Question 1

A fan is listed at Rs. 150 and is offered to customers at a first discount of 20%; what additional discount on the reduced price must be given so that the final net selling price becomes Rs. 108?

Accepted Answer

Introduction / Context: This question tests the concept of successive discounts. A product is first discounted by a certain percentage on its list price and then an additional discount is applied on the already reduced price. The final selling price is known, and we must determine the second discount. A key point is that successive discounts do not simply add the percentages, because each discount is applied on a different base price. Given Data / Assumptions: List price of the fan is Rs. 150. First discount is 20% on the list price. After both discounts, net selling price should be Rs. 108. We must find the second discount percentage applied on the reduced price after the first discount. The second discount is applied on the already reduced price, not on the original price. Concept / Approach: First we compute the price after the initial 20% discount. Then we assume the second discount rate as d percent on this reduced price and form an equation equating the final price to Rs. 108. From this equation we determine the value of d. Finally, we compare the obtained value with the options. If none of the options match exactly, the correct choice is "None of these". This shows the importance of calculating precisely rather than selecting approximate values. Step-by-Step Solution: Marked price (list price) of the fan is Rs. 150.First discount is 20%, so price after first discount = 150 * (1 - 0.20) = 150 * 0.8 = Rs. 120.Let the additional discount on Rs. 120 be d%.Then net price after second discount = 120 * (1 - d / 100).We are told that this final net price must be Rs. 108.So 120 * (1 - d / 100) = 108.Divide both sides by 120: 1 - d / 100 = 108 / 120 = 0.9.Thus d / 100 = 0.1 so d = 10%. Verification / Alternative check: Confirm with actual prices. After first discount, price is 120. If second discount is 10%, then discount amount is 10% of 120 that is 12, so final price is 120 - 12 = 108. This matches the required final selling price. Therefore the correct additional discount is exactly 10%. Since the options list 11 1/9%, 15%, 8%, and None of these, and 10% is not present, the only correct option is "None of these". Why Other Options Are Wrong: 11 1/9% would produce a second discount of about 13.33 on 120, resulting in a final price different from 108. 15% would reduce 120 by 18, giving a final price of 102, which is too low. 8% would reduce 120 by 9.6 to 110.4, which is higher than 108 and not acceptable. Common Pitfalls: A common mistake is to think that an overall discount from 150 to 108 is 42 rupees, which is 28% of 150, and then assume the second discount is 8% because 20% plus 8% equals 28%. This ignores the fact that the second discount applies to 120, not to 150. Another pitfall is misreading the question to think that the second discount is also on the list price. Always identify the exact base to which each percentage is applied and set up equations accordingly. Final Answer: The required additional discount is 10%, which is not listed among the numerical options, so the correct choice is None of these.

Question 2

In a college there are 1650 students and the difference between the number of boys and the number of girls is 400; what is the percentage of girls in the college?

Accepted Answer

Introduction / Context: This question is a straightforward application of solving simultaneous equations and then converting the result into a percentage. It involves a group of students separated into boys and girls with a known total and a known difference. These types of problems are very common in aptitude tests and help in practicing basic algebra and percentage conversion. Given Data / Assumptions: Total number of students in the college = 1650. The difference between the number of boys and the number of girls is 400. We assume the number of boys is greater than the number of girls, which is natural because the difference is stated without sign. We are asked for the percentage of girls in the college. Concept / Approach: Let the number of boys be B and the number of girls be G. We know B + G equals the total students and B - G equals the given difference. Solving these two linear equations gives us B and G. After that, the percentage of girls is (G / total) * 100. This approach is standard for all such total and difference type questions that appear in exams. Step-by-Step Solution: Let the number of boys be B and the number of girls be G.We are given B + G = 1650.We are also given B - G = 400.Add the two equations: (B + G) + (B - G) = 1650 + 400.This gives 2B = 2050, so B = 2050 / 2 = 1025.Substitute B = 1025 into B + G = 1650 to find G.So 1025 + G = 1650 implies G = 1650 - 1025 = 625.Percentage of girls = (G / total) * 100 = (625 / 1650) * 100.Compute 625 / 1650 which is approximately 0.3788, so percentage is about 37.88%.Rounded to the nearest whole percent, this is 38%. Verification / Alternative check: Check the numbers with the original conditions. Boys = 1025, girls = 625. The total is 1025 + 625 = 1650, which matches the given total students. The difference is 1025 - 625 = 400, which matches the stated difference. The percentage of girls is 625 divided by 1650, multiplied by 100, giving close to 37.88%. In exam options, we choose the nearest value, which is 38%, confirming our calculation. Why Other Options Are Wrong: 49% would mean girls are roughly 808, but that would give a different difference between boys and girls from the given 400. 34% and 43% correspond to different student counts for girls and cannot satisfy both the total and the difference equations simultaneously. Common Pitfalls: Some candidates directly assume boys are 1025 and girls are 625 by guesswork or miscalculate the division when converting to percentage. Others might mistakenly compute the percentage of boys instead of girls or mix up the total and difference equations. Writing equations clearly and solving step by step, followed by a quick check of totals and differences, is the best way to avoid errors. Final Answer: The percentage of girls in the college is approximately 38%.

Question 3

A shepherd had n goats in the year 2000; in 2001 the number increased by 40%, in 2002 it decreased to 70% of that, in 2003 it grew again by 30%, and in 2004 he sold 10% of the goats leaving him with 34,398 goats; what was the overall percentage incr…

Accepted Answer

Introduction / Context: This question is about successive percentage increases and decreases applied to a population of goats over several years. After a sequence of changes, a certain number of goats remain. From this final value we must infer the initial number and then compute the net percentage increase over the whole period. Such problems are excellent practice for understanding compound effects of percentages and are common in topics like population growth, interest, and depreciation. Given Data / Assumptions: Initial number of goats in year 2000 is n. In 2001, number increases by 40% (multiplied by 1.4). In 2002, number decreases to 70% of that year 2001 value (multiplied by 0.7). In 2003, number increases by 30% (multiplied by 1.3). In 2004, 10% of goats are sold, leaving 90% of the previous count (multiplied by 0.9). After all these changes, the final number of goats is 34,398. We must find the overall percentage increase from n to 34,398. Concept / Approach: Successive percentage changes can be represented by successive multiplication factors. The overall effect is the product of these factors. If the overall factor times n gives the final number, we can find n by dividing the final number by that factor. Once we know both the initial and final values, we compute net percentage increase by the formula: (final - initial) / initial * 100. This method avoids doing year by year tracking in absolute numbers and instead uses multiplication factors efficiently. Step-by-Step Solution: Increase of 40% in 2001 corresponds to multiplying by 1.4.Decrease to 70% in 2002 corresponds to multiplying by 0.7.Increase of 30% in 2003 corresponds to multiplying by 1.3.Sale of 10% in 2004 corresponds to keeping 90%, so multiply by 0.9.Overall multiplier from 2000 to final is 1.4 * 0.7 * 1.3 * 0.9.Compute stepwise: 1.4 * 0.7 = 0.98, 0.98 * 1.3 = 1.274, and 1.274 * 0.9 = 1.1466.Thus final number = 1.1466 * n = 34,398.So n = 34,398 / 1.1466 = 30,000.Net increase = final - initial = 34,398 - 30,000 = 4,398 goats.Percentage increase = (4,398 / 30,000) * 100 = 14.66%. Verification / Alternative check: We can verify by moving forward from 30,000 goats year by year. After 40% increase: 30,000 * 1.4 = 42,000. After 2002 decrease to 70%: 42,000 * 0.7 = 29,400. After 30% increase in 2003: 29,400 * 1.3 = 38,220. After selling 10% in 2004, remaining goats are 38,220 * 0.9 = 34,398. This matches the given final number perfectly. Calculating net increase again: increase is 4,398 goats on 30,000, giving a percentage of 4,398 / 30,000 * 100 = 14.66%, consistent with the earlier calculation. Why Other Options Are Wrong: 16.66% and 20% are higher than the correct value and would require a larger final number relative to 30,000. 33.33% implies a much larger growth factor that does not match the sequence of mixed increase and decrease percentages provided in the problem. Common Pitfalls: One common mistake is to simply add and subtract percentages, for example 40% minus 30% minus 10% and so on, rather than multiplying the appropriate factors. Another mistake is to treat the decrease to 70% as a 30% decrease on the original 30,000 instead of the intermediate value. To avoid these errors, write each year change as a clear multiplication factor and then multiply them. When the final value is given, always solve back for the initial and compute the net percentage correctly. Final Answer: The overall percentage increase in the number of goats over the entire duration is 14.66%.

Question 4

Subbu purchases a tape recorder at nine-tenths of its selling price and then sells it at 8% more than its selling price; what is his gain percentage on the cost price?

Accepted Answer

Introduction / Context: This problem is a variation on profit and loss where the cost price and selling price are defined in terms of a listed selling price. Subbu first buys the tape recorder cheaper than this selling price and then sells it for more than that selling price. The question asks for his profit percentage based on his actual cost. This type of question helps students understand how to manipulate ratios and percentages even when actual rupee values are not given explicitly. Given Data / Assumptions: Let the listed selling price be S rupees. Subbu purchases the tape recorder at 9/10 of S, so his cost price is (9/10) * S. He sells it at 8% more than the selling price S, that is at 1.08 * S. We need to find the profit percentage based on his cost price. Concept / Approach: Profit percentage is calculated as (Profit / Cost Price) * 100. Here, instead of working with actual rupees, we represent everything in terms of S. We compute his cost price, compute his final selling price, take the difference as profit, and then divide this profit by cost price to get the profit percentage. This approach shows that the actual numeric value of S is irrelevant for the percentage as long as cost and selling price are expressed in consistent terms. Step-by-Step Solution: Let the given selling price be S rupees.Subbu buys the tape recorder at 9/10 of S, so cost price CP = (9/10) * S = 0.9 * S.He sells it at 8% more than S, so final selling price SP = 1.08 * S.Profit = SP - CP = 1.08 * S - 0.9 * S = 0.18 * S.Profit percentage = (Profit / CP) * 100 = (0.18 * S / 0.9 * S) * 100.Simplify 0.18 / 0.9 = 0.2.Thus profit percentage = 0.2 * 100 = 20%. Verification / Alternative check: For a numeric check, choose S as Rs. 1000. Then Subbu purchases the tape recorder at 9/10 of 1000, which is Rs. 900. He sells it at 8% more than 1000, that is 1.08 * 1000 = Rs. 1080. Profit is 1080 - 900 = 180. Profit percentage is 180 / 900 * 100 = 20%. This matches perfectly with the algebraic solution, confirming that the profit percentage is 20% independent of the actual value of S. Why Other Options Are Wrong: 9% and 10% are significantly smaller than the derived 20% and result from incorrect ratios such as 0.18 divided by 2 or dividing by 1.8 instead of 0.9. 180% is far too large and would require profit to be almost double the cost price, which is not the case here. Common Pitfalls: Students sometimes misread the phrase "9/10 of its selling price" and set cost price equal to 9/10 of the final selling price instead of the listed one. Another error is to compare profit with S rather than with the cost price. Remember that profit percentage is always calculated with respect to cost price, not with respect to selling price, unless a question explicitly asks for something else. Carefully expressing both cost and selling prices in terms of S before computing their difference helps avoid such mistakes. Final Answer: Subbu gain on the tape recorder is 20% of his cost price.

Question 5

A man spends 20% of his monthly salary on his children education, 50% of his salary on food and transport, pays house rent of Rs. 1000, and is left with Rs. 2000; what is his total monthly salary?

Accepted Answer

Introduction / Context: This is a standard salary distribution problem that tests the ability to interpret percentage expenses and fixed expenses together. The man spends parts of his salary in percentages on certain categories, pays a fixed rent amount and has a known remaining balance. We must reconstruct his total salary from this information. Such questions are common in the percentage and arithmetic sections of aptitude exams. Given Data / Assumptions: Let the monthly salary be S rupees. 20% of S is spent on children education. 50% of S is spent on food and transport. Rent is Rs. 1000. After all these expenses, he is left with Rs. 2000. We need to find S. Concept / Approach: The total salary must equal the sum of all expenses plus the remaining amount after expenses. The expenses include two percentage based items and one fixed amount. We first express the percentage based expenses in terms of S, then add them along with the fixed rent and the final remaining balance. This sum must equal S. Solving the resulting linear equation gives the value of S. This method is clear and avoids miscounting any part of the salary. Step-by-Step Solution: Let monthly salary be S rupees.Spending on children education = 20% of S = 0.20 * S.Spending on food and transport = 50% of S = 0.50 * S.Rent is Rs. 1000.Remaining amount after expenses is Rs. 2000.Total of all outflow and remaining amount must equal S.So equation is 0.20S + 0.50S + 1000 + 2000 = S.Combine like terms: 0.70S + 3000 = S.Rearrange to get S - 0.70S = 3000, so 0.30S = 3000.Hence S = 3000 / 0.30 = 10,000. Verification / Alternative check: Check the distribution using S = 10,000. Education spending is 20% of 10,000, which is 2000. Food and transport spending is 50% of 10,000, which is 5000. Rent is 1000. Total expenses are 2000 + 5000 + 1000 = 8000. The remaining amount is 10,000 - 8000 = 2000, which matches the given remaining amount. This confirms that a salary of Rs. 10,000 per month is consistent with all the information given in the question. Why Other Options Are Wrong: A salary of Rs. 7000 would not allow both percentage allocations and fixed rent while leaving Rs. 2000, because 70% alone would already be 4900 leaving too little for rent and savings. Rs. 14,000 or Rs. 20,000 would leave larger remaining amounts than Rs. 2000 once all expenses are calculated, so they cannot satisfy the final condition. Common Pitfalls: Some students add the percentage figures incorrectly or neglect the fixed rent when forming the equation. Others mistakenly assume that 20% plus 50% plus the remaining 2000 must directly equal the salary, ignoring the rent. To avoid such errors, always write an explicit equation where salary equals the sum of all expenditures and the final remaining amount, and then solve carefully for S. Final Answer: The man total monthly salary is Rs. 10,000.

Question 6

Eighty five is what percentage of 352, expressed to two decimal places?

Accepted Answer

Introduction / Context: This question is a basic percentage calculation where a smaller number is compared to a larger one, and we are asked what percentage the smaller number represents of the larger number. Such direct fraction to percentage conversions are very common in aptitude and data interpretation sections, and fluency with them helps in tackling more complex ratio and percentage problems later. Given Data / Assumptions: Smaller number is 85. Larger number is 352. We must find what percentage 85 is of 352. The answer should be accurate to two decimal places, as seen in the options. Concept / Approach: To find what percentage one number is of another, we use the formula: Percentage = (part / whole) * 100. Here, 85 is the part and 352 is the whole. After forming the fraction 85 / 352, we convert it to a decimal and multiply by 100 to get the percentage. Finally, we compare this with the options to select the closest matching value. Step-by-Step Solution: We need to compute (85 / 352) * 100.First, calculate the fraction 85 / 352.This division gives approximately 0.241477 when rounded to six decimal places.Multiply this by 100 to convert to a percentage: 0.241477 * 100 = 24.1477.Rounded to two decimal places, this is 24.14%.Therefore, 85 is approximately 24.14% of 352. Verification / Alternative check: For a quick check, estimate using nearby proportions. Half of 352 is 176, so 85 is slightly less than half of 176, and half of 352 is 50%. Since 85 is much less than 176 and a little less than 88 (which would be exactly one quarter of 352), it is reasonable that the percentage is slightly less than 25%. Among the options, 24.14% is the only value just under 25%, while the others are either far below or above in comparison. This rough reasoning supports the calculated answer. Why Other Options Are Wrong: 21.5% and 23.68% are both lower than 24.14% and do not match the exact division result. 25.47% is higher than the true value and would correspond to a larger part than 85 for the same whole of 352. Common Pitfalls: Sometimes students inadvertently compute 352 as a percentage of 85 instead of 85 as a percentage of 352, leading to the reciprocal fraction. Another mistake is to round off too early in the calculation, which can slightly shift the percentage. It is better to keep sufficient decimal places during intermediate calculations and only round to two decimal places at the end, especially when answer options are close to each other. Final Answer: Eighty five is approximately 24.14% of 352.

Question 7

A batsman scores 180 runs in an innings, including 11 boundaries and 8 sixes; what percentage of his total score does he make by running between the wickets?

Accepted Answer

Introduction / Context: This question comes from cricket based percentage problems in aptitude exams. A batsman score consists of runs from boundaries and sixes plus runs made by physically running between the wickets. The goal is to calculate the proportion of his total score that comes from running. This type of problem checks understanding of decomposition of a total into parts and then converting one part into a percentage of the whole. Given Data / Assumptions: Total runs scored by the batsman = 180. He hits 11 boundaries, each boundary gives 4 runs. He hits 8 sixes, each six gives 6 runs. All remaining runs are made by running between the wickets. We must find the percentage of total runs that come from running. Concept / Approach: The total score is the sum of runs scored via boundaries, sixes, and running. We first compute the runs earned from boundaries and from sixes. The difference between the total score and this sum gives the runs made by running between the wickets. Then, to find the percentage contribution of running, we divide these running runs by total runs and multiply by 100. This is an application of the basic percentage formula in a sports context. Step-by-Step Solution: Runs from boundaries = number of boundaries * 4 = 11 * 4 = 44.Runs from sixes = number of sixes * 6 = 8 * 6 = 48.Total runs from boundaries and sixes = 44 + 48 = 92.Total score is 180 runs.Runs made by running between wickets = total runs - runs from boundaries and sixes = 180 - 92 = 88.Now compute the percentage of runs scored by running: (88 / 180) * 100.88 / 180 simplifies to 0.4888 recurring approximately.Multiplying by 100 gives approximately 48.88%. Verification / Alternative check: To verify quickly, note that half of 180 is 90, so 88 is very close to half of his total runs. Therefore, the percentage should be very close to 50%. Among the options, 48.88% is just under 50% and matches our calculated value. This sanity check with approximate values helps confirm that the detailed arithmetic performed is correct and no calculation step has been missed or misapplied. Why Other Options Are Wrong: 51.42% and 62.45% are both above 50%, implying that more than half or much more than half of the runs are scored by running, which is inconsistent with 88 being slightly less than 90. 47.32% is somewhat close to 48.88% but still not equal to the accurate ratio of 88 out of 180, which is exactly 0.4888 recurring. Common Pitfalls: Some students mistakenly add up 11 and 8 to get 19 and then treat 19 as runs instead of boundaries or sixes. Others forget to multiply by 4 and 6 respectively or miscalculate the total runs from boundaries and sixes. Another common error is to divide the total by the running runs instead of running runs by total. Always clearly separate each contribution to the total score and double check the arithmetic for the multiplication and subtraction steps before converting to a percentage. Final Answer: The batsman scores 48.88% of his total runs by running between the wickets.

Question 8

The price of pulses increases by 25%; by what percentage should a family reduce its consumption of pulses so that its total expenditure on pulses remains unchanged?

Accepted Answer

Introduction / Context: This question is another example of how price and quantity interact to determine expenditure. When the price of a commodity rises, a family that wants to keep expenditure unchanged must reduce quantity consumed in a specific proportion. Here the price of pulses has increased by 25% and we must calculate the required percentage reduction in consumption to keep the overall spending constant. Given Data / Assumptions: Let the original price of pulses be P per unit. Let the original quantity consumed be Q units. Original expenditure is P * Q. Price increases by 25%, so new price is 1.25 * P. The family wants the expenditure on pulses to remain the same as before. Let the new quantity consumed be q units. Concept / Approach: The fundamental relation is Expenditure = Price * Quantity. If the price increases and expenditure stays constant, quantity must decrease accordingly. We express both original and new expenditure in terms of P, Q and q, then equate them. Solving for q in terms of Q gives the factor by which quantity must be adjusted. The percentage reduction in consumption is then found by comparing Q and q. Step-by-Step Solution: Original expenditure = P * Q.After price increase, new price = 1.25 * P.Let new quantity be q, so new expenditure = 1.25 * P * q.Since expenditure is to remain the same, we set 1.25 * P * q = P * Q.Cancel P from both sides to get 1.25 * q = Q.So q = Q / 1.25.Write 1.25 as 5 / 4, hence q = Q * (4 / 5) = 0.8 * Q.This means new quantity is 80% of the original quantity.Therefore, percentage reduction in consumption = 100% - 80% = 20%. Verification / Alternative check: Take a simple example. Suppose original price P is Rs. 100 per unit and quantity consumed Q is 1 unit. Expenditure is Rs. 100. After a 25% price increase, new price becomes 125 per unit. To keep expenditure at Rs. 100, the family can now buy q = 100 / 125 = 0.8 units. This is exactly 80% of the original quantity, indicating a 20% reduction. This consistent result confirms the algebraic solution and the final percentage reduction. Why Other Options Are Wrong: 16% and 72% are unrelated to the actual ratio required to offset a 25% price increase while keeping expenditure constant. 84% would mean the family almost stops consuming pulses, which clearly does not correspond to just a 25% increase in price with unchanged expenditure. Common Pitfalls: Many students mistakenly think that a 25% increase in price means a 25% decrease in consumption to keep expenditure fixed, but this is incorrect because the product of price and quantity must remain the same. Others incorrectly compute the new quantity factor as 1 / (1 + 0.25) and then fail to convert it properly into a percentage reduction. Always set up the equation with original and new expenditures equal and carefully solve for the new quantity before converting to a percent change. Final Answer: The family must reduce its consumption of pulses by 20% to keep expenditure unchanged.

Question 9

In measuring the sides of a rectangle, one side is taken 20% in excess of its actual length and the other side is taken 10% less than its actual length; what is the percentage error in the calculated area compared with the true area?

Accepted Answer

Introduction / Context: This question deals with the effect of errors in measurements on the calculated area of a rectangle. Small percentage errors in each dimension can combine to produce a net percentage error in the area that is not simply the sum or difference of those percentages. Such problems commonly appear in the error and approximation section of quantitative aptitude and help build an understanding of how multiplication of dimensions affects area estimates. Given Data / Assumptions: Let the true length of one side be L. Let the true breadth of the rectangle be B. Area of the rectangle is L * B. In measurement, length is overestimated by 20%, so measured length is 1.2 * L. In measurement, breadth is underestimated by 10%, so measured breadth is 0.9 * B. We must find the percentage error in the area calculated using these wrong measurements. Concept / Approach: The true area is L * B. The wrong measured area is (1.2 * L) * (0.9 * B). The ratio of wrong area to true area gives the factor by which the area is overestimated or underestimated. Percentage error can be found by subtracting 1 from this ratio and multiplying by 100. If the ratio is greater than 1, the error is excess; if the ratio is less than 1, the error is deficit. Step-by-Step Solution: True area A is L * B.Measured length is 1.2 * L due to 20% excess.Measured breadth is 0.9 * B due to 10% deficit.Measured area Am is (1.2 * L) * (0.9 * B) = 1.2 * 0.9 * L * B.Compute multiplier: 1.2 * 0.9 = 1.08.So measured area Am = 1.08 * L * B = 1.08 * A.Thus the measured area is 1.08 times the true area.Percentage error = (Am - A) / A * 100 = (1.08A - A) / A * 100 = 0.08 * 100 = 8%.Since the factor is greater than 1, the error is 8% in excess. Verification / Alternative check: Take an easy numeric example. Suppose true length L is 10 units and true breadth B is 10 units. True area is 10 * 10 = 100 square units. Measured length would be 20% greater, so 12 units. Measured breadth would be 10% less, so 9 units. Measured area becomes 12 * 9 = 108 square units. The difference is 108 - 100 = 8 square units. Percentage error is 8 / 100 * 100 = 8%, and since the measured area is larger, this is an 8% excess. This simple example confirms the general calculation. Why Other Options Are Wrong: 9% deficit and 11% deficit both assume the measured area is smaller than the true area, which contradicts the computed factor of 1.08. 12% excess would require the multiplier to be 1.12 rather than 1.08, which is not the result of multiplying 1.2 and 0.9. Common Pitfalls: A common error is to simply add the percentage changes, for example 20% minus 10%, and conclude that the net error is 10%, which is wrong. Another mistake is to treat the wrong measurements separately and not realize that area is the product of both length and breadth. The correct method is always to convert percentage changes into multiplicative factors, multiply them, and then interpret the final factor relative to 1 to determine both the magnitude and direction of the error. Final Answer: The calculated area is in 8% excess compared with the true area.

Question 10

Twelve percent of Kalyan monthly salary is equal to sixteen percent of Nandu monthly salary, and Sarnesh monthly salary is half of Nandu monthly salary; if Sarnesh annual salary is Rs. 1.08 lacs, what is Kalyan monthly salary?

Accepted Answer

Introduction / Context: This salary based problem tests simultaneous use of percentage relationships and simple ratios. Three persons, Kalyan, Nandu, and Sarnesh, have salaries linked by given conditions. We are told the annual salary of one person and asked to find the monthly salary of another. This is typical of algebra and percentage problems in aptitude tests and trains students to translate word relations into equations. Given Data / Assumptions: Let Kalyan monthly salary be K. Let Nandu monthly salary be N. Let Sarnesh monthly salary be S. Twelve percent of K equals sixteen percent of N. Sarnesh monthly salary is half of Nandu salary, so S = N / 2. Sarnesh annual salary is Rs. 1.08 lacs, that is Rs. 108,000 per year. We must find the value of K in rupees per month. Concept / Approach: The approach is to use the annual salary information of Sarnesh to first determine his monthly salary, and from that find Nandu monthly salary. With N determined, we then use the percentage relation between K and N to solve for K. The key equation for K and N is 12% of K equals 16% of N, which can be written as 0.12 * K = 0.16 * N. Solving this gives K as a multiple of N, which we then evaluate using the value of N from Sarnesh salary relation. Step-by-Step Solution: Annual salary of Sarnesh is Rs. 108,000.His monthly salary S is 108,000 divided by 12, so S = 108,000 / 12 = 9,000.We are told S = N / 2, so Nandu monthly salary N = 2 * S = 18,000.Relation between K and N is 12% of K equals 16% of N.Write this as 0.12 * K = 0.16 * N.Rearrange for K: K = (0.16 / 0.12) * N.Compute 0.16 / 0.12 = 4 / 3.So K = (4 / 3) * N = (4 / 3) * 18,000 = 24,000. Verification / Alternative check: Verify all conditions with K = 24,000 and N = 18,000. Twelve percent of K is 0.12 * 24,000 = 2,880. Sixteen percent of N is 0.16 * 18,000 = 2,880. The two values match, confirming the relation between Kalyan and Nandu. For Sarnesh, Nandu salary is 18,000, so half of that is 9,000, which is the monthly salary we obtained by dividing his annual salary 108,000 by 12. All relations are satisfied, which confirms that K = 24,000 is correct. Why Other Options Are Wrong: 15,000, 36,000, and 12,500 do not satisfy the equation 0.12 * K = 0.16 * N when N is 18,000. For each of these values, either the percentage equality fails or they are not in the correct ratio compared to N salary as determined from Sarnesh income. Common Pitfalls: Some students mistakenly use annual salaries directly in the percentage relations without converting them to monthly values, which mixes time units and breaks the logic. Others mis-handle the fraction 0.16 / 0.12 and get 1.2 or another incorrect ratio. It is always good practice to write the fraction in simplest integer form, here 16 / 12 simplifies to 4 / 3, before applying it to the known value of N. Consistent units and careful fraction simplification help avoid such mistakes. Final Answer: Kalyan monthly salary is Rs. 24,000.

Question 11

Last year, Vivek saved 10% of his annual income. This year, he earned 5% more than last year and saved 12% of his annual income. The amount saved this year was what percentage of the amount saved last year?

Accepted Answer

Introduction / Context: This question tests percentage comparison between two years, focusing on how a change in income and a change in saving percentage together affect the amount saved. Such problems are common in aptitude exams because they combine basic percentage increase with percentage of a quantity, and then ask you to express one amount as a percentage of another. Given Data / Assumptions: Last year, Vivek saved 10% of his annual income. This year, his annual income increased by 5% compared to last year. This year, he saved 12% of his annual income. We assume income is positive and there are no other changes. We need the amount saved this year as a percentage of the amount saved last year. Concept / Approach: The key idea is to use a convenient assumed value for last year's income, convert the given percentages to actual amounts, and then compare the new saving with the old saving. Because everything is in percentages, we can assume last year's income as 100 units (or any other value). Then we apply the income increase and the saving percentages. Finally, we compute the ratio: (saving this year) / (saving last year) and convert this ratio into a percentage. Step-by-Step Solution: Step 1: Assume last year's income = I. Step 2: Last year's saving = 10% of I = 0.10 * I. Step 3: This year's income increased by 5%, so new income = 1.05 * I. Step 4: This year, Vivek saves 12% of his new income, so saving this year = 0.12 * (1.05 * I) = 0.126 * I. Step 5: Now compare the two savings. Ratio = (saving this year) / (saving last year) = (0.126 * I) / (0.10 * I). Step 6: The I cancels out, so ratio = 0.126 / 0.10 = 1.26. Step 7: A ratio of 1.26 means the new saving is 1.26 times the old saving, that is 126% of the previous amount. Verification / Alternative check: Take I = 100 as a concrete number. Then last year he saved 10% of 100 = 10 units. This year income is 5% more, so 105, and he saves 12% of that, that is 12.6 units. Now express 12.6 as a percentage of 10: (12.6 / 10) * 100 = 126%. This matches our earlier result, so the calculation is consistent. Why Other Options Are Wrong: 87% would indicate that this year's saving is less than last year's, which is impossible because both income and saving percentage increased. 98% suggests almost no change, which again contradicts the combined effect of 5% higher income and a higher saving rate. 138% would mean the saving increased a lot more than it actually did; the exact ratio is 1.26, not 1.38. Common Pitfalls: Many learners mistakenly add percentages directly (for example, 10% saving becoming 12% saving and income rising by 5%) and try to combine them incorrectly. Some also forget that a 5% increase in income affects the base on which the saving percentage is calculated. Others simply compare 12% with 10% and say 120% instead of carefully computing the effect of the higher base income. Always translate the words into algebra or numbers systematically. Final Answer: The amount saved this year is 126% of the amount saved last year.

Question 12

The marked price of a watch is Rs. 8400. It is sold for Rs. 6896 after two successive discounts. If the first discount is 12%, what is the percentage of the second discount?

Accepted Answer

Introduction / Context: This is a successive discount problem that tests understanding of how multiple percentage discounts work on a marked price. Instead of simply adding or subtracting percentages, we must apply each discount step by step on the current price. This is a common type of question in commercial arithmetic and shopkeeper related aptitude problems. Given Data / Assumptions: Marked price (MP) of the watch = Rs. 8400. First discount = 12% on the marked price. After both discounts, the selling price (SP) becomes Rs. 6896. The second discount is applied on the already reduced price after the first discount. We must find the second discount percentage. Concept / Approach: Successive discounts are applied multiplicatively, not additively. The first discount reduces the marked price to MP * (1 - first discount). The second discount then reduces this new price further to a final selling price. To find the missing second discount, compute the price after the first discount, then set up an equation where that price multiplied by (1 - second discount) equals the final selling price. Solve for the second discount rate and convert it to a percentage. Step-by-Step Solution: Step 1: Marked price MP = 8400. Step 2: First discount = 12%, so price after first discount = 8400 * (1 - 0.12) = 8400 * 0.88. Step 3: Compute 8400 * 0.88 = 7392. So, reduced price after the first discount = Rs. 7392. Step 4: Let the second discount be d (expressed as a decimal). Then final selling price SP = 7392 * (1 - d). Step 5: We are given SP = 6896. Set up the equation 7392 * (1 - d) = 6896. Step 6: Divide both sides by 7392: 1 - d = 6896 / 7392. Step 7: 6896 / 7392 is approximately 0.9329. Hence d = 1 - 0.9329 = 0.0671 approximately. Step 8: Convert to percentage: d% = 0.0671 * 100 ≈ 6.71%. The closest option is 6.709%. Verification / Alternative check: We can verify by taking the approximate second discount 6.709%. Compute the final selling price again: 7392 * (1 - 0.06709) = 7392 * 0.93291 ≈ 6896. This matches the given selling price, so the computed second discount is correct. Small rounding differences are expected due to decimal approximation, but the option 6.709% fits this calculation very closely. Why Other Options Are Wrong: 7.2%, 9.14% and 7.13% all give final prices that differ more significantly from Rs. 6896 when applied after the first discount. They either reduce the price too much or not enough. Only 6.709% produces a selling price that agrees with the given value within normal rounding tolerance. Common Pitfalls: A frequent error is to treat successive discounts as simply 12% + x% and compare that total to the overall reduction from 8400 to 6896. That approach ignores that each discount is applied on a different base value. Another mistake is to round intermediate values too aggressively, which can push the final percentage away from the correct option. Always apply discounts sequentially and keep at least four decimal places during calculation when multiple percentages are involved. Final Answer: The second discount rate is approximately 6.709%.

Question 13

A wooden duster has dimensions 5 cm × 5 cm × 7 cm and is gently lowered into a cylindrical vessel of water with radius 5 cm and height 10 cm. After inserting the duster, the water level rises by 2.14 cm. What percentage of the volume of the duster r…

Accepted Answer

Introduction / Context: This question involves volume displacement and percentage comparison. By placing a solid object (the duster) into water, some part of the object is submerged and displaces water, which raises the water level. Using the concept that the volume of water displaced equals the submerged volume of the object, we can determine what fraction of the object is under water and therefore what fraction remains above the surface. Given Data / Assumptions: Dimensions of the duster: 5 cm × 5 cm × 7 cm. Shape of the container: a cylinder with radius 5 cm and height 10 cm. Rise in water level after inserting the duster = 2.14 cm. The duster floats partially submerged; it is not completely sunk to the bottom in the sense of being fully under water. We neglect any water spillage or absorption and assume perfect displacement. Concept / Approach: The principle of displacement tells us that the volume of water displaced equals the volume of the object that is submerged. The rise in water level within a cylindrical vessel directly gives the displaced volume as base area * height rise. Once we know the submerged volume, we compare it with the total volume of the duster to find the submerged fraction. The remaining fraction, multiplied by 100, gives the percentage of the duster that remains above the water surface. Step-by-Step Solution: Step 1: Calculate total volume of the duster. Volume of duster = 5 * 5 * 7 = 175 cubic centimetres. Step 2: Calculate volume of water displaced. Base area of cylinder = pi * r^2 = pi * 5^2 = 25 * pi square centimetres. Height rise = 2.14 cm. Displaced volume = base area * height rise = 25 * pi * 2.14 cubic centimetres. Step 3: Approximate pi as 3.14. Displaced volume ≈ 25 * 3.14 * 2.14 ≈ 168 cubic centimetres (approximately). Step 4: This displaced volume equals the submerged volume of the duster, so submerged volume ≈ 168 cubic centimetres. Step 5: Fraction submerged = 168 / 175 ≈ 0.96, that is 96%. Step 6: Therefore, fraction above the water = 1 - 0.96 = 0.04, which is 4%. Verification / Alternative check: We can verify the fraction using exact symbolic reasoning. Let submerged fraction be x. Then x * (volume of duster) = volume of displaced water. So x * 175 = 25 * pi * 2.14. Solve for x: x = (25 * pi * 2.14) / 175. Note that 25 / 175 = 1 / 7, so x = (pi * 2.14) / 7. Taking pi ≈ 3.14, x ≈ (3.14 * 2.14) / 7 ≈ 6.7196 / 7 ≈ 0.96. This again shows that 96% of the duster volume is submerged and 4% remains above water. Why Other Options Are Wrong: 96% would represent the proportion submerged, not the part above the water. The question explicitly asks for the portion above the layer, that is the complement. 97% and 3% do not correspond to the accurate calculations and would require a different displaced volume than the one derived from the given rise of 2.14 cm. Common Pitfalls: A common mistake is to interpret the percentage as the submerged part instead of the part above the water. Another error is in calculating the displaced volume, either by forgetting to square the radius in the area formula or by misusing the value of pi. Some students also confuse the absolute volume difference (175 - 168) with the percent directly, without dividing by the total volume first. Always work systematically: compute volumes, find the fraction, then convert to a percentage and interpret correctly in context. Final Answer: Approximately 4% of the duster's volume remains above the water surface.

Question 14

In a village in India, 5% of the people died due to dengue. Of the remaining population, 15% left the village out of fear. If the population after these events is now 3553, what was the population of the village at the beginning?

Accepted Answer

Introduction / Context: This problem involves successive percentage decreases on a population. First, a certain percentage of people die, and then a percentage of the survivors leave the village. After both of these reductions, the final population is known. The goal is to reverse these steps and determine the original population before any changes happened. This is a classic reverse percentage problem. Given Data / Assumptions: 5% of the people died due to dengue. 15% of the remaining people left the village. The final population after both changes is 3553. We assume the population counts are whole numbers and that only these two events affected the population. Concept / Approach: When a quantity is reduced by a percentage, the remaining part is (100 - percentage)% of the original. When two percentage changes are applied successively, the overall effect is multiplicative. To find the original value from the final value, we divide by the net remaining factor. Here we first calculate the remaining fraction after each step, multiply these to get the overall remaining fraction, and then divide the final population by that fraction to obtain the initial population. Step-by-Step Solution: Step 1: Let the initial population be P. Step 2: First, 5% die. So remaining population after deaths = 95% of P = 0.95 * P. Step 3: Next, 15% of the remaining people leave the village. Step 4: Remaining fraction after this second step = 100% - 15% = 85% of the current population. Step 5: Thus, final population = 0.85 * (0.95 * P) = 0.8075 * P. Step 6: We are told final population = 3553, so 0.8075 * P = 3553. Step 7: Solve for P: P = 3553 / 0.8075. Step 8: Carry out the division: 3553 / 0.8075 = 4400. Step 9: Therefore, the initial population of the village was 4400. Verification / Alternative check: Start with 4400 and apply the given reductions to confirm. First 5% die: 5% of 4400 = 0.05 * 4400 = 220; remaining = 4400 - 220 = 4180. Next, 15% of 4180 leave: 0.15 * 4180 = 627. Remaining = 4180 - 627 = 3553. This matches the given final population, confirming the calculation is correct. Why Other Options Are Wrong: 3100 and 2800 are too low; if we apply the same percentage reductions to them, the result is far below 3553. 5600 is too high; after the same successive percentage drops, the final population would exceed 3553 significantly. Only 4400 reproduces the exact final population of 3553 when the given percentages are applied. Common Pitfalls: One common mistake is to add the percentages and assume a single 20% reduction (5% + 15%), which is incorrect because the second percentage is applied on a reduced base. Another mistake is trying to work from the beginning rather than from the end; reversing the process by dividing by the net remaining fraction is more straightforward. Also, some learners forget to convert percentages into decimals properly before multiplying or dividing. Final Answer: The original population of the village was 4400.

Question 15

In a school with 1500 boys, 35% are Hindus, 30% are Sikhs, 15% are Muslims and the remaining are Christians. How many Christian boys are there in the school?

Accepted Answer

Introduction / Context: This question is a straightforward percentage distribution problem. The total number of boys in a school is given, along with the percentage distribution across different religious groups. We are asked to find how many boys belong to the Christian group, which is the remaining percentage after accounting for the other groups. Given Data / Assumptions: Total number of boys in the school = 1500. 35% of the boys are Hindus. 30% of the boys are Sikhs. 15% of the boys are Muslims. The remaining boys are Christians. We assume the percentages cover all boys exactly. Concept / Approach: The central idea is that the sum of all percentage groups must be 100%. First, we add the given percentages for Hindus, Sikhs and Muslims to find their combined share. Then we subtract this sum from 100% to get the percentage of Christian boys. Finally, we apply this percentage to the total number of boys to find the required count. This is a typical application of percentage and complementary values in basic arithmetic. Step-by-Step Solution: Step 1: Total boys = 1500. Step 2: Percentage of Hindu boys = 35%. Step 3: Percentage of Sikh boys = 30%. Step 4: Percentage of Muslim boys = 15%. Step 5: Combined percentage of Hindu, Sikh and Muslim boys = 35% + 30% + 15% = 80%. Step 6: Since all boys must belong to one of these groups, the percentage of Christian boys = 100% - 80% = 20%. Step 7: Number of Christian boys = 20% of 1500 = 0.20 * 1500. Step 8: Compute 0.20 * 1500 = 300. Step 9: Therefore, there are 300 Christian boys in the school. Verification / Alternative check: We can check by computing the number in each group. Hindu boys = 35% of 1500 = 525. Sikh boys = 30% of 1500 = 450. Muslim boys = 15% of 1500 = 225. Christian boys (if 300) give a total of 525 + 450 + 225 + 300 = 1500, which matches the total boys given. Hence the distribution is consistent and the answer is correct. Why Other Options Are Wrong: If there were 250 or 200 Christian boys, the total count after adding all religious groups would be less than 1500. If there were 350 Christian boys, the total would exceed 1500. Therefore, only 300 satisfies both the percentage and total count conditions. Common Pitfalls: Students sometimes incorrectly add percentages as 35 + 30 + 15 + required = 1500, confusing percentage values with absolute counts. Another frequent error is to mistakenly take the Christian percentage as one of the listed numbers rather than computing it as the complement of the sum of the other percentages. Always remember that percentages should sum to 100% when all groups are included. Final Answer: The number of Christian boys in the school is 300.

Question 16

Evaluate the expression 21% of 500 plus 31% of 1100 minus 30% of 790. What is the value of this numerical expression?

Accepted Answer

Introduction / Context: This is a direct numerical computation involving multiple percentages applied to different numbers. Problems like this measure comfort with converting percentage expressions into arithmetic operations and performing accurate calculations. There is no tricky conceptual twist; accuracy and systematic working are the main skills required. Given Data / Assumptions: We must compute: 21% of 500 + 31% of 1100 - 30% of 790. Percent means “per hundred”, so x% of y is (x / 100) * y. All numbers are in standard units; no special units are attached. Concept / Approach: The approach is to handle each percentage piece separately. First, convert each percentage to its decimal form (for example, 21% becomes 0.21). Then multiply it with the corresponding base number. After obtaining the three numeric values, add the first two results and subtract the third as indicated. Keeping calculations neat and stepwise helps avoid simple arithmetic errors. Step-by-Step Solution: Step 1: Compute 21% of 500. 21% of 500 = (21 / 100) * 500 = 0.21 * 500. 0.21 * 500 = 105. Step 2: Compute 31% of 1100. 31% of 1100 = (31 / 100) * 1100 = 0.31 * 1100. 0.31 * 1100 = 341. Step 3: Compute 30% of 790. 30% of 790 = (30 / 100) * 790 = 0.30 * 790. 0.30 * 790 = 237. Step 4: Combine the results: 21% of 500 + 31% of 1100 - 30% of 790 = 105 + 341 - 237. Step 5: 105 + 341 = 446. Step 6: 446 - 237 = 209. Step 7: Therefore, the final value of the expression is 209. Verification / Alternative check: We can quickly cross check using approximate mental arithmetic. 20% of 500 is around 100, so 21% should be slightly more than 100; 31% of 1100 is around one third, about 360, but exactly 341; and 30% of 790 is roughly 240. Then 100 plus 340 minus 240 is close to 200. This sanity check supports the precise calculation of 209, as it lies within the expected range. Why Other Options Are Wrong: 290 and 309 are higher than the result we got and would require either an error in one of the percentage multiplications or incorrect addition and subtraction. 390 is far too large and would correspond to neglecting the subtraction term almost entirely. Careful calculation clearly produces 209 as the only correct value. Common Pitfalls: Students may forget to divide by 100 when converting percentages, effectively treating 21% of 500 as 21 * 500 instead of 0.21 * 500. Another common mistake is mixing up the order of operations or incorrectly subtracting the final term. Writing each intermediate result clearly and performing arithmetic in a stepwise fashion significantly reduces these risks. Final Answer: The value of the expression is 209.

Question 17

Sairam spent Rs. 38,460 on renovating his home and Rs. 24,468 on buying a TV set, after which he still had 28% of his total original amount left. What was the total amount of money he had initially?

Accepted Answer

Introduction / Context: This question involves reverse percentage and basic algebra. Sairam spends fixed rupee amounts on renovation and on a TV, and is left with 28% of his original total. By treating the original amount as an unknown and expressing the remaining percentage in terms of that unknown, we can set up a simple linear equation and solve for his starting amount. Given Data / Assumptions: Amount spent on renovation = Rs. 38,460. Amount spent on TV set = Rs. 24,468. He still has 28% of his original total amount left after both purchases. No other expenditures or incomes are involved. Concept / Approach: Let the original total amount be T. The total spent is the sum of the two given expenditures. The remaining money is 28% of T, that is 0.28 * T. We know that original total T equals the sum of money spent plus the remaining money. Hence we can write an equation T = (total spent) + 0.28 * T, and solve for T. This is a standard reverse percentage technique used in many aptitude questions. Step-by-Step Solution: Step 1: Let the original total amount be T. Step 2: Total amount spent = 38,460 + 24,468. Step 3: Compute total spent: 38,460 + 24,468 = 62,928. Step 4: Remaining money = 28% of T = 0.28 * T. Step 5: Original total T must equal money spent plus remaining money. So T = 62,928 + 0.28 * T. Step 6: Rearrange the equation: T - 0.28 * T = 62,928. Step 7: This gives 0.72 * T = 62,928. Step 8: Solve for T: T = 62,928 / 0.72. Step 9: 62,928 / 0.72 = 87,400. Step 10: Therefore, the original amount was Rs. 87,400. Verification / Alternative check: Check by working forward from Rs. 87,400. Sairam spends 38,460 and 24,468, which totals 62,928. Remaining amount = 87,400 - 62,928 = 24,472. Now check if this is 28% of 87,400: 28% of 87,400 = 0.28 * 87,400 = 24,472. This matches exactly, confirming that the original amount is correctly found as Rs. 87,400. Why Other Options Are Wrong: If the original amount were Rs. 68,700, 28% of that would not leave enough to cover the given expenditures. Similar mismatches occur for Rs. 74,000 and Rs. 94,060; when you compute 28% of those values and add the spent amount, the totals do not align with the required consistency condition that T equals spent plus remaining. Only Rs. 87,400 satisfies the equation exactly. Common Pitfalls: One frequent error is to treat 28% as the spent portion instead of the remaining portion. Another pitfall is failing to set up the equation correctly, for example writing T = 0.28 * T - spent instead of T = spent + 0.28 * T. Some students also mistakenly divide the spent amount by 0.28 instead of 0.72. Always remember: if 28% remains, then 72% has been spent, and the spent amount equals 0.72 * T, while the total is the sum of spent and remaining portions. Final Answer: Sairam originally had Rs. 87,400.

Question 18

In the expression 63.5% of 8924.2 plus x% of 5324.4 equals 6827.5862, what is the value of x (in percent)?

Accepted Answer

Introduction / Context: This question asks you to find an unknown percentage in a linear equation involving percentages. One term of the expression is a known percentage of a given number, and the second term is an unknown percentage of another number. Their sum is equal to a specified total. This is a typical algebraic percentage problem where we convert percentage phrases into algebraic expressions and then solve for the unknown percentage. Given Data / Assumptions: We have 63.5% of 8924.2 plus x% of 5324.4 equals 6827.5862. That is, 63.5% of 8924.2 + x% of 5324.4 = 6827.5862. We must determine the value of x (the unknown percentage) in whole percent, matching the nearest option. Concept / Approach: The phrase “63.5% of 8924.2” translates to 0.635 * 8924.2, and “x% of 5324.4” translates to (x / 100) * 5324.4. We first compute the known term 0.635 * 8924.2, then subtract that from 6827.5862 to isolate the unknown term. After that, we divide by 5324.4 and multiply by 100 to find x as a percentage. Finally, we compare the result to the given options and choose the closest match. Step-by-Step Solution: Step 1: Compute the known percentage term. 63.5% of 8924.2 = 0.635 * 8924.2. 0.635 * 8924.2 ≈ 5666.867. Step 2: Use the given equation. 5666.867 + (x / 100) * 5324.4 = 6827.5862. Step 3: Subtract 5666.867 from both sides to isolate the unknown term. (x / 100) * 5324.4 = 6827.5862 - 5666.867. Step 4: Compute the right side: 6827.5862 - 5666.867 ≈ 1160.7192. Step 5: Now (x / 100) * 5324.4 ≈ 1160.7192. Step 6: Solve for x / 100: (x / 100) = 1160.7192 / 5324.4. Step 7: Compute 1160.7192 / 5324.4 ≈ 0.218, that is 21.8%. Step 8: Thus x ≈ 21.8. The nearest integer percentage among the options is 22. Verification / Alternative check: Take x = 22% and recompute the total. 22% of 5324.4 = 0.22 * 5324.4 ≈ 1171.368. Add the earlier term 5666.867: 5666.867 + 1171.368 ≈ 6838.235. This is reasonably close to 6827.5862, and the small difference can be attributed to rounding issues in intermediate steps and the approximate nature of the given decimal values. Given the answer choices, 22% fits best with the computed approximate value of 21.8%. Why Other Options Are Wrong: If x were 28%, the second term would be much larger and the total would exceed the target value significantly. Similarly, 18% or 12% would yield totals that are too small. Only 22% is close to the mathematically derived approximate value of 21.8%, making it the correct choice from the provided options. Common Pitfalls: One common mistake is to confuse percentage points with percentage values, or to misplace the decimal when converting from percent to a decimal factor. Another pitfall is to round intermediate results too early, which can cause noticeable discrepancies when recomputed. It is advisable to keep several decimal places in calculations until the final step and then match the result to the closest option. Final Answer: The value of x is approximately 22%.

Question 19

Simplify the expression a% of b plus b% of a. Which of the following is equal to a% of b + b% of a?

Accepted Answer

Introduction / Context: This is an algebraic percentage identity problem. It asks you to rewrite a% of b plus b% of a in a simpler, more compact form. Such identities are very useful in mental arithmetic and shortcut methods in quantitative aptitude, particularly in problems involving percentages and ratios between two quantities. Given Data / Assumptions: We have two variables a and b. The expression to simplify is: a% of b + b% of a. We need to choose the correct equivalent expression from the options. All variables are real numbers and percentages are interpreted in the usual way. Concept / Approach: The phrase “a% of b” means (a / 100) * b. Similarly, “b% of a” means (b / 100) * a. By converting both terms into algebraic form, we can add them and factor common terms. The core idea is to carefully handle the structures and see what factor emerges from the sum. Once simplified, we compare the resulting expression with the given options to identify an equivalent percentage form. Step-by-Step Solution: Step 1: Write a% of b in algebraic form. a% of b = (a / 100) * b = (ab) / 100. Step 2: Write b% of a in algebraic form. b% of a = (b / 100) * a = (ba) / 100. Step 3: Add the two expressions. a% of b + b% of a = (ab / 100) + (ba / 100). Step 4: Since ab = ba, we can combine them: (ab / 100) + (ab / 100) = 2ab / 100. Step 5: Now express 2ab / 100 back in percentage form. 2ab / 100 = (2a / 100) * b = 2a% of b. Step 6: Therefore, a% of b + b% of a = 2a% of b. Verification / Alternative check: Take a simple numerical example to check. Let a = 10 and b = 50. Then a% of b = 10% of 50 = 5. b% of a = 50% of 10 = 5. Sum = 10. Now compute 2a% of b = 2 * 10% of 50 = 20% of 50 = 10. The numerical check confirms that 2a% of b is equal to a% of b + b% of a for these sample values. This supports the algebraic derivation. Why Other Options Are Wrong: 2a% of 2b would be (2a / 100) * 2b = 4ab / 100, which is twice as large as 2ab / 100. 2a% of 2a and 2b% of 2b involve multiplying by a or b again and do not match the required dimensions of ab / 100. They produce completely different algebraic structures and cannot be equal to the sum of a% of b and b% of a. Common Pitfalls: A typical mistake is mishandling the percentage notation, for example treating a% as a / 10 instead of a / 100. Another error is to confuse 2a% of b with (a + b)% of something without checking the underlying algebra. When in doubt, always translate percentage statements into basic fraction or decimal context and manipulate them using standard algebraic rules. Final Answer: a% of b + b% of a is equal to 2a% of b.

Question 20

The marked price of a shirt is Rs. 1280. It is being sold for Rs. 900 after a single discount. What is the discount percentage on the shirt?

Accepted Answer

Introduction / Context: This problem is a standard marked price and discount percentage question. We are given the marked price, the final selling price and asked to compute the discount percentage. Such questions are frequently used in aptitude tests to verify understanding of percentage decrease and comparison of two monetary values. Given Data / Assumptions: Marked price (MP) of the shirt = Rs. 1280. Selling price (SP) after discount = Rs. 900. There is a single discount applied on the marked price. We must find the discount percentage. Concept / Approach: The discount amount is the difference between the marked price and the selling price. The discount percentage is then calculated as (discount amount / marked price) * 100. It is important to divide by the marked price, not the selling price, because discount is defined relative to the original quoted price of the product. Step-by-Step Solution: Step 1: Compute the discount amount. Discount = MP - SP = 1280 - 900. Discount = 380. Step 2: Express the discount as a fraction of the marked price. Discount fraction = 380 / 1280. Step 3: Simplify 380 / 1280. 380 / 1280 = 0.296875. Step 4: Convert this fraction into a percentage. Discount percentage = 0.296875 * 100 = 29.6875%. Step 5: Rounded to two decimal places, this is approximately 29.68%. Verification / Alternative check: We can reverse-check by applying 29.68% discount to the marked price. A 29.68% discount on 1280 is approximately 0.2968 * 1280 ≈ 380.9. Subtracting this from 1280 gives a selling price near 899.1, which, considering rounding, is consistent with the given selling price of Rs. 900. This confirms that 29.68 is the correct discount percentage among the options. Why Other Options Are Wrong: A discount of 31.31% would give a lower selling price than 900. Similarly, 25.57% and 34.36% discounts would produce selling prices that do not match Rs. 900 when applied to 1280. Only around 29.7% discount bridges the gap from 1280 down to 900 accurately. Common Pitfalls: A common error is to divide the discount amount by the selling price instead of by the marked price, leading to an incorrect discount percentage. Another error is rounding too early and then using the rounded number to check against the options, which might cause confusion when values are close. Always base the percentage on the marked price and keep enough decimal accuracy until the final step. Final Answer: The discount percentage on the shirt is approximately 29.68.

Percentage Questions