - The retailer marks the price above CP by an unknown percentage m. - A discount of 28 percent is offered on this marked price (MP). - After giving this discount, the final selling price equals CP. - We must find the percentage markup m. Concept / Approach: The marked price is related to cost price by MP = CP * (1 + m / 100). The selling price (SP) is related to marked price by SP = MP * (1 - discount / 100). Here, SP is given indirectly as equal to CP. By equating these expressions, we can solve for m. Using CP as a variable makes the solution general, but setting CP = 1 or 100 can simplify mental calculations. Step-by-Step Solution: Let CP = 1 unit for simplicity. Let the markup be m percent, so MP = 1 * (1 + m / 100) = 1 + m / 100. Discount offered = 28 percent, so SP = MP * (1 - 28 / 100) = (1 + m / 100) * 0.72. Given that SP equals CP, we have (1 + m / 100) * 0.72 = 1. Therefore, 1 + m / 100 = 1 / 0.72. Compute 1 / 0.72 = 100 / 72 = 25 / 18. So, 1 + m / 100 = 25 / 18. Thus, m / 100 = 25 / 18 - 1 = (25 - 18) / 18 = 7 / 18. Therefore, m = 100 * 7 / 18 = 700 / 18 = 38.88 percent (approximately). Verification / Alternative check: Assume CP = Rs. 100 for an easier numeric check. With markup 38.88 percent, MP = 100 + 38.88 = Rs. 138.88. Now apply a 28 percent discount: SP = 138.88 * 0.72. Compute 138.88 * 0.72 which is approximately 100. This shows that the selling price returns to the cost price, confirming that the markup must be about 38.88 percent. Why Other Options Are Wrong: 18.25 percent and 22 percent: These markups are too low; after a 28 percent discount the selling price would be less than cost, leading to a loss. 28 percent: This corresponds numerically to the discount but not to the required markup to end at cost price. 25 percent: Again, this is insufficient markup to offset a 28 percent discount. Common Pitfalls: One common mistake is assuming that a 28 percent markup will cancel a 28 percent discount, which is incorrect because the discount is applied on the marked price, not on cost price. Another error is to try to add or subtract percentages directly without setting up the equations. Always write the relationships between CP, MP, and SP clearly to avoid confusion in such problems. Final Answer: The retailer must mark up the goods by approximately 38.88 percent over the cost price to break even after giving a 28 percent discount.

Question 17

Two numbers are 30% and 60% 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 relative percentage comparison between numbers defined in terms of a common reference. The two numbers are expressed as percentages of a third number, and we must find by how much the smaller one must be increased so that it equals the larger one. Such questions are typical in ratio and percentage topics in aptitude exams. Given Data / Assumptions: - There is a third number, call it T. - First number A is 30 percent less than T. - Second number B is 60 percent less than T. - We must find the percentage increase needed for B to become equal to A. Concept / Approach: A number that is 30 percent less than T is equal to 70 percent of T. Similarly, one that is 60 percent less than T is 40 percent of T. Once A and B are expressed as fractions of T, we can compare them directly. The required percentage increase is the difference between A and B divided by B, multiplied by 100. It is important to consider B as the base, since we are increasing B. Step-by-Step Solution: Let the third number be T. A is 30% less than T, so A = T * (1 - 30 / 100) = 0.70T. B is 60% less than T, so B = T * (1 - 60 / 100) = 0.40T. We need to increase B so that it becomes equal to A. Required increase = A - B = 0.70T - 0.40T = 0.30T. Percentage increase based on B = (Required increase / B) * 100. So, percentage increase = (0.30T / 0.40T) * 100 = (0.75) * 100 = 75 percent. Verification / Alternative check: Take a simple value T = 100. Then A = 70 and B = 40. To raise B from 40 to 70, the increase needed is 30. As a percentage of B, this is (30 / 40) * 100 = 75 percent. The numerical method exactly matches our algebraic result, so the required percentage increase is confirmed as 75 percent. Why Other Options Are Wrong: 42.86 percent and 50 percent: These percentages are too low; applying them to 40 does not give 70. 30 percent: This confuses the absolute difference 30 with the percentage increase, ignoring the correct base of 40. 60 percent: This is closer but still does not give the exact required value. Sixty percent of 40 is 24, which would give 64, not 70. Common Pitfalls: A typical mistake is taking the average of 30 and 60 or directly subtracting them, rather than correctly comparing the actual numbers A and B. Another pitfall is computing the percentage relative to the larger number instead of the smaller one. Remember, if we are increasing B, then B is the base for the percentage change formula. Final Answer: The smaller number must be increased by 75 percent to become equal to the larger number.

Question 18

The value of a vehicle at the end of each year becomes 3/5 of its value at the beginning of that year. If the initial value is Rs. 10,000, what will be its value at the end of three years?

Accepted Answer

Introduction / Context: This question is about successive depreciation, where an asset loses a fixed fraction of its value every year. The phrase that the value becomes 3/5 of its value at the beginning of the year means that 2/5 has been depreciated. We must apply this depreciation repeatedly over three years. This is a standard application of repeated percentage or fractional change in quantitative aptitude. Given Data / Assumptions: - Initial value of the vehicle is Rs. 10,000. - At the end of each year, the value becomes 3/5 of the value at the beginning of that year. - We need the value at the end of three years. - Depreciation is assumed to be applied once per year, on the current value. Concept / Approach: If each year the value is multiplied by a fixed factor, here 3/5, then after n years the value is the initial value multiplied by (3/5)^n. This is similar to compound interest, but with a depreciation factor instead of a growth factor. We must calculate the value after three years using this repeated multiplication. Step-by-Step Solution: Initial value V0 = Rs. 10,000. At the end of year 1, value V1 = V0 * (3 / 5) = 10,000 * 3 / 5. Compute V1 = 10,000 * 0.6 = Rs. 6000. At the end of year 2, value V2 = V1 * (3 / 5) = 6000 * 3 / 5. Compute V2 = 6000 * 0.6 = Rs. 3600. At the end of year 3, value V3 = V2 * (3 / 5) = 3600 * 3 / 5. Compute V3 = 3600 * 0.6 = Rs. 2160. Therefore, at the end of three years, the vehicle is worth Rs. 2160. Verification / Alternative check: We can also compute in one step using powers. Each year multiplies value by 3/5, so after three years the factor is (3 / 5)^3. Compute (3 / 5)^3 = 27 / 125 = 0.216. Multiply this by the initial value: 10,000 * 0.216 = 2160. This matches the step-by-step calculation and confirms the answer. Why Other Options Are Wrong: Rs. 6000 and Rs. 3600: These are the values after one and two years respectively, not after three years. Rs. 415 and Rs. 2560: These values do not correspond to multiplying by (3 / 5)^3; they represent incorrect arithmetic or wrong interpretation of the fraction. Common Pitfalls: A common mistake is to treat depreciation as subtracting a fixed amount each year instead of multiplying by a factor. Another error is to misinterpret the statement and think that 3/5 is the amount depreciated rather than the remaining value. Always read carefully whether the fraction or percentage refers to the remaining value or to the amount lost. Using the compound factor approach is the most reliable method for successive depreciation problems. Final Answer: The value of the vehicle at the end of three years will be Rs. 2160.

Question 19

Prashanth sold an album for $735 and made a profit of 22.5%. Later the cost price of the album increased by $120, so he also increased his selling price by $120. What is his new profit percentage?

Accepted Answer

Introduction / Context: This question is a profit and loss problem with a change in both cost price and selling price. First, we are told about an original sale with a known profit percentage. Then the cost increases by a fixed amount and the selling price is also increased by the same amount. We must compute the new profit percentage based on the new cost and new selling price. Given Data / Assumptions: - Original selling price (SP1) is 735 dollars. - Original profit percentage is 22.5 percent. - The cost price then increases by 120 dollars. - The new selling price (SP2) is old selling price plus 120 dollars, that is SP2 = 735 + 120. - We must find the new profit percentage based on the new cost price and new selling price. Concept / Approach: The basic relationship is profit = selling price - cost price, and profit percentage = (profit / cost price) * 100. First we find the original cost price from the given original selling price and profit percentage. Then we add 120 to this cost to get the new cost price. Similarly, we add 120 to the old selling price to get the new selling price. Finally, we compute the new profit percentage using the new figures. Step-by-Step Solution: Let original cost price be CP1. Given original profit is 22.5%, so SP1 = CP1 * (1 + 22.5 / 100). So 735 = CP1 * 1.225. Therefore, CP1 = 735 / 1.225 = 600. Now cost price increases by 120 dollars, so new cost price CP2 = 600 + 120 = 720. New selling price SP2 = 735 + 120 = 855. New profit = SP2 - CP2 = 855 - 720 = 135. New profit percentage = (135 / 720) * 100. Compute 135 / 720 = 0.1875, so profit percentage = 18.75%. Verification / Alternative check: We can confirm the original profit: with CP1 = 600 and SP1 = 735, profit1 = 135. Profit percentage1 = (135 / 600) * 100 = 22.5%, which matches the given condition, so CP1 is correct. For the new situation, CP2 = 720 and SP2 = 855, profit2 = 135 again, but now relative to a larger cost price. New profit percentage = 135 / 720 * 100 = 18.75%. This confirms the calculation. Why Other Options Are Wrong: 19.32%, 21.41%, and 22%: These values do not match the exact ratio of 135 to 720 and arise from incorrect arithmetic or misunderstanding that profit amount remains the same but base cost changes. 20%: This is a nice round number but does not match the precise computation of 135 / 720. Common Pitfalls: One mistake is assuming the profit percentage remains the same because both cost price and selling price are increased by the same absolute amount. This is not true because profit percentage depends on the ratio of profit to the new cost price, which has changed. Another error is to misinterpret 22.5 percent as a fraction that is difficult to handle, but converting the percentage into a decimal (1.225) makes the calculation straightforward. Final Answer: Prashanth's new profit percentage after the cost and selling price both increase by $120 is 18.75 percent.

Question 20

What percentage of 85 is equal to 120?

Accepted Answer

Introduction / Context: This is a straightforward percentage problem where we express one number as a percentage of another. Such questions directly check the basic skill of converting a ratio into a percentage, which is fundamental in many areas of quantitative aptitude, data interpretation and everyday calculations. Given Data / Assumptions: - We have two numbers: 120 and 85. - We need to find what percentage 120 is of 85. Concept / Approach: To find what percent one number A is of another number B, we use the formula: Percentage = (A / B) * 100. Here, A is 120 and B is 85. We simply compute the fraction 120 / 85 and multiply by 100 to get the required percentage. The result will be more than 100 percent because 120 is greater than 85. Step-by-Step Solution: Let the required percentage be p. Then p = (120 / 85) * 100. First compute the fraction: 120 / 85. 120 / 85 simplifies to 24 / 17. Now compute (24 / 17) * 100. 24 * 100 = 2400. So p = 2400 / 17. Compute 2400 / 17 which is approximately 141.17 percent. Verification / Alternative check: We can check reasonableness. Since 85 * 1.4 = 119, which is very close to 120, we expect the answer to be close to 140 percent. More precisely, 85 * 1.4117 is roughly 120. That matches the calculated percentage of about 141.17 percent and confirms the correctness of the answer. Why Other Options Are Wrong: 133.33% and 135%: These values are too small; they would produce numbers significantly less than 120 when multiplied by 85 as percents. 145.54% and 139.29%: These are close but still do not match the exact ratio 2400 / 17, which is approximately 141.17%. Common Pitfalls: A common error is to compute 85 as a percentage of 120 instead of the other way around, which would invert the fraction. Another pitfall is rounding too early or trying to handle the calculation mentally without simplifying the fraction first. Writing the fraction clearly as 24 / 17 and then dividing helps reach the accurate value. Final Answer: The number 120 is approximately 141.17 percent of 85.

Percentage Questions