Question 1

A retailer offers a discount of 28% on the marked price of his goods and thereby sells them at cost price. What is the percentage mark-up on the cost price?

Accepted Answer

Introduction / Context: This question tests understanding of the relationship between mark up and discount. The retailer sets a marked price above the cost price and then offers a discount, yet ends up selling at exactly the cost price. From this, you have to back calculate the percentage by which the marked price exceeds the cost price. This is a classic successive percentage and profit loss concept. Given Data / Assumptions: Let the cost price (CP) of the goods be C. Let the marked price (MP) be M. The retailer offers a discount of 28% on MP. After discount, the selling price equals the cost price: SP = CP. We must find the percentage mark up, that is how much MP exceeds CP in percentage terms. Concept / Approach: A discount of 28% means the selling price is 72% of the marked price, or SP = 0.72M. We are told this SP equals the cost price C. Therefore, C = 0.72M. To find the mark up, we want (M - C) / C * 100. Since C = 0.72M, we can express M in terms of C or C in terms of M and compute the percentage. Working in terms of C keeps the interpretation of markup straightforward. Step-by-Step Solution: Step 1: From the discount, we have SP = 72% of MP = 0.72M. Step 2: Given that SP equals CP, we have C = 0.72M. Step 3: Rearrange to express M in terms of C: M = C / 0.72. Step 4: Mark up = M - C = (C / 0.72) - C. Step 5: Factor out C: Mark up = C * (1 / 0.72 - 1). Step 6: Compute 1 / 0.72 = 1.3888... so mark up fraction = 1.3888... - 1 = 0.3888... . Step 7: Convert to percentage: 0.3888... * 100 ≈ 38.88%. Verification / Alternative check: Assume the cost price C is Rs. 100 for simplicity. Mark up of 38.88% means MP ≈ 138.88. Now apply a 28% discount on 138.88. Discount = 0.28 * 138.88 ≈ 38.8864. Selling price ≈ 138.88 - 38.8864 ≈ 100, which equals our assumed cost price. This confirms the mark up percentage is correct. Why Other Options Are Wrong: 18.25% and 22%: These are much smaller than the actual 38.88% needed to offset a large 28% discount and still reach cost price. 28%: This would imply that discount and mark up cancel linearly, which is not the case with successive percentages. 33.33%: This corresponds to a factor of 4/3, which does not match the required 1 / 0.72. Common Pitfalls: Many learners incorrectly assume that a 28% discount requires a 28% markup to break even, but markup and discount are applied on different bases (cost vs marked price), so they are not symmetrical. Another pitfall is mismanaging the inversion 1 / 0.72, leading to approximate but incorrect values. Writing clear equations prevents confusion. Final Answer: The retailer has marked up the goods by approximately 38.88% over the cost price.

Question 2

When a number is increased by 38, it becomes 119% of its original value. What is the original number?

Accepted Answer

Introduction / Context: This is a classic reverse percentage problem involving an absolute increase that transforms a number into a specified percentage of itself. Such questions test how well you can translate verbal percentage descriptions into equations and then solve for the unknown original value. Given Data / Assumptions: Let the original number be N. When it is increased by 38, the new number is N + 38. This new number is equal to 119% of the original number. We must determine the value of N. Concept / Approach: The key idea is to express “119% of the original number” in algebraic form as 1.19N. Since the increased value N + 38 equals 1.19N, we set up the equation N + 38 = 1.19N. Then, we solve this equation by moving terms involving N to one side and constants to the other. This is a simple linear equation once the percentage is converted to decimal form. Step-by-Step Solution: Step 1: Let the original number be N. Step 2: After increasing by 38, the new number is N + 38. Step 3: The new number is 119% of the original, so write N + 38 = 119% of N = 1.19N. Step 4: Rearrange the equation: N + 38 = 1.19N implies 38 = 1.19N - N = 0.19N. Step 5: Solve for N: N = 38 / 0.19. Step 6: Compute N = 200. Verification / Alternative check: Check the answer by substituting N = 200 back into the situation. Increase 200 by 38 to get 238. Now, calculate 119% of 200: (119 / 100) * 200 = 238. The increased value exactly matches 119% of the original, confirming the solution is correct. Why Other Options Are Wrong: 342, 456, 570, 150: If any of these were the original number, then adding 38 and checking whether the result is 119% of the assumed original quickly shows that the equality does not hold. For example, if N = 150, N + 38 = 188, but 119% of 150 is 178.5, not 188. Common Pitfalls: Some students mistakenly treat 119% as 0.19 instead of 1.19, or they write 119N instead of 1.19N. Others forget that the 38 is an absolute increase, not a percentage increase. Always convert percentage phrases carefully and set up the equation with correct decimal values. Final Answer: The original number is 200.

Question 3

A shop offers a deal: buy 4 dupattas at Rs. 1300 each and get 1 dupatta free. What is the effective discount percentage on each dupatta compared to the marked price?

Accepted Answer

Introduction / Context: This problem deals with effective discount in a promotional offer of the type “buy some items and get one free.” Although no percentage is directly mentioned, such offers can be converted into equivalent percentage discounts by comparing the total amount paid with the total number of items received. This is a common scenario in retail marketing and exam questions. Given Data / Assumptions: Marked price of one dupatta = Rs. 1300. Customer buys 4 dupattas and gets 1 dupatta free. Customer pays only for 4 dupattas but receives 5 dupattas in total. We need the effective discount percentage on each dupatta relative to its marked price. Concept / Approach: The total marked price of 5 dupattas would be 5 times the marked price, but the customer pays only for 4. So, the customer effectively pays a lower average price per dupatta than the marked price. To find the effective discount percent, compute the average price per dupatta that the customer pays, compare it to the marked price, and convert the difference into a percentage of the marked price. Step-by-Step Solution: Step 1: Marked price per dupatta = Rs. 1300. Step 2: For 5 dupattas at marked price, total would be 5 * 1300 = Rs. 6500. Step 3: Under the offer, the customer pays only for 4 dupattas: 4 * 1300 = Rs. 5200. Step 4: Effective price per dupatta = total paid / total dupattas received = 5200 / 5 = Rs. 1040. Step 5: Discount per dupatta = marked price - effective price = 1300 - 1040 = Rs. 260. Step 6: Effective discount percentage = (260 / 1300) * 100 = 20%. Verification / Alternative check: We can also compare the ratio of effective price to marked price: 1040 / 1300 = 0.8. This means the customer is paying 80% of the marked price per dupatta. Hence the discount is 100% - 80% = 20%. This matches the earlier calculation, confirming that the effective discount per dupatta is 20%. Why Other Options Are Wrong: 12.5%: Would correspond to paying 87.5% of the marked price; the offer here is more generous than that. 25% and 33.33%: These are larger discounts than actually provided by the buy 4 get 1 free arrangement at this price point. 50%: This would mean paying only half the marked price per dupatta, which is clearly not the case here. Common Pitfalls: A frequent mistake is to assume the discount is 1 item out of 5, that is 20%, without checking if the prices are equal or more complex. In this question the prices are equal, so that logic works, but one must still demonstrate it through calculations. Another error is to calculate discount based on total marked price of 4 items instead of 5 items received, which leads to confusion. Final Answer: The effective discount on each dupatta is 20% of the marked price.

Question 4

The price of an article is reduced by 25%. By what percentage must the new reduced price be increased to restore the article to its original price?

Accepted Answer

Introduction / Context: This question illustrates that a decrease of a certain percentage does not require the same percentage increase to return to the original value. Understanding this asymmetry is crucial for handling successive percentage change problems correctly, especially in pricing, profit loss, and salary revision contexts. Given Data / Assumptions: The original price of an article is some amount P. The price is reduced by 25%. The reduced price must then be increased by some percentage to get back to P. We must find that required percentage increase. Concept / Approach: A 25% reduction means the new price is 75% of the original price, that is 0.75P. To restore the price back to P, we need to find a percentage increase x such that 0.75P * (1 + x / 100) = P. Solving this equation gives the required percentage increase. This clearly shows why the increase must be larger than 25%. Step-by-Step Solution: Step 1: Let the original price be P. Step 2: After a 25% reduction, new price = P - 0.25P = 0.75P. Step 3: Let the required percentage increase be x%. Step 4: After increasing by x%, price becomes 0.75P * (1 + x / 100). Step 5: To restore to original price, set 0.75P * (1 + x / 100) = P. Step 6: Divide both sides by P: 0.75 * (1 + x / 100) = 1. Step 7: Divide both sides by 0.75: 1 + x / 100 = 1 / 0.75 = 4 / 3. Step 8: So x / 100 = 4 / 3 - 1 = 1 / 3. Step 9: Multiply by 100: x = 100 / 3 ≈ 33.33%. Verification / Alternative check: Assume P = 100 for easy numbers. After a 25% decrease, the price becomes 75. Increase 75 by 33.33%: 33.33% of 75 is approximately 25, so 75 + 25 = 100. This returns the price back to its original value, confirming that a 33.33% increase is correct. Why Other Options Are Wrong: 36.31%, 57%, 71.25%: These are larger than necessary and overshoot the original price when applied to 0.75P. 25%: This is a common incorrect guess assuming symmetry, but 25% of 75 is only 18.75, which does not restore the price to 100 when added. Common Pitfalls: The most common error is to think that a 25% decrease followed by a 25% increase returns to the starting value. This is incorrect because the increment is applied to a smaller base. Another pitfall is incorrectly setting up the equation or forgetting to divide by 0.75 when solving for the required percentage. Final Answer: The new price must be increased by 33.33% to restore the original price.

Question 5

An item is sold at a price of Rs. 855 after a discount of 5% on its marked price. What is the marked price of the item?

Accepted Answer

Introduction / Context: This problem is another example of reversing a given percentage discount to find the original marked price. Knowing how to move from selling price back to marked price is very important for understanding how discounts affect prices and for solving many commercial math problems. Given Data / Assumptions: Let the marked price be M rupees. A discount of 5% is applied to M. After discount, the selling price (SP) is Rs. 855. We must find the value of M. Concept / Approach: A discount of 5% means that the customer pays 95% of the marked price. Therefore, SP = 95% of M = 0.95M. The given SP is 855, so we form the equation 0.95M = 855 and solve for M by dividing by 0.95. This is a standard reverse percentage calculation. Step-by-Step Solution: Step 1: Let M be the marked price. Step 2: After a 5% discount, the customer pays 95% of M, so SP = 0.95M. Step 3: Given that SP = 855, we have 0.95M = 855. Step 4: Solve for M: M = 855 / 0.95. Step 5: Compute M = 900. Step 6: So, the marked price of the item is Rs. 900. Verification / Alternative check: Check the result by applying the 5% discount on the marked price of Rs. 900. Discount = 5% of 900 = (5 / 100) * 900 = 45. Selling price = 900 - 45 = 855, which matches the given selling price. This confirms that the marked price of Rs. 900 is correct. Why Other Options Are Wrong: Rs. 812.25 and Rs. 897.75: These values do not produce a selling price of 855 when a 5% discount is applied. Rs. 814: After a 5% discount, the resulting selling price would be lower than 773.3, not 855. Rs. 950: A 5% discount on 950 gives 902.5, which is not equal to 855. Common Pitfalls: Sometimes students attempt to subtract 5 from 855 or divide 855 by 5 directly, both of which do not represent the correct concept. Always remember that a percentage discount is applied to the marked price, and to reverse it, you must divide by the remaining percentage (in decimal form), not by the discount percentage. Final Answer: The marked price of the item is Rs. 900.

Question 6

In a hotel, 60% of the people had vegetarian lunch, 30% had non vegetarian lunch and 15% had both types of lunch. If a total of 96 people were present, how many people did not eat either type of lunch?

Accepted Answer

Introduction / Context: This problem is a classic example of the use of set theory and percentages in practical situations. It involves two overlapping groups, those who had vegetarian lunch and those who had non vegetarian lunch, with some people counted in both groups. We are asked to find the number of people who belong to neither group, using inclusion exclusion principles. Given Data / Assumptions: Total number of people in the hotel = 96. 60% had vegetarian lunch. 30% had non vegetarian lunch. 15% had both vegetarian and non vegetarian lunches. We assume everyone is included in the total of 96 and that percentages are of this total. We must find how many did not eat either type of lunch. Concept / Approach: To work with overlapping categories, we use the principle of inclusion exclusion. The percentage of people who had at least one type of lunch equals the sum of the individual percentages minus the percentage of people counted twice (those who had both types). Once we find the percentage of people who had at least one type of lunch, we subtract this from 100% to find the percentage who had neither type. Then we apply this percentage to the total number of people. Step-by-Step Solution: Step 1: Let V be the set of people who had vegetarian lunch, N the set who had non vegetarian lunch. Step 2: Percentage in V = 60%, percentage in N = 30%. Step 3: Percentage in both V and N (intersection) = 15%. Step 4: Percentage in at least one of the two sets = 60% + 30% - 15% = 75%. Step 5: Therefore, percentage who had neither type of lunch = 100% - 75% = 25%. Step 6: Number of people who had neither type = 25% of 96 = (25 / 100) * 96 = 24. Verification / Alternative check: Compute actual numbers for each category. Vegetarian: 60% of 96 = 57.6 (conceptually, but the percentage is applied to the total distribution and is assumed exact for this problem). Non vegetarian: 30% of 96 = 28.8. Both: 15% of 96 = 14.4. While these numbers are fractional, the net effect using inclusion exclusion on percentages is exact. Using percentages only, we already found 25% of 96 is 24, which is an integer and matches typical exam expectations. Why Other Options Are Wrong: 30, 36, 42, 18: Each of these corresponds to different percentages of 96 (about 31.25%, 37.5%, 43.75%, 18.75%), none matching the required 25% remaining after accounting for the overlapping lunch groups. Substituting any of these into the total leads to incorrect percentages for those who had at least one type of lunch. Common Pitfalls: A frequent error is to simply add 60% and 30% to get 90% and then subtract from 100% to infer that 10% had neither, ignoring the overlap. This double counts those who had both types of lunch. Always remember to subtract the intersection percentage when using inclusion exclusion. Another pitfall is being confused by fractional counts; working entirely in percentages avoids this. Final Answer: 24 people did not eat either type of lunch.

Question 7

2 is what percentage of 50?

Accepted Answer

Introduction / Context: This is a simple percentage question that asks you to express one number as a percentage of another. Such problems are foundational in percentage arithmetic and form the basis for more complex comparisons, growth rates, and ratio interpretations in quantitative exams. Given Data / Assumptions: The part value is 2. The base or whole value is 50. We must find what percent 2 is of 50. Percent means per hundred, so we compare by scaling the fraction to a denominator of 100. Concept / Approach: To find what percentage one number is of another, we use the formula: Percentage = (part / whole) * 100. Here, the part is 2 and the whole is 50. We compute the fraction 2 / 50 and then multiply by 100 to express it as a percentage. This process reinforces the meaning of percentages as normalized fractions. Step-by-Step Solution: Step 1: Write the required percentage as (part / whole) * 100. Step 2: Substitute the values: (2 / 50) * 100. Step 3: Simplify the fraction 2 / 50 = 1 / 25. Step 4: Compute (1 / 25) * 100. Step 5: 100 / 25 = 4. Step 6: Therefore, 2 is 4% of 50. Verification / Alternative check: We can also reverse check by taking 4% of 50 and seeing whether we get 2. Compute 4% of 50: (4 / 100) * 50 = 2. Since this returns the original part value, our percentage calculation is confirmed. Why Other Options Are Wrong: 2%: 2% of 50 is 1, not 2. 2.5%: 2.5% of 50 is 1.25, not 2. 3%: 3% of 50 is 1.5, which is still less than 2. 5%: 5% of 50 is 2.5, which is greater than 2. Common Pitfalls: A common mistake is to divide 50 by 2 instead of 2 by 50, which yields 25 and then mistakenly interpreting this as 25%. Remember that the numerator should be the part and the denominator the whole. Another error is forgetting to multiply by 100 at the end and leaving the answer as a decimal fraction only. Final Answer: 2 is 4% of 50.

Question 8

What is one quarter (1/4) of 12?

Accepted Answer

Introduction / Context: This is a basic fraction question that checks understanding of the phrase one quarter and how to compute a fraction of a whole number. Mastery of simple fractional operations like taking one half, one third, or one quarter is essential for more advanced problems involving fractions, percentages, and ratios. Given Data / Assumptions: We have the whole number 12. The fraction we need is one quarter, which is 1 / 4. We must compute 1 / 4 of 12. All values are positive integers, and standard fraction multiplication rules apply. Concept / Approach: One quarter of a number means dividing the number into four equal parts and taking one part. Mathematically, one quarter of a number N is (1 / 4) * N. For integer values, this boils down to dividing the number by 4. This interpretation also reinforces the notion that a fraction denotes a part of a whole. Step-by-Step Solution: Step 1: Interpret one quarter of 12 as (1 / 4) * 12. Step 2: Multiply 1 by 12 to get 12, then divide by 4: 12 / 4. Step 3: Compute 12 / 4 = 3. Step 4: Therefore, 1 / 4 of 12 is 3. Verification / Alternative check: Think of 12 objects, for example 12 apples. If you split them equally into 4 groups, each group will contain 12 / 4 = 3 apples. Taking one of these groups corresponds exactly to one quarter of the total, confirming that one quarter of 12 is 3. Why Other Options Are Wrong: 4: This would be one third of 12, not one quarter. 2: This would be one sixth of 12, not one quarter. 1: This would be one twelfth of 12, not one quarter. 6: This would be one half of 12, not one quarter. Common Pitfalls: Some learners confuse one quarter with 0.25, then incorrectly apply it. While 0.25 * 12 also equals 3, if rounding is used or misunderstood, errors can occur. Others may think of one quarter as dividing by 2 twice but may make a slip in one of the steps. Keeping the fraction form 1 / 4 and directly dividing by 4 is usually the safest approach. Final Answer: One quarter of 12 is 3.

Question 9

If the ratio of the marked price to the selling price of an article is 11 : 10, what is the discount percentage on the marked price?

Accepted Answer

Introduction / Context: This question uses a ratio between marked price and selling price to derive the discount percentage. Instead of being given actual rupee values, we work with proportional values. Understanding how to translate ratios into percentages is a key skill in commercial arithmetic, especially in discount and profit loss problems. Given Data / Assumptions: Ratio of marked price (MP) to selling price (SP) = 11 : 10. We assume MP and SP are positive amounts in the same units. The discount is the difference between MP and SP, taken as a percentage of MP. We must find the discount percentage. Concept / Approach: If MP : SP = 11 : 10, we can assume a convenient common factor, for example MP = 11 units and SP = 10 units. The discount is then MP minus SP, which is 1 unit. Discount percentage is defined as (discount / MP) * 100. This gives (1 / 11) * 100. The result is a recurring decimal, and we usually round to two decimal places. Step-by-Step Solution: Step 1: From the ratio 11 : 10, assume MP = 11 units and SP = 10 units. Step 2: Discount = MP - SP = 11 - 10 = 1 unit. Step 3: Discount percentage = (discount / MP) * 100. Step 4: Substitute values: Discount% = (1 / 11) * 100. Step 5: Compute 100 / 11 ≈ 9.09. Step 6: Therefore, the discount percentage is approximately 9.09%. Verification / Alternative check: Consider an actual example with MP = Rs. 110 and SP = Rs. 100 (multiplying both ratio terms by 10). Discount = 110 - 100 = Rs. 10. Discount percentage = (10 / 110) * 100 = 1000 / 11 ≈ 90.91 / 10 = 9.09%. This confirms that the ratio 11 : 10 corresponds to a discount of about 9.09%. Why Other Options Are Wrong: 11.11%: This would correspond to a ratio more like 9 : 8. Now MP : SP = 9 : 8 implies a discount of 1 / 9 which is about 11.11%, not 1 / 11. 12.5%: This would correspond to a discount of 1 / 8 and a ratio of 8 : 7, unlike the given 11 : 10. 17.28%: This is not consistent with any simple ratio close to 11 : 10. 5%: This would correspond to MP : SP ≈ 100 : 95, which is not 11 : 10. Common Pitfalls: One common error is to reverse the ratio and treat SP : MP as 11 : 10, which leads to a completely different calculation. Another mistake is to compute the discount percentage as (1 / 10) * 100 when using the ratio 11 : 10, incorrectly taking SP as the base instead of MP. Always remember that discount percentage is based on the marked price, not the selling price. Final Answer: The discount percentage on the marked price is approximately 9.09%.

Question 10

A trader sells a cycle at a loss of 10 percent. If he had increased the selling price by Rs. 200, he would have made a profit of 6 percent instead of a loss. What is the cost price of the cycle?

Accepted Answer

Introduction / Context:This problem tests the core percentage profit and loss concept where the selling price is changed and we need to recover the original cost price. Such questions are very common in bank exams, SSC, railways and other aptitude tests, so understanding the structure of the equation is extremely useful.Given Data / Assumptions:The trader first sells the cycle at a loss of 10 percent.If the selling price is increased by Rs. 200, there is a profit of 6 percent.We assume cost price is a positive real number.Market conditions, taxes and other charges are ignored.Concept / Approach:We express both selling prices in terms of the same unknown cost price. Loss of 10 percent means the selling price is 90 percent of cost price. Profit of 6 percent means the selling price is 106 percent of cost price. The difference between these two selling prices is given as Rs. 200. Solving this linear equation gives the cost price.Step-by-Step Solution:Let cost price = C.First selling price at 10 percent loss = 0.9 * C.New selling price at 6 percent profit = 1.06 * C.Difference between these selling prices = 1.06 * C - 0.9 * C = 0.16 * C.Given that this difference is Rs. 200, so 0.16 * C = 200.Therefore, C = 200 / 0.16 = 1250.Verification / Alternative check:If cost price is Rs. 1250, then 10 percent loss selling price is 0.9 * 1250 = 1125. At 6 percent profit, selling price is 1.06 * 1250 = 1325. The difference between 1325 and 1125 is exactly Rs. 200, so our computation is consistent with the conditions in the question.Why Other Options Are Wrong:Rs. 1200, Rs. 1205 and Rs. 1275 do not satisfy the equation 0.16 * C = 200. If you substitute any of these values, the difference between the two selling prices will not be exactly Rs. 200, so they cannot be correct cost prices.Common Pitfalls:Candidates often subtract percentage values directly or try to apply the difference 6 + 10 = 16 percent without linking it to the cost price. The key is to express both selling prices in terms of cost price and then equate their difference to the given rupee difference. Forgetting that loss and profit percentages are always on cost price is another frequent mistake.Final Answer:The cost price of the cycle is Rs. 1250.

Question 11

The salary of A is 50 percent more than the salary of B. By what percentage is B's salary less than A's salary?

Accepted Answer

Introduction / Context:This is a classic percentage comparison question where one quantity is more than another by a certain percent and we are asked to reverse the comparison. Many students wrongly assume the percentages will be the same in both directions, which is not true in percentage arithmetic.Given Data / Assumptions:A's salary is 50 percent more than B's salary.We need the percentage by which B's salary is less than A's salary.Salaries are positive numbers and we use simple percentage definitions.Concept / Approach:When A is 50 percent more than B, it means A = B + 0.5 * B = 1.5 * B. To find how much B is less than A, we compare the difference (A - B) with A, not with B. This is where many candidates go wrong, because the base for the percentage changes when we reverse the comparison.Step-by-Step Solution:Assume B's salary = 100 units for ease of calculation.Then A's salary = 100 + 50 = 150 units.Difference between A and B = 150 - 100 = 50 units.Required percentage = (difference / A) * 100.So percentage by which B is less than A = (50 / 150) * 100 = 33.33 percent (approximately).Verification / Alternative check:We can scale salaries to any convenient real values. For example, if B earns Rs. 20,000, then A earns 1.5 * 20,000 = Rs. 30,000. The difference is Rs. 10,000. The percentage by which B is less than A is 10,000 / 30,000 * 100 = 33.33 percent, which matches the earlier calculation.Why Other Options Are Wrong:Values like 24.5 percent, 29.5 percent and 31.3 percent come from using wrong bases or approximate mental arithmetic. None of these percentages correctly represent the ratio of the difference to A's salary. Only 33.3 percent is consistent with the exact ratio 1/3.Common Pitfalls:A very common error is to say that if A is 50 percent more than B, then B is 50 percent less than A, which is incorrect because percentage change always depends on the reference quantity. Another mistake is to divide the difference by B instead of A while answering the reversed question.Final Answer:B's salary is approximately 33.3 percent less than A's salary.

Question 12

The price of an item is first reduced by 25 percent. By what percentage must the reduced price be increased to obtain the original price again?

Accepted Answer

Introduction / Context:This question checks understanding of successive percentage changes. A decrease followed by an increase does not lead back to the starting value when the percentages are numerically equal, because the bases are different. This is a very standard trap in aptitude tests.Given Data / Assumptions:The original price of an item is reduced by 25 percent.We then want to increase the reduced price to get back to the original price.We need the required percentage increase on the reduced price.Concept / Approach:Percentage decrease and increase act on the current value, not on the original value in both steps. The natural method is to assume a convenient original price, apply the decrease to get the reduced price and then compute the required increase that brings us back to the assumed original price.Step-by-Step Solution:Assume original price = 100 units.After a 25 percent decrease, reduced price = 100 - 25 = 75 units.We want to reach back to 100 units from 75 units.Increase required = 100 - 75 = 25 units.Required percentage increase = (25 / 75) * 100 = 33.33 percent (approximately).Verification / Alternative check:If the actual original price were Rs. 400, then a 25 percent reduction gives 400 - 0.25 * 400 = Rs. 300. To return from Rs. 300 to Rs. 400, we need an increase of Rs. 100. The percentage is 100 / 300 * 100 = 33.33 percent, which confirms the same value.Why Other Options Are Wrong:Values like 43.3 percent, 50 percent and 55.3 percent are obtained when the candidate mistakenly applies the percentage on the wrong base or treats the two percentages symmetrically. They do not convert 75 back to 100, so they cannot be correct.Common Pitfalls:The most common mistake is to think that if a quantity is reduced by 25 percent, an increase of 25 percent will bring it back to its original value. This is wrong because the 25 percent increase is applied on the smaller reduced value, not on the original value. Always identify the correct base.Final Answer:The reduced price must be increased by approximately 33.33 percent to restore the original price.

Question 13

In an examination, 35 percent of the students failed in Hindi, 45 percent failed in English and 20 percent failed in both subjects. What percentage of students passed in both Hindi and English?

Accepted Answer

Introduction / Context:This problem tests knowledge of set theory and percentage, specifically the use of the inclusion exclusion principle for two sets. The sets here represent failures in Hindi and English, and we are required to determine the percentage that passes in both subjects.Given Data / Assumptions:35 percent students fail in Hindi.45 percent students fail in English.20 percent students fail in both Hindi and English.Every student either passes or fails in each subject.Concept / Approach:Let H be the set of students who fail in Hindi and E be the set of students who fail in English. We know percentages of H, E and H intersection E. Using the formula for union of two sets, we can find the percentage that fail in at least one subject. The complement of that union gives the percentage that pass in both subjects.Step-by-Step Solution:Percentage failing in Hindi = 35 percent.Percentage failing in English = 45 percent.Percentage failing in both = 20 percent.Percentage failing in at least one subject = 35 + 45 - 20 = 60 percent.Therefore, percentage passing in both subjects = 100 - 60 = 40 percent.Verification / Alternative check:Assume there are 100 students. Then 35 fail Hindi, 45 fail English and 20 fail both. Using a Venn diagram or simple counting, at least one failure occurs in 60 students, leaving 40 students who do not belong to either failure set. Hence they pass both subjects, confirming the answer of 40 percent.Why Other Options Are Wrong:Values 45 percent, 35 percent and 20 percent come from misreading the question or from directly using the given failure percentages without applying inclusion exclusion and complement correctly. Only 40 percent matches the count of students who pass both subjects.Common Pitfalls:Common mistakes include adding all three percentages 35, 45 and 20 or subtracting wrongly. Some candidates also confuse the meaning of failing both with passing both. Always use the standard formula for two sets and then take the complement for those who pass both.Final Answer:40 percent of the students pass in both Hindi and English.

Question 14

A dealer marks his goods 20 percent above the cost price and then allows a discount on the marked price so that his net profit is 8 percent. What is the rate of discount?

Accepted Answer

Introduction / Context:This question combines markup and discount concepts applied on cost price and marked price. It is important to understand which percentage is applied to which base. Many students confuse profit percentage with discount percentage, so a clear stepwise calculation is essential.Given Data / Assumptions:Goods are marked 20 percent above cost price.Some discount is given on the marked price.Net profit obtained by the dealer is 8 percent on cost price.We are asked to find the discount rate.Concept / Approach:First express the marked price in terms of the cost price using the markup. Then express the selling price using the profit percentage. Finally relate selling price to marked price using a discount factor and solve for the discount rate. All calculations are easier if we assume a convenient cost price like 100 units.Step-by-Step Solution:Let cost price = 100 units.Marked price = cost price * 1.20 = 120 units.Required profit = 8 percent, so selling price = 100 * 1.08 = 108 units.Let discount rate be d percent on marked price.Then selling price = marked price * (1 - d/100) = 120 * (1 - d/100).So 120 * (1 - d/100) = 108, therefore (1 - d/100) = 108 / 120 = 0.9.Hence d/100 = 0.1 and discount rate d = 10 percent.Verification / Alternative check:With 10 percent discount on marked price of 120, the selling price becomes 120 * 0.9 = 108. Since cost price is 100, profit is 8 units, which is 8 percent of 100. This confirms the correctness of our calculation.Why Other Options Are Wrong:Discounts like 12 percent, 6 percent and 4 percent produce selling prices that do not correspond to exactly 8 percent profit on cost price. For each wrong discount, the selling price either gives less or more than 8 percent profit.Common Pitfalls:One common mistake is to subtract profit percent directly from markup percent (20 minus 8) and assume 12 percent is the discount. This is wrong because markup and discount are applied on different bases. Always convert everything to actual rupee or unit values from a chosen cost price before solving.Final Answer:The rate of discount is 10 percent.

Question 15

The selling price of an article is eight fifths of its cost price. What is the percentage gain made on this article?

Accepted Answer

Introduction / Context:This is a direct profit percentage calculation based on a given ratio of selling price to cost price. Problems of this type are frequently asked to test whether the candidate can quickly convert a ratio into a profit percentage.Given Data / Assumptions:Selling price (SP) is 8/5 of cost price (CP).We need to calculate the profit percentage.Cost price and selling price are positive real values.Concept / Approach:Profit is defined as SP minus CP. Profit percentage is given by (profit / CP) * 100. With SP expressed as a multiple of CP, the algebra becomes simple. To avoid dealing with fractions later, we can assume a convenient value for CP that clears the denominator.Step-by-Step Solution:Let cost price CP = 5 units (to match denominator of 5).Then selling price SP = (8/5) * CP = (8/5) * 5 = 8 units.Profit = SP - CP = 8 - 5 = 3 units.Profit percentage = (profit / CP) * 100 = (3 / 5) * 100.So profit percentage = 60 percent.Verification / Alternative check:We can also use a different cost price, for example CP = 10. Then SP = (8/5) * 10 = 16, profit = 6 and profit percentage = 6 / 10 * 100 = 60 percent. This confirms that the result is independent of the particular cost price chosen.Why Other Options Are Wrong:Profit percentages 20 percent, 28 percent and 68 percent do not match the ratio 8/5. They would correspond to different SP to CP ratios. Only 60 percent correctly reflects that the selling price is 1.6 times the cost price, yielding a 0.6 CP profit.Common Pitfalls:Candidates sometimes confuse the ratio and directly treat 8/5 as 8 percent or 5 percent, which is wrong. Another common error is to compute (SP / CP) * 100 instead of (SP - CP) / CP * 100, which gives 160 percent, not the profit percentage. Always apply the correct profit formula.Final Answer:The gain percentage on the article is 60 percent.

Question 16

The salary of Manoj is 40 percent less than the salary of Subhash. By what percentage is the salary of Subhash more than the salary of Manoj?

Accepted Answer

Introduction / Context:This question again illustrates the asymmetric nature of percentage comparisons. If one value is a certain percentage less than another, the reverse comparison uses a different percentage because the base changes. Understanding this is crucial for percentage based aptitude questions.Given Data / Assumptions:Manoj's salary is 40 percent less than Subhash's salary.We must find how many percent Subhash's salary is more than Manoj's salary.All salaries are positive values.Concept / Approach:If a quantity is 40 percent less than another, it means it is 60 percent of that other quantity. So we first express Manoj's salary in terms of Subhash's salary. Then we compute the ratio of the difference to Manoj's salary to obtain the required percentage by which Subhash earns more.Step-by-Step Solution:Assume Subhash's salary = 100 units.Manoj's salary is 40 percent less, so Manoj's salary = 100 - 40 = 60 units.Difference between their salaries = 100 - 60 = 40 units.Required percentage = (difference / Manoj's salary) * 100.So percentage by which Subhash earns more = (40 / 60) * 100 = 66.67 percent (approximately).Verification / Alternative check:If we take Subhash earning Rs. 50,000, then Manoj earns 60 percent of that, that is Rs. 30,000. The difference is Rs. 20,000. The percentage difference relative to Manoj is 20,000 / 30,000 * 100 = 66.67 percent, confirming our result.Why Other Options Are Wrong:Percentages like 60 percent, 62.5 percent and 65 percent come from rough approximations or from using the wrong base. Only 66.67 percent correctly matches the exact fraction 2/3 derived from the salary ratio 100:60 = 5:3.Common Pitfalls:Many learners simply reuse 40 percent when reversing the comparison, which is incorrect. Others divide the difference by Subhash's salary again, getting 40 percent. Always pay attention to which salary is used as the reference when converting a difference into a percentage.Final Answer:The salary of Subhash is approximately 66.67 percent more than the salary of Manoj.

Question 17

The sum of three numbers is 67. The ratio of the first number to the second number is 3 : 5, and the ratio of the second number to the third number is 4 : 7. What is the value of the second number?

Accepted Answer

Introduction / Context:This is a ratio and sum problem where we are asked to determine individual values of three numbers that are linked by ratios. Such questions evaluate comfort with manipulating ratios and converting them into actual numerical values using a common multiplying factor.Given Data / Assumptions:Sum of three numbers = 67.First : Second = 3 : 5.Second : Third = 4 : 7.All three numbers are assumed to be positive real numbers.Concept / Approach:We have two ratio relations, each involving the second number. We convert them into a single ratio involving all three numbers by finding a common representation for the second number. After that, we know the total sum, so we can solve for the common multiplying factor and hence each individual number.Step-by-Step Solution:Let first : second = 3 : 5, so first = 3k and second = 5k.Let second : third = 4 : 7, so second = 4m and third = 7m.Equate the second number: 5k = 4m.Take k = 4t and m = 5t so that second = 20t.Then first = 3k = 12t and third = 7m = 35t.Sum of numbers = 12t + 20t + 35t = 67t.Given sum is 67, so 67t = 67, hence t = 1.Therefore, the second number = 20t = 20.Verification / Alternative check:Numbers are 12, 20 and 35. Check the ratios: 12 : 20 simplifies to 3 : 5 and 20 : 35 simplifies to 4 : 7. Their sum equals 12 + 20 + 35 = 67, which matches the given total. Thus the second number is correctly found as 20.Why Other Options Are Wrong:Values 24, 18 and 16 do not satisfy the stated ratios when combined with any corresponding first and third numbers while keeping the sum equal to 67. Only 20 works consistently with both ratio conditions and the total sum constraint.Common Pitfalls:Learners sometimes mix up the order of ratios or try to add ratios directly. Another pitfall is forgetting to match the middle term of both ratios by using a common multiple. Without that step, the derived numbers will not satisfy both ratio relationships simultaneously.Final Answer:The value of the second number is 20.

Question 18

The marked price of an article is Rs. 520 and the selling price is Rs. 384.80. What is the discount percentage given on the article?

Accepted Answer

Introduction / Context:This question is a straightforward application of the discount percentage formula based on marked price and selling price. It checks basic comfort with percentage reduction calculations in a shopping or profit and loss context.Given Data / Assumptions:Marked price (MP) of the article = Rs. 520.Selling price (SP) of the article = Rs. 384.80.Required: discount percentage on the marked price.Concept / Approach:Discount is the reduction from marked price to selling price. Discount amount is MP minus SP, and discount percentage is (discount / MP) * 100. The formula always uses marked price as the base, not the selling price.Step-by-Step Solution:Marked price MP = 520.Selling price SP = 384.80.Discount amount = MP - SP = 520 - 384.80 = 135.20.Discount percentage = (discount / MP) * 100.So discount percentage = (135.20 / 520) * 100.Compute the fraction: 135.20 / 520 = 0.26.Therefore, discount percentage = 0.26 * 100 = 26 percent.Verification / Alternative check:We can check by reversing the percentage. If the discount is 26 percent on Rs. 520, the discount amount should be 0.26 * 520 = Rs. 135.20 and selling price should be 520 - 135.20 = Rs. 384.80, which matches the given selling price. This confirms the calculation.Why Other Options Are Wrong:Values like 24.42, 35.13 and 29 represent incorrect ratios of the discount to the marked price. They arise from calculation mistakes such as dividing by selling price instead of marked price or by making rounding errors.Common Pitfalls:Some students mistakenly divide the discount amount by the selling price to compute the discount percentage. Others may approximate the values too early and lose accuracy. Always use marked price as the base for discount percentage and perform division carefully.Final Answer:The discount percentage allowed on the article is 26 percent.

Question 19

A student accidentally multiplies a number by 4/7 instead of 7/4. What is the percentage error in the result compared with the correct value?

Accepted Answer

Introduction / Context:This problem tests understanding of percentage error when the wrong multiplier is used. It is a classic exam question where the candidate must compare an incorrect result with a correct one and then express the difference as a percentage of the correct value.Given Data / Assumptions:Correct operation: multiply a number by 7/4.Incorrect operation: multiply the same number by 4/7.We must find the percentage error relative to the correct result.Concept / Approach:Let the original number be N. The correct result is N * 7/4. The wrong result is N * 4/7. The ratio of wrong result to correct result simplifies the calculation. Percentage error is (correct - wrong) / correct * 100, assuming the wrong result is smaller than the correct one in this case.Step-by-Step Solution:Correct result = N * 7/4.Wrong result = N * 4/7.Ratio wrong / correct = (N * 4/7) / (N * 7/4) = (4/7) * (4/7) = 16/49.So wrong result is 16/49 of correct result.Percentage error = (correct - wrong) / correct * 100.This equals (1 - 16/49) * 100 = (33/49) * 100.Compute 33/49 * 100 ≈ 67.35 percent.Verification / Alternative check:Take N = 28 for easy computation. Correct result = 28 * 7/4 = 28 * 1.75 = 49. Wrong result = 28 * 4/7 = 16. Difference = 49 - 16 = 33. Percentage error = 33 / 49 * 100 ≈ 67.35 percent. This matches the algebraic method.Why Other Options Are Wrong:Values 206.25 percent and 103.13 percent correspond to other incorrect comparisons such as wrong over correct or using inverse fractions wrongly. The value 33.67 percent is roughly half of the correct error, again from using wrong ratios.Common Pitfalls:Many candidates compute the ratio correct / wrong or subtract fractions wrongly, leading to absurdly large or very small percentages. Some also forget that percentage error must be based on the correct value as the denominator.Final Answer:The percentage error in the calculation is approximately 67.35 percent.

Question 20

Two numbers P and Q are respectively 20 percent and 28 percent less than a number X. By what percentage is Q smaller than P?

Accepted Answer

Introduction / Context:This question checks the ability to compare two derived quantities that are each a given percentage less than a common base. The important point is to convert each percentage reduction into an explicit expression and then compare the resulting values directly.Given Data / Assumptions:P is 20 percent less than X.Q is 28 percent less than X.We need the percentage by which Q is smaller than P.X is a positive real number.Concept / Approach:First express P and Q in terms of X using percentage reduction. Then compute the difference P minus Q and express this difference as a percentage of P. That gives the percentage by which Q is smaller than P. Choosing a convenient value for X makes the arithmetic more straightforward.Step-by-Step Solution:Let X = 100 units for simplicity.P is 20 percent less than X, so P = 100 - 20 = 80 units.Q is 28 percent less than X, so Q = 100 - 28 = 72 units.Difference between P and Q = 80 - 72 = 8 units.Required percentage by which Q is smaller than P = (difference / P) * 100.So percentage = (8 / 80) * 100 = 10 percent.Verification / Alternative check:If we take X = 250, then P = 250 * 0.8 = 200 and Q = 250 * 0.72 = 180. Difference is 20. The percentage 20 / 200 * 100 = 10 percent, which again shows that Q is 10 percent smaller than P regardless of the particular value of X.Why Other Options Are Wrong:Options 23 percent, 12 percent and 13 percent do not match the exact ratio of the difference to P. They result from dividing by X or by Q instead of P or from misreading the percentage reductions.Common Pitfalls:Candidates sometimes subtract the two reduction percentages directly, saying 28 minus 20 equals 8 percent, which is not the answer being asked. The question wants the relative difference between P and Q, not between their reductions from X. Always check which quantities you are comparing.Final Answer:Q is 10 percent smaller than P.

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