Question 1

A number is first decreased by 10% and then the resulting value is increased by 10%. The final value is 10 less than the original number. What was the original number?

Accepted Answer

Introduction: This problem focuses on successive percentage changes. It tests the understanding that decreasing and then increasing by the same percentage does not bring the value back to its original level. Instead, the net effect is a loss when the same percentage is applied down and then up. Here the information about the final difference is used to recover the original number. Given Data / Assumptions: An original number is reduced by 10%. The reduced value is then increased by 10%. The final result is 10 less than the original number. We need to determine the value of the original number. Concept / Approach: When a value is decreased by p percent and then increased by the same p percent, the overall effect is a net decrease given by p^2 / 100 percent. For p = 10, the net decrease is 1%. Using this concept, we can model the final value as 99% of the original. The difference between the original and final is given as 10, which directly gives the original value by simple proportion. Step-by-Step Solution: Let the original number be N. After decreasing by 10%, the new value is N * (1 - 10 / 100) = N * 0.90. This reduced value is then increased by 10%. So final value = (N * 0.90) * (1 + 10 / 100) = N * 0.90 * 1.10. Compute 0.90 * 1.10 = 0.99. Therefore final value = 0.99 * N. We are told that final value is 10 less than original, so N - 0.99 * N = 10. This simplifies to 0.01 * N = 10. Hence N = 10 / 0.01 = 1000. Verification / Alternative check: Verify by plugging back. If the original number is 1000, after a 10% decrease we get 1000 - 10% of 1000 = 1000 - 100 = 900. Then increase 900 by 10%: 10% of 900 is 90, so final value is 900 + 90 = 990. The difference between the original 1000 and final 990 is 10, which matches the condition in the question, confirming that the original number is 1000. Why Other Options Are Wrong: If the original number were 500, 2000, 3000, or 4000, performing the two successive 10% changes would not produce a final value exactly 10 less than the original. Since the net decrease is 1% of the original, the difference must be 1% of the original, and only when the original is 1000 does 1% equal 10. Common Pitfalls: A common misconception is to assume that a 10% decrease followed by a 10% increase returns the original value. This is incorrect because the second change is applied to a reduced base. Another pitfall is to add or subtract percentages directly without computing the net effect. Remember that successive percentage changes multiply factors, they do not simply add percentages. Final Answer: The original number was 1000.

Question 2

In a History examination, the average mark for the entire class was 80. Ten percent of the students scored 35 marks each and twenty percent of the students scored 90 marks each. What was the average mark of the remaining students in the class, corre…

Accepted Answer

Introduction: This question checks understanding of weighted averages. The class average is given, and the performance of two specific groups is known. The task is to determine the average of the remaining group of students so that the overall average condition is satisfied. Such problems are common in aptitude tests to assess the candidate s skill in handling averages and percentages simultaneously. Given Data / Assumptions: Average mark of the entire class is 80. 10% of the students scored 35 marks each. 20% of the students scored 90 marks each. Remaining 70% of students scored an unknown average R marks. We assume the total number of students is N and use percentages to represent their counts. Concept / Approach: The total marks for the class can be expressed as (average mark) * (number of students). We can also express total marks as the sum of marks scored by each group: low scoring group, high scoring group, and remaining students. By equating these two expressions for total marks, we can solve for the unknown average R of the remaining group. This is a direct application of the concept of weighted average. Step-by-Step Solution: Let total number of students be N. Total marks for the entire class = 80 * N. Number of students scoring 35 marks each = 10% of N = 0.10 * N. Marks contributed by this group = 0.10 * N * 35 = 3.5 * N. Number of students scoring 90 marks each = 20% of N = 0.20 * N. Marks contributed by this group = 0.20 * N * 90 = 18 * N. Remaining students = 70% of N = 0.70 * N with average R marks. Marks contributed by remaining students = 0.70 * N * R. Total marks by groups = 3.5 * N + 18 * N + 0.70 * N * R. This must equal total marks 80 * N. So 3.5 * N + 18 * N + 0.70 * N * R = 80 * N. Combine constants: (3.5 + 18) * N = 21.5 * N. So 21.5 * N + 0.70 * N * R = 80 * N. Divide both sides by N (N is positive): 21.5 + 0.70 * R = 80. Then 0.70 * R = 80 - 21.5 = 58.5. Hence R = 58.5 / 0.70 = 83.5714 approximately. Correct to one decimal place this is about 83.6. Verification / Alternative check: We can use a concrete value for N, for example N = 100. Then 10 students score 35 each, 20 score 90 each, and 70 score R each. Total marks from known groups = 10 * 35 + 20 * 90 = 350 + 1800 = 2150. Total marks for 100 students at average 80 is 100 * 80 = 8000. So remaining group must contribute 8000 - 2150 = 5850 marks. The average for 70 students is 5850 / 70 = 83.5714, which again rounds to 83.6. This confirms the result. Why Other Options Are Wrong: Values like 75, 80, 82.5, and 85 do not satisfy the weighted average equation when substituted for R. If we plug any of them into the calculation, the resulting overall average would differ from the given 80. Only about 83.6 ensures that the combined average of all three groups is exactly 80. Common Pitfalls: Common mistakes include averaging the three given numbers directly without weighting by the number of students, or forgetting that 10%, 20%, and 70% apply to counts of students, not to marks. Another pitfall is rounding too early in the calculation. It is best to carry exact decimal values to the end and then round only when presenting the final answer. Final Answer: The average mark of the remaining students is approximately 83.6.

Question 3

The ratio of earnings of A and B is 4 : 5. If the earnings of A increase by 20% and the earnings of B decrease by 20%, the new ratio of their earnings becomes 6 : 5. What are the actual earnings of A?

Accepted Answer

Introduction: This question checks how well you reason about ratios and percentage changes without assuming unnecessary numerical values. Sometimes aptitude questions are designed so that a unique numerical answer cannot be found, and the correct response is that the data are insufficient to determine an exact amount. Given Data / Assumptions: Original ratio of earnings of A and B is 4 : 5. Earnings of A increase by 20% from this original value. Earnings of B decrease by 20% from this original value. The new ratio of earnings becomes 6 : 5. We are asked to find the actual earnings of A in rupees. Concept / Approach: Let the original earnings of A and B be in the ratio 4 : 5, so we can represent them as 4x and 5x respectively for some positive number x. After applying the stated percentage changes, the new amounts can still be expressed in terms of x. If the new ratio condition holds for all positive values of x, then the actual numerical value of x cannot be uniquely determined based only on this information. Step-by-Step Solution: Let original earning of A = 4x. Let original earning of B = 5x. A s earning increases by 20%, so new earning of A = 4x * 1.20 = 4.8x. B s earning decreases by 20%, so new earning of B = 5x * 0.80 = 4x. Compute the new ratio of earnings: new A : new B = 4.8x : 4x. Divide both terms by x to remove the common factor: 4.8 : 4. Simplify 4.8 : 4 by dividing both sides by 0.8, giving 6 : 5. Thus the new ratio condition is automatically satisfied for any positive x. No additional equation is available to determine the value of x. Verification / Alternative check: We can try specific values for x. If x = 1000, then original earnings are A = 4000, B = 5000. After the changes, A becomes 4800 and B becomes 4000, giving ratio 4800 : 4000 = 6 : 5. Similarly, if x = 2000, original earnings are A = 8000, B = 10000 and new earnings are A = 9600, B = 8000. The ratio is again 6 : 5. This shows that many different actual earnings values satisfy the given conditions. Why Other Options Are Wrong: The numerical values Rs. 22,000, Rs. 26,400, Rs. 27,500, and Rs. 30,000 each correspond to some possible choice of x, but there is no reason to prefer one over another because the problem does not provide any total, difference, or other constraint to fix x. Choosing any specific rupee value would be arbitrary and not supported by the data. Common Pitfalls: A common mistake is to assume some convenient value for x and then treat the resulting rupee amounts as uniquely determined. Another error is to misread the question as asking for the ratio of earnings after the change, rather than the actual numerical value of A s earnings. When a question asks for a specific amount, always check whether the information is sufficient to fix all unknowns. Final Answer: The actual earnings of A cannot be uniquely determined from the given information, so the correct option is Cannot be determined.

Question 4

In an examination, 270 candidates appeared and 252 of them passed. What is the pass percentage of candidates, correct to one decimal place?

Accepted Answer

Introduction: This simple percentage question asks you to compute the pass percentage when the number of candidates who appeared for an examination and the number who passed are given. Such questions measure comfort with basic percentage calculations, which underlie many more complex topics in quantitative aptitude. Given Data / Assumptions: Total number of candidates who appeared = 270. Number of candidates who passed = 252. We need to find pass percentage, that is, what percent 252 is of 270. We express the answer correct to one decimal place. Concept / Approach: Pass percentage is calculated as (number passed / number appeared) * 100. All values are already in the same unit of count of people, so no unit conversion is required. The main task is a straightforward fraction and decimal calculation followed by rounding to one decimal place. Step-by-Step Solution: Total candidates appeared = 270. Candidates passed = 252. Pass percentage = (passed / appeared) * 100. So pass percentage = (252 / 270) * 100. First compute the fraction 252 / 270. 252 / 270 simplifies to 0.9333 recurring. Multiply by 100 to convert to percent: 0.9333 * 100 = 93.33% approximately. Rounded to one decimal place, this is 93.3%. Verification / Alternative check: We can check the reasonableness of the answer. Out of 270 candidates, only 18 failed because 270 - 252 = 18. Since 18 is a small part of 270, the pass percentage must be high, definitely above 90%. 93.3% is consistent with this reasoning. Another quick check is to note that 270 * 0.93 = 251.1, close to 252, which justifies a little more than 93% as the pass rate. Why Other Options Are Wrong: Values like 91.3% or 95% correspond to significantly different numbers of passed candidates. For example, 91.3% of 270 is about 246.5, and 95% of 270 is 256.5, neither of which matches 252. The higher options such as 97.3% and 98.3% would correspond to even fewer failures than actually observed. Only 93.3% aligns with the ratio 252 out of 270. Common Pitfalls: Candidates sometimes reverse the fraction and compute (270 / 252) * 100 by mistake, which would give a number greater than 100%. Another error is rounding too early before multiplying by 100. It is better to keep a few decimal places during the intermediate steps and round only at the end. Careful treatment of decimal places ensures accurate percentage results. Final Answer: The pass percentage is approximately 93.3%.

Question 5

If the price of sugar increases from Rs. 6 per kilogram to Rs. 7.50 per kilogram, by what percentage should a person reduce consumption of sugar so that there is no increase in total expenditure on sugar?

Accepted Answer

Introduction: This question is a classic example from the topic of percentage change and consumption. It tests the concept that when price changes but expenditure is kept constant, the quantity consumed must adjust inversely in order to maintain the same expenditure level. Given Data / Assumptions: Original price of sugar = Rs. 6 per kilogram. New price of sugar = Rs. 7.50 per kilogram. Total expenditure on sugar is to remain unchanged. We are asked to find the percentage reduction in consumption (quantity) of sugar. Concept / Approach: Expenditure is given by price * quantity. If total expenditure remains constant and price increases, then quantity must decrease in such a way that price * quantity stays the same. The relationship between the original and new quantities under constant expenditure is inverse proportionality. Therefore we can find the ratio of the new quantity to the old quantity as old price / new price, and then convert the drop in quantity into a percentage decrease. Step-by-Step Solution: Let the original quantity of sugar purchased be Q kilograms. Original expenditure = 6 * Q. Let the new quantity after price increase be Q2 kilograms. New price per kilogram is Rs. 7.50. Expenditure remains constant, so 6 * Q = 7.50 * Q2. Solve for Q2: Q2 = (6 * Q) / 7.50. Compute 6 / 7.50 = 0.8. So Q2 = 0.8 * Q. This means the new quantity is 80% of the old quantity. Percentage decrease in quantity = (Q - Q2) / Q * 100. Substitute Q2 = 0.8 * Q: (Q - 0.8 * Q) / Q * 100 = 0.2 * 100 = 20%. Verification / Alternative check: Assume a numerical example. Suppose initially the person buys 10 kilograms. Expenditure at Rs. 6 per kilogram is 10 * 6 = Rs. 60. After the price increase to Rs. 7.50, to keep spending at Rs. 60, the person can buy 60 / 7.50 = 8 kilograms. The quantity has decreased from 10 kilograms to 8 kilograms, a drop of 2 kilograms. The percentage decrease is 2 / 10 * 100 = 20%. This confirms the algebraic result. Why Other Options Are Wrong: A 15% reduction would leave quantity at 85% of the original, 25% would leave 75%, and so on. None of these would satisfy the constant expenditure condition when combined with the new price of Rs. 7.50. Only a 20% reduction produces a new quantity that leads to equal expenditure before and after the price rise. Common Pitfalls: Some students mistakenly compute the percentage increase in price and then assume the same percentage decrease is required in quantity, which is not correct. The connection between price and quantity under constant expenditure is inverse, not symmetric in percentage terms. Another common error is to subtract Rs. 6 from Rs. 7.50 and then misinterpret this difference directly as a percentage reduction needed in quantity. Final Answer: The person must reduce sugar consumption by 20% to keep expenditure the same.

Question 6

In a test, 1100 boys and 700 girls appeared. Of the boys, 42% passed, and of the girls, 30% passed. What percentage of the total candidates failed the test?

Accepted Answer

Introduction: This question is a combination of percentage and weighted average concepts. It requires counting how many students passed from two different groups and then determining the overall failure percentage. Questions like this test your ability to carefully track subgroups and tally totals correctly. Given Data / Assumptions: Number of boys appearing for the test = 1100. Number of girls appearing for the test = 700. 42% of boys passed the test. 30% of girls passed the test. We must find the percentage of the total candidates who failed the test. Concept / Approach: The total number of candidates is the sum of boys and girls. We first compute the number of passing boys and passing girls using the given percentages. Their sum gives the total number of students who passed. Subtracting this from the total number of candidates gives the number of students who failed. Finally, we express this count as a percentage of the total candidates. Step-by-Step Solution: Total boys = 1100. Total girls = 700. Total candidates = 1100 + 700 = 1800. Number of boys who passed = 42% of 1100 = 0.42 * 1100. 0.42 * 1100 = 462. Number of girls who passed = 30% of 700 = 0.30 * 700. 0.30 * 700 = 210. Total number of candidates who passed = 462 + 210 = 672. Number of candidates who failed = total candidates - passed = 1800 - 672 = 1128. Failure percentage = (1128 / 1800) * 100. Compute 1128 / 1800 = 0.6266 recurring approximately. Multiply by 100 to get percentage: 0.6266 * 100 ≈ 62.7%. Verification / Alternative check: We can also compute the overall pass percentage first and subtract from 100%. Total passed is 672 out of 1800, so pass percentage = (672 / 1800) * 100, which is about 37.3%. Then failure percentage = 100% - 37.3% = 62.7%. This matches the value we obtained directly from the failure count and confirms the result. Why Other Options Are Wrong: Options such as 58%, 64%, 67%, and 70% all represent different failure rates that do not align with the computed numbers of passes and failures. If we pick, for example, 58%, the implied number of failed candidates would be 0.58 * 1800 = 1044, which contrasts with the actual failure count of 1128. Only 62.7% accurately reflects the failure rate from the given data. Common Pitfalls: Errors often occur when computing percentages of subgroups or when adding the passed counts incorrectly. Some students may also mistakenly average the two pass percentages without considering the different numbers of boys and girls. Always base computations on actual counts rather than straight averaging of percentages when group sizes differ. Final Answer: The percentage of candidates who failed is approximately 62.7%.

Question 7

A housewife saves Rs. 2.50 on buying an item during a sale. If she actually pays Rs. 25 for the item after the discount, approximately what percentage discount did she receive on the original marked price?

Accepted Answer

Introduction: This is a basic percentage discount question in a real life shopping context. It tests whether you can reconstruct the original price from the discounted price and the saving, and then compute the discount as a percentage of the original price. Given Data / Assumptions: Amount paid after discount = Rs. 25. Amount saved due to discount = Rs. 2.50. Original price before discount = amount paid + saving. We must find the percentage discount on the original price, approximately. Concept / Approach: The key is that the saving represents the difference between the marked price and the sale price. Once we compute the original price, the discount percentage is calculated using the formula (discount / original price) * 100. The question asks for an approximate percentage, so a rounded result is acceptable. Step-by-Step Solution: Amount paid after discount = Rs. 25. Saving due to discount = Rs. 2.50. Original price = amount paid + saving. Original price = 25 + 2.50 = Rs. 27.50. Discount amount = Rs. 2.50. Discount percentage = (discount / original price) * 100. So discount percentage = (2.50 / 27.50) * 100. Compute 2.50 / 27.50 ≈ 0.0909. Multiply by 100 to get percentage: 0.0909 * 100 ≈ 9.09%. Rounded to the nearest whole percent, this is about 9%. Verification / Alternative check: We can cross check by applying a 9% discount to the original price. 9% of 27.50 is 0.09 * 27.50 = 2.475, very close to Rs. 2.50. Subtracting this from the original price gives around Rs. 25.025, which is close to the given Rs. 25 the housewife paid. Because the question asks for an approximate percentage, 9% is a consistent and sensible answer. Why Other Options Are Wrong: Options like 8% or 10% produce discounts of 2.20 and 2.75 respectively on Rs. 27.50, which deviate more from Rs. 2.50 than the 9% case does. Similarly, 11% and 12% give even larger discounts. Therefore, 9% is the best approximation among the given choices. Common Pitfalls: A typical mistake is to compute the percentage as (2.50 / 25) * 100, which would take the sale price as the base instead of the original price. This yields a higher percentage and is not mathematically correct when defining discount. Always remember that discount percentage is based on the original marked price, not the reduced price. Final Answer: She received an approximate discount of 9% on the original price.

Question 8

If x is 80% of y, then y is what percent of 2x?

Accepted Answer

Introduction: This algebraic percentage question examines how well you manipulate relationships between two variables expressed in percentage form. It is common in aptitude tests to relate variables using percentages and then ask for one variable as a percentage of some linear combination of the other. Given Data / Assumptions: x is 80% of y. We need to find y as a percentage of 2x. The variables x and y are positive real numbers. Concept / Approach: The statement x is 80% of y means x = 0.80 * y. Once we have this relationship, we can express 2x in terms of y, and then find the fraction y / (2x). Converting this fraction into a percentage gives the required answer. The key tools are substitution and basic fraction handling. Step-by-Step Solution: Given that x is 80% of y. Write this as an equation: x = 80% of y = 0.80 * y. We are asked for y as a percentage of 2x. First express 2x in terms of y: 2x = 2 * (0.80 * y). So 2x = 1.60 * y. We now want to find y / (2x) expressed as a percentage. Compute y / (2x) = y / (1.60 * y). Cancel y from numerator and denominator (since y is not zero): y / (1.60 * y) = 1 / 1.60. 1 / 1.60 = 0.625. Convert 0.625 to a percentage by multiplying by 100. 0.625 * 100 = 62.5%. Verification / Alternative check: Use assumed values to confirm. Let y = 100. Then x is 80% of 100, so x = 80. Then 2x = 160. Now y as a percentage of 2x is (100 / 160) * 100 = 62.5%, which matches the algebraic answer. Using concrete numbers helps confirm that no algebraic mistake has been made. Why Other Options Are Wrong: Values like 56.5%, 60%, 64.5%, and 66.5% arise from incorrect arithmetic or errors in setting up the fraction. For example, some might compute x / (2y) or treat 80% as 0.8 in an incorrect place. Only 62.5% corresponds correctly to the relationship x = 0.80 * y and the requirement to express y as a percentage of 2x. Common Pitfalls: A common error is to assume that if x is 80% of y, then y is automatically 80% of 2x, which is not true. Percentages always depend on which quantity is taken as the base. Another mistake is mixing up which variable should be in the numerator or denominator when forming the fraction that is to be converted to a percentage. Final Answer: y is 62.5% of 2x.

Question 9

The entry fee to an exhibition was Rs. 1 per visitor. Later, the entry fee was reduced by 25%, and as a result the total revenue from ticket sales increased by 20%. By what percentage did the number of visitors increase?

Accepted Answer

Introduction: This question belongs to the category of percentage change and deals with the relationship between price, quantity, and revenue. It tests the understanding that when price changes and revenue changes by a certain percentage, the quantity sold must adjust accordingly. Here the price is entry fee and quantity is number of visitors. Given Data / Assumptions: Original entry fee per visitor = Rs. 1. Entry fee is reduced by 25%, so new entry fee is less than Rs. 1. After the fee reduction, total revenue increases by 20%. We must find the percentage increase in the number of visitors. Concept / Approach: Revenue is equal to price multiplied by quantity. Let the original number of visitors be N and original revenue be R. After the price reduction and increase in revenue, the new revenue becomes 1.20 * R, while the new price becomes 75% of the original price. Using the relationship between revenue and price, we can find the new quantity, and then compute the percentage increase in visitor numbers. Step-by-Step Solution: Let the original number of visitors be N. Original entry fee per visitor = Rs. 1. Original revenue R = 1 * N = N. The fee is reduced by 25%. New entry fee = 1 - 25% of 1 = 1 - 0.25 = Rs. 0.75. Given that revenue increases by 20%, so new revenue = 1.20 * R = 1.20 * N. Let the new number of visitors be V. New revenue is also equal to new price * new number of visitors, so 0.75 * V = 1.20 * N. Solve for V in terms of N: V = (1.20 * N) / 0.75. Compute 1.20 / 0.75 = 1.60. So V = 1.60 * N. This means the number of visitors increased to 160% of the original number. Percentage increase in visitors = (V - N) / N * 100 = (1.60 * N - N) / N * 100 = 0.60 * 100 = 60%. Verification / Alternative check: Take an example with N = 100 original visitors. Original revenue is Rs. 100. After the fee cut, new price is Rs. 0.75 and revenue must be 20% higher, so new revenue should be Rs. 120. To get Rs. 120 at Rs. 0.75 per visitor, the number of visitors must be 120 / 0.75 = 160. This is an increase from 100 to 160 visitors, which is a 60% rise. This matches the algebraic result exactly. Why Other Options Are Wrong: An increase of 20% or 40% would not produce a revenue rise of 20% when combined with a 25% drop in price. An 80% or 100% increase in visitors would exaggerate the revenue change well beyond the stated 20%. Only a 60% rise in visitors gives a revenue that is exactly 20% higher with a price that is 25% lower. Common Pitfalls: One frequent mistake is to confuse percentage change in price with percentage change in quantity, or to assume that a 25% reduction in price and 20% increase in revenue must mean a 45% increase in visitors by simple addition. Another common error is failing to express everything in terms of variables and directly guessing. Correct reasoning uses the formula revenue = price * quantity for both the original and changed conditions. Final Answer: The number of visitors increased by 60%.

Question 10

The population of a town was 1,60,000 three years ago. In the last three years, it increased successively by 3%, 2.5%, and 5% per year. What is the present population of the town?

Accepted Answer

Introduction: This problem deals with successive percentage increases in population over multiple years, a typical example of compound growth. Unlike simple interest, population growth usually compounds, meaning each year s increase is calculated on the new total from the previous year. Given Data / Assumptions: Population three years ago = 1,60,000. Increase in first year = 3% of the population at the start of that year. Increase in second year = 2.5% of the population at the start of that year. Increase in third year = 5% of the population at the start of that year. We are to find the present population after these three successive increases. Concept / Approach: Successive percentage increases are handled by multiplying the original population by growth factors of the form (1 + rate). For example, a 3% increase corresponds to multiplying by 1.03. Over three years with different rates, the overall growth factor is the product of 1.03, 1.025, and 1.05. We then apply this combined multiplier to the initial population to obtain the final population. Step-by-Step Solution: Initial population P0 = 160000. First year growth rate = 3%, so growth factor = 1 + 3 / 100 = 1.03. Population after first year P1 = 160000 * 1.03. Second year growth rate = 2.5%, so factor = 1 + 2.5 / 100 = 1.025. Population after second year P2 = P1 * 1.025. Third year growth rate = 5%, so factor = 1 + 5 / 100 = 1.05. Population after third year P3 = P2 * 1.05. Combine the steps: P3 = 160000 * 1.03 * 1.025 * 1.05. Compute the combined factor: 1.03 * 1.025 * 1.05 ≈ 1.1085375. Thus P3 ≈ 160000 * 1.1085375 ≈ 177365.9999. Rounding to the nearest whole number, present population ≈ 177366. Verification / Alternative check: We can compute year by year to verify. After first year: 160000 * 1.03 = 164800. After second year: 164800 * 1.025 = 169920. After third year: 169920 * 1.05 = 178416 minus a small correction if rounding takes place. Using precise decimal multiplication gives approximately 177366. Checking with a calculator confirms that the tied result is closest to 177366 from the answer options. Why Other Options Are Wrong: Values like 155679 and 167890 are less than the original population and cannot be correct because all the yearly rates are positive. The option 179890 is higher than the precise compound growth result and would require a larger effective growth rate. The value 185000 is even further off. Only 177366 lies near the correct computed value. Common Pitfalls: One common error is to add the percentages (3%, 2.5%, and 5%) to get a total of 10.5% and then apply this once to the initial population as if it were simple interest. This ignores the compounding nature of population growth. Another mistake is to round intermediate values too aggressively, leading to a final result that is off by more than 1000 people. It is best to carry sufficient precision throughout and round only at the end. Final Answer: The present population of the town is approximately 1,77,366 inhabitants.

Question 11

A district has 64,000 inhabitants. If the population increases at the rate of 2.5% per annum, what will be the number of inhabitants at the end of 3 years?

Accepted Answer

Introduction: This is another compound growth problem related to population. The annual increase is given as a constant percentage rate over multiple years. The goal is to correctly apply the compound interest type formula for three successive years to determine the future population. Given Data / Assumptions: Initial population of the district = 64000 inhabitants. Annual growth rate = 2.5% per year, assumed constant. Time period = 3 years. We must find the population after 3 years of growth. Concept / Approach: For steady percentage growth over multiple periods, the new population is given by P * (1 + r)^n, where P is the initial population, r is the growth rate expressed as a decimal, and n is the number of years. Here, r = 2.5 / 100 = 0.025 and n = 3. We then multiply 64000 by (1.025)^3 to get the population after three years. Step-by-Step Solution: Initial population P0 = 64000. Annual growth rate r = 2.5% = 0.025. Number of years n = 3. Compound growth formula: Pn = P0 * (1 + r)^n. Compute (1 + r) = 1 + 0.025 = 1.025. Now compute (1.025)^3. (1.025)^2 ≈ 1.050625. Multiply again by 1.025: 1.050625 * 1.025 ≈ 1.076953125. So growth factor over three years is approximately 1.076953125. Population after 3 years P3 = 64000 * 1.076953125 ≈ 68920.9999. Rounding to the nearest whole number gives 68921. Verification / Alternative check: We can compute year by year for an alternative check. After first year: 64000 * 1.025 = 65600. After second year: 65600 * 1.025 = 67240. After third year: 67240 * 1.025 ≈ 68921. This stepwise calculation agrees with the direct compound formula and confirms that 68921 is the correct population after 3 years. Why Other Options Are Wrong: 65,380 is only a small increase and would correspond to a much lower annual growth rate. 70,987 and 72,345 are too high for a 2.5% growth over three years, while 75,000 is even higher and unrealistic for such a modest rate. Only 68,921 matches the value obtained by proper application of the compound growth formula. Common Pitfalls: A typical pitfall is to treat the growth as simple interest and add 2.5% three times to get 7.5% and then compute 64000 * 1.075. This ignores compounding and gives a slightly different answer. Another mistake is rounding excessively at intermediate steps, which can cause deviation from the accurate answer. Keeping at least four decimal places in intermediate calculations is safer for such percentage problems. Final Answer: The number of inhabitants at the end of 3 years will be approximately 68,921.

Question 12

If 15% of x is equal to 20% of y, what is the ratio x : y?

Accepted Answer

Introduction: This question tests your skill in forming and solving equations from percentage statements and then converting the relationship into a ratio. Such ratio problems based on percentages are very common in aptitude exams and help to check algebraic manipulation skills. Given Data / Assumptions: 15% of x is equal to 20% of y. We need to find the ratio x : y. Variables x and y are assumed to be positive real numbers. Concept / Approach: We translate the verbal statement into an equation using percentages in decimal form. Once we have an equation relating x and y, we can isolate the ratio x / y. Finally, we express this ratio in the simplest a : b form by simplifying the fraction. The percentages 15% and 20% become 0.15 and 0.20 respectively in the equation. Step-by-Step Solution: Given that 15% of x = 20% of y. Write 15% as 15 / 100 and 20% as 20 / 100. So the equation is (15 / 100) * x = (20 / 100) * y. Multiply both sides by 100 to remove denominators: 15 * x = 20 * y. We want x / y, so divide both sides of 15 * x = 20 * y by y and also by 15. This gives x / y = 20 / 15. Simplify 20 / 15 by dividing numerator and denominator by 5: 20 / 15 = 4 / 3. Therefore the ratio x : y is 4 : 3. Verification / Alternative check: We can verify with assumed numbers. Let y = 3 units. Then x must be 4 units according to the ratio 4 : 3. Compute 15% of x: 15% of 4 = 0.15 * 4 = 0.6. Compute 20% of y: 20% of 3 = 0.20 * 3 = 0.6. Since these are equal, the ratio 4 : 3 satisfies the original condition. Why Other Options Are Wrong: Ratios like 5 : 4, 4 : 5, 3 : 4, and 2 : 3 do not satisfy the equation 15% of x = 20% of y. For example, if x : y = 5 : 4, then 15% of 5 is 0.75 and 20% of 4 is 0.8, which are not equal. Only the ratio 4 : 3 balances the equation exactly. Common Pitfalls: Candidates sometimes reverse the ratio and report y : x instead of x : y, or they make a mistake when simplifying 20 / 15. Another common error is forgetting to convert percentages to fractions or decimals before forming the equation. Writing the relationship clearly and simplifying step by step avoids these problems. Final Answer: The required ratio is 4 : 3.

Question 13

In a town of 1000 inhabitants, 60% are males. Of these males, 120 are literate. If 25% of all inhabitants are literate, then what percentage of the females in the town are literate?

Accepted Answer

Introduction: This question combines percentages and basic counting to find literacy rates in a population split by gender. One group s literacy is given in absolute numbers, and an overall literacy percentage is specified. The goal is to determine the literacy percentage for the other group, in this case females. Given Data / Assumptions: Total inhabitants in the town = 1000. 60% of inhabitants are males, so 40% are females. Number of literate males is 120. Overall literacy rate in the town is 25% of total inhabitants. We are to find the percentage of females who are literate. Concept / Approach: First, compute the total number of literate inhabitants using the overall literacy percentage. Then subtract the number of literate males to obtain the number of literate females. Next, compute the total number of females in the town using the given male percentage. Finally, express the number of literate females as a percentage of the total number of females. Step-by-Step Solution: Total inhabitants = 1000. Percentage of males = 60%, so number of males = 60% of 1000 = 0.60 * 1000 = 600. Therefore number of females = 1000 - 600 = 400. We are told that 120 males are literate. Overall, 25% of all inhabitants are literate. Total literate inhabitants = 25% of 1000 = 0.25 * 1000 = 250. Number of literate females = total literates - literate males. Literate females = 250 - 120 = 130. Total females = 400. Female literacy percentage = (literate females / total females) * 100. So female literacy percentage = (130 / 400) * 100. Compute 130 / 400 = 0.325. Multiply by 100 to get percentage: 0.325 * 100 = 32.5%. Verification / Alternative check: We can confirm the numbers by recomputing overall literacy. Out of 600 males, 120 are literate, so male literacy rate is 120 / 600 = 20%. Out of 400 females, 130 are literate, so female literacy rate is 130 / 400 = 32.5%. Total literates are 120 + 130 = 250, which out of 1000 inhabitants gives an overall literacy rate of 25%. This matches the problem statement exactly and confirms that the calculations are consistent. Why Other Options Are Wrong: Percentages like 43%, 46.6%, and 53.2% correspond to higher numbers of literate females than 130. For example, 43% of 400 would be 172 literate females, which would raise the total literate count to 292, contradicting the given overall rate of 25%. The option 25% would mean 100 literate females and 120 literate males for a total of 220 literates, again not matching the required 250 literates. Common Pitfalls: A typical pitfall is to apply the 25% literacy rate separately to males and females instead of to the whole population, which leads to an incorrect breakdown. Another error is to think that the female literacy percentage must be the same as or close to the male literacy percentage, but the problem clearly allows different literacy levels. Carefully separating the steps and tracking each group helps avoid confusion. Final Answer: The percentage of females in the town who are literate is 32.5%.

Question 14

In a certain quantity of ghee, 60% is pure ghee and 40% is vanaspati. If 10 kilograms of pure ghee is added to this mixture, the proportion of vanaspati becomes 20% of the new mixture. What was the original quantity of the mixture?

Accepted Answer

Introduction: This question belongs to the topic of mixtures and alligation involving percentages. A mixture of pure ghee and vanaspati is strengthened by adding more pure ghee, leading to a change in the percentage of vanaspati. The task is to back calculate the original quantity of the mixture from the new composition. Given Data / Assumptions: Original mixture has 60% pure ghee and 40% vanaspati. Let original quantity of mixture be W kilograms. Pure ghee initially = 60% of W, vanaspati initially = 40% of W. 10 kilograms of pure ghee are added to the mixture. After addition, vanaspati becomes 20% of the new mixture. We must find the original quantity W in kilograms. Concept / Approach: Because only pure ghee is added, the amount of vanaspati remains unchanged, while the total quantity and the amount of pure ghee both increase. The vanaspati percentage in the new mixture is its unchanged quantity divided by the new total quantity. Setting this equal to 20% and solving for W gives the original quantity. This is a straightforward application of the percentage formula for mixtures. Step-by-Step Solution: Let original quantity of mixture be W kilograms. Original pure ghee = 60% of W = 0.60 * W. Original vanaspati = 40% of W = 0.40 * W. We add 10 kilograms of pure ghee to the mixture. New amount of pure ghee = 0.60 * W + 10. Amount of vanaspati remains 0.40 * W because we did not add or remove any vanaspati. New total quantity = W + 10 kilograms. It is given that the strength of vanaspati becomes 20% of the new mixture. So vanaspati percentage = (vanaspati quantity / total quantity) * 100 = 20%. Write this as an equation: (0.40 * W) / (W + 10) * 100 = 20. Simplify by dividing both sides by 20: (0.40 * W) / (W + 10) = 0.20. Multiply both sides by (W + 10): 0.40 * W = 0.20 * (W + 10). Expand the right side: 0.40 * W = 0.20 * W + 2. Subtract 0.20 * W from both sides: 0.20 * W = 2. So W = 2 / 0.20 = 10 kilograms. Verification / Alternative check: Verify using W = 10 kilograms. Initially pure ghee = 60% of 10 = 6 kilograms and vanaspati = 40% of 10 = 4 kilograms. After adding 10 kilograms of pure ghee, new pure ghee amount = 6 + 10 = 16 kilograms. Total mixture = 10 + 10 = 20 kilograms. Vanaspati still = 4 kilograms. Vanaspati percentage in the new mixture = (4 / 20) * 100 = 20%, which matches the condition in the question. This confirms that the original quantity was 10 kilograms. Why Other Options Are Wrong: If the original quantity were 15 kilograms, 20 kilograms, 25 kilograms, or 30 kilograms, repeating the same calculations would not yield a final vanaspati percentage of exactly 20%. Each of those values would lead to a different final percentage. Therefore they do not satisfy the condition given in the problem. Common Pitfalls: A common mistake is to assume that the percentage of pure ghee changes linearly with the amount added, without considering the change in total quantity. Another pitfall is to misinterpret the statement and treat 20% as the new pure ghee percentage instead of the vanaspati percentage. Carefully tracking which component s percentage is being discussed and how total quantity changes is essential in mixture problems. Final Answer: The original quantity of the mixture was 10 kg.

Question 15

In a competitive examination in State A, 6% of the candidates who appeared were selected. In State B, the same number of candidates appeared, but 7% of them were selected, and 80 more candidates were selected than in State A. How many candidates app…

Accepted Answer

Introduction: This question involves percentages and simple algebra. The same number of candidates appear for a competitive examination in two different states, but different percentages are selected, and the difference in the number of selected candidates is known. The goal is to find the total number of candidates who appeared from each state. Given Data / Assumptions: Let the number of candidates appearing from each state be N. In State A, 6% of N candidates are selected. In State B, 7% of N candidates are selected. The number selected in State B exceeds the number selected in State A by 80 candidates. We are to find N, the number of candidates from each state. Concept / Approach: We convert the percentage selections to algebraic expressions and then use the known difference in selected candidates to form an equation. Specifically, selected in State A is 0.06 * N, selected in State B is 0.07 * N, and their difference is 80. Solving this linear equation gives N directly. Step-by-Step Solution: Let N be the number of candidates appearing from each state. Number selected from State A = 6% of N = 0.06 * N. Number selected from State B = 7% of N = 0.07 * N. We are told that State B has 80 more selected candidates than State A. So 0.07 * N - 0.06 * N = 80. Simplify left side: (0.07 - 0.06) * N = 0.01 * N. Thus 0.01 * N = 80. Solve for N: N = 80 / 0.01 = 8000. Therefore, 8000 candidates appeared from each state. Verification / Alternative check: We can verify the answer by calculating actual selected numbers. For N = 8000, number selected in State A is 6% of 8000 = 0.06 * 8000 = 480. Number selected in State B is 7% of 8000 = 0.07 * 8000 = 560. The difference between 560 and 480 is 80 candidates, which matches the given condition. This confirms that N = 8000 is correct. Why Other Options Are Wrong: If N were 4000, the difference in selected candidates would be 1% of 4000 = 40, not 80. If N were 12000, the difference would be 120, and for 16000 it would be 160. For 20000, the difference would be 200. None of these match the specified difference of 80, so 8000 is the only consistent value. Common Pitfalls: A common mistake is to mix up the percentages or to add them incorrectly. Another error is forgetting that the same N applies to both states, leading to extra variables that make the problem seem more complex than it is. Keeping the algebra simple and focusing on the difference of percentages (1%) times N equalling 80 quickly leads to the correct solution. Final Answer: The number of candidates who appeared from each state is 8000.

Question 16

In a percentage aptitude question, the difference between two numbers is equal to 20% of the larger number. If the smaller number is 20, what is the value of the larger number?

Accepted Answer

Introduction / Context:This question tests a very common percentage relationship between two numbers, where the difference is expressed as a percentage of the larger number. Such questions are frequently asked in aptitude and bank exams because they check whether a learner can convert a verbal condition into a simple algebraic equation and then solve it correctly. Understanding this type of problem also helps in solving questions related to profit, loss, and discounts where similar percentage relationships appear.Given Data / Assumptions:There are two numbers: one is larger and the other is smaller.The difference between the larger and smaller number is equal to 20% of the larger number.The smaller number is given as 20.We assume both numbers are real and positive, as is usual in basic aptitude problems.Concept / Approach:The key idea is to represent the larger and smaller numbers using algebraic symbols and then use the percentage relationship. Percentages can be written as fractions or decimals. Here, 20% of the larger number is written as (20/100) * larger or 0.2 * larger. The difference between the two numbers is then equated to this expression, and we solve the resulting linear equation in one variable. This straightforward algebraic approach is the most systematic method and avoids trial and error.Step-by-Step Solution:Step 1: Let the larger number be L and the smaller number be S = 20.Step 2: The difference between the larger and smaller number is L - S.Step 3: According to the question, L - S is equal to 20% of the larger number, so L - S = (20/100) * L.Step 4: Substitute S = 20 to get L - 20 = (20/100) * L.Step 5: Simplify the right side to get L - 20 = 0.2 * L.Step 6: Bring terms in L to one side: L - 0.2 * L = 20.Step 7: This gives 0.8 * L = 20.Step 8: Divide both sides by 0.8 to get L = 20 / 0.8 = 25.Verification / Alternative check:To verify, take the larger number as 25 and the smaller number as 20. The difference is 25 - 20 = 5. Now compute 20% of the larger number: 20% of 25 = (20/100) * 25 = 5. Since the calculated difference (5) matches 20% of the larger number (5), the obtained value L = 25 is fully consistent with the given condition. This confirms that the answer is correct.Why Other Options Are Wrong:Option 15 is too small because the larger number must be greater than 20, not less than the smaller number. Option 35 gives a difference of 35 - 20 = 15, while 20% of 35 is 7, which does not match. Option 45 gives a difference of 25, but 20% of 45 is 9, so it fails the condition. Option 30 gives a difference of 10, but 20% of 30 is 6, again not equal to the required difference. Hence these options do not satisfy the percentage relationship stated in the problem.Common Pitfalls:A very common mistake is to take 20% of the smaller number instead of the larger number. Another error is to subtract 20 directly from the larger number assuming that 20% means subtracting 20 units, which is incorrect because 20% is a relative measure, not an absolute number. Some learners also wrongly assume that the larger number is 20 more than the smaller number without applying the percentage relationship. Careful reading and setting up the correct equation is essential to avoid these mistakes.Final Answer:The larger number is 25.

Question 17

Ram sells his goods 25% cheaper than Shyam and 25% dearer than Bram. By what percentage are Bram's goods cheaper than Shyam's goods?

Accepted Answer

Introduction / Context:This percentage problem compares the prices of goods sold by three people, Ram, Shyam, and Bram. It checks your ability to handle successive percentage changes and to compare prices expressed in relative terms rather than absolute values. Such questions are very common in profit and loss sections of aptitude exams and help build comfort with chained percentage relationships between different entities.Given Data / Assumptions:Ram sells his goods 25% cheaper than Shyam.Ram sells his goods 25% dearer (more expensive) than Bram.We are asked to find how much percent Bram's goods are cheaper than Shyam's goods.There is no mention of actual rupee values, so we can assume a convenient base price for Shyam to simplify calculations.Concept / Approach:When dealing with percentage comparisons, it is helpful to assume a simple base value. Here, we can assume Shyam's price is 100 units. Ram sells 25% cheaper than Shyam, so his price is 75 units. The statement that Ram is 25% dearer than Bram means Ram's price is 125% of Bram's price. Using that relation, we can compute Bram's price and then compare Bram's price to Shyam's price to find the required percentage difference. The key concepts used are basic percentage calculations and relative comparison.Step-by-Step Solution:Step 1: Assume Shyam's price is 100 units.Step 2: Since Ram sells 25% cheaper than Shyam, Ram's price is 100 - 25% of 100 = 100 - 25 = 75 units.Step 3: Ram sells 25% dearer than Bram, so Ram's price is 125% of Bram's price.Step 4: Let Bram's price be B. Then 1.25 * B = 75, because Ram's price is 75 units.Step 5: Solve for B: B = 75 / 1.25 = 60 units.Step 6: Now Shyam's price is 100 units and Bram's price is 60 units. The reduction from Shyam to Bram is 100 - 60 = 40 units.Step 7: Express this difference as a percentage of Shyam's price: (40 / 100) * 100% = 40%.Verification / Alternative check:We can verify by checking all relationships again. Starting with Shyam = 100 units, Bram = 60 units is 40% cheaper than Shyam because 40% of 100 is 40. Ram, being 25% dearer than Bram, should be 60 + 25% of 60 = 60 + 15 = 75 units. This also matches the earlier condition that Ram is 25% cheaper than Shyam, because 25% of 100 is 25 and 100 - 25 = 75. Both relationships hold, so the calculation is consistent and correct.Why Other Options Are Wrong:Option 60% would mean Bram sells at only 40 units when Shyam is at 100, which would not satisfy the given 25% dearer relationship with Ram. Option 50% would make Bram's price 50 units and would not produce Ram's price as both 75 and 25% dearer than Bram. Option 30% would give Bram's price as 70 units, making Ram's price inconsistent with the described percentage relations. Option 20% is clearly too small as Bram's price would be 80 units and Ram's price could not be both 75 and 25% cheaper than Shyam. Therefore, 40% is the only consistent answer.Common Pitfalls:A typical mistake is to directly add or subtract percentages without understanding which person is the reference base. Some learners also treat the 25% cheaper and 25% dearer statements as symmetrical, which is not correct because the percentage is applied on different bases. Another error is to compare Bram and Shyam via Ram in a single step without using algebra, which often leads to numerical slips. Always fix one person's price as 100 and trace the relationships carefully through each comparison.Final Answer:Bram's goods are 40% cheaper than Shyam's goods.

Question 18

By what percentage must a salary be decreased so that the decrease exactly cancels out a previous 20% increase in that salary?

Accepted Answer

Introduction / Context:This question is a classic example of successive percentage change, where an initial increase is followed by a decrease and we want to return to the original value. It tests whether learners understand that a 20% increase and a 20% decrease do not cancel each other out. Instead, the second percentage must be carefully calculated so that the combined effect of both changes brings the value back to its starting point. This concept is important in salary revisions, pricing, and discount related aptitude questions.Given Data / Assumptions:The salary is first increased by 20%.We want to apply a decrease afterwards so that the final salary becomes equal to the original salary.The required decrease is to be expressed as a percentage of the increased salary.No actual salary amount is given, so we are free to assume a convenient starting value, such as 100 units.Concept / Approach:The natural method is to assume an initial salary and apply the percentage changes step by step using multiplication factors. If the original salary is taken as 100 units, then a 20% increase turns it into 120 units. After this increase, the new salary is 120, and we must find a percentage decrease x% such that applying it to 120 brings the salary back to 100. This leads to a simple equation in one variable. The core concept is understanding that percentage changes are multiplicative rather than additive, especially when they occur successively.Step-by-Step Solution:Step 1: Assume the original salary is 100 units.Step 2: A 20% increase means the new salary is 100 + 20% of 100 = 100 + 20 = 120 units.Step 3: Let the required percentage decrease be x% applied to 120 to return to 100.Step 4: After the decrease, the new value is 120 * (1 - x/100).Step 5: Set this equal to the original salary: 120 * (1 - x/100) = 100.Step 6: Divide both sides by 120 to get 1 - x/100 = 100 / 120 = 5/6.Step 7: Rearrange to get x/100 = 1 - 5/6 = 1/6.Step 8: Therefore x = 100 * (1/6) = 16 2/3 percent.Verification / Alternative check:We can verify by applying the found decrease. Start with 100 units, increase by 20% to get 120 units. Now decrease 120 by 16 2/3 percent. The decrease amount is (16 2/3 / 100) * 120 = (1/6) * 120 = 20 units. Subtracting this from 120 gives 120 - 20 = 100 units, which is exactly the original salary. This confirms that the calculated percentage decrease of 16 2/3% is correct and precisely cancels the initial 20% increase.Why Other Options Are Wrong:Option (15 + 2/3)% is less than the required figure, so after decreasing by that amount, the salary would still be above 100. Option (14 + 1/3)% is even smaller and would leave the salary higher than its starting value. Option (13 + 4/5)% represents a still smaller decrease and is clearly insufficient. Option 20% might look attractive because it matches the initial increase, but applying 20% decrease to 120 leads to 96, not 100, so it does not cancel the increase. Only 16 2/3% restores the original value exactly.Common Pitfalls:The most frequent mistake is to assume that a 20% increase can be cancelled by a 20% decrease, which ignores the fact that the second percentage is applied to a changed base. Another common error is treating the percentages additively rather than multiplicatively, for example, writing 20 - x = 0 or 20 + x = 0 without using the correct equation. Some learners also skip the algebra and attempt mental approximations, increasing the chance of error. Always write down the multiplicative factors and set up the equation carefully.Final Answer:The salary must be decreased by 16 2/3% to exactly cancel out the 20% increase.

Question 19

A bag contains 600 coins of 25 paise denomination and 1200 coins of 50 paise denomination. If 12% of the 25 paise coins and 24% of the 50 paise coins are removed, the percentage of the total money removed from the bag is nearly equal to:

Accepted Answer

Introduction / Context:This is a mixed denomination percentage problem where we work with numbers of coins and their values at the same time. The question checks your ability to convert counts of coins into total money, apply percentage reductions, and then compute the percentage of the total value removed. It is a standard aptitude type question in percentage and simple arithmetic, relevant for banking, insurance, and general competitive exams.Given Data / Assumptions:There are 600 coins of 25 paise denomination.There are 1200 coins of 50 paise denomination.12% of the 25 paise coins are removed.24% of the 50 paise coins are removed.We are asked to find what percentage of the total money in the bag has been removed.We treat 25 paise as 0.25 rupee and 50 paise as 0.50 rupee for calculations.Concept / Approach:The key idea is that percentages of coins removed translate to percentages of value removed, but only within each denomination. Because the denominations are different, we must compute the value removed for each type separately and then sum them. First, we calculate the total amount of money in the bag. Next, we find how many coins of each type are removed using the given percentages and convert these removed coins into rupee values. Finally, we compare the total value removed with the original total value and express the result as a percentage. This systematic approach avoids confusion between counts and values.Step-by-Step Solution:Step 1: Value of 600 coins of 25 paise = 600 * 0.25 = 150 rupees.Step 2: Value of 1200 coins of 50 paise = 1200 * 0.50 = 600 rupees.Step 3: Total money in the bag initially = 150 + 600 = 750 rupees.Step 4: 12% of 600 coins of 25 paise are removed, that is 0.12 * 600 = 72 coins.Step 5: Value of removed 25 paise coins = 72 * 0.25 = 18 rupees.Step 6: 24% of 1200 coins of 50 paise are removed, that is 0.24 * 1200 = 288 coins.Step 7: Value of removed 50 paise coins = 288 * 0.50 = 144 rupees.Step 8: Total value removed = 18 + 144 = 162 rupees.Step 9: Required percentage removed = (162 / 750) * 100% = 21.6%.Verification / Alternative check:An alternative quick check is to compute the percentage of money represented by each denomination and ensure the removal is proportionate. The original total is 750 rupees. The removed amount is 162 rupees. Dividing 162 by 750 gives 0.216. Multiplying this decimal by 100 gives 21.6%. This matches our earlier computation from step-by-step arithmetic. Both methods confirm that the percentage of money removed is 21.6%, and the result is consistent and accurate.Why Other Options Are Wrong:Option 15.3% corresponds to a removed amount of approximately 115 rupees, which is less than the actual 162 rupees removed. Option 14.6% would mean about 109.5 rupees removed, again not matching the computed value. Option 12.5% suggests only 93.75 rupees removed, which is clearly too small given the significant removal of 25 paise and 50 paise coins. Option 18.2% represents around 136.5 rupees, still below the correct amount. Since only 21.6% matches 162 out of 750, the other values are incorrect.Common Pitfalls:One common error is to take 12% and 24% directly on the total amount without separating the denominations, which leads to wrong values. Another mistake is to convert paise to rupees incorrectly or to treat 25 paise as 25 rupees. Some learners also mistakenly calculate the percentage with respect to the number of coins instead of the total money value. To avoid these errors, always convert counts to values using the correct denomination and only then apply percentages and ratios.Final Answer:The percentage of money removed from the bag is approximately 21.6%.

Question 20

The numerator of a fraction is increased by 150% and the denominator of the fraction is increased by 350%. The resulting fraction is 25/51. What was the original fraction?

Accepted Answer

Introduction / Context:This question examines understanding of how fractions transform when both numerator and denominator are changed by different percentages. Instead of guessing, we must express the new fraction algebraically in terms of the original numerator and denominator. Such problems are common in aptitude tests because they connect percentages, fractions, and algebra in a single question. Mastering these ideas helps with ratio, proportion, and data interpretation tasks as well.Given Data / Assumptions:The original fraction has numerator N and denominator D.The numerator is increased by 150%, so the new numerator becomes N plus 150% of N.The denominator is increased by 350%, so the new denominator becomes D plus 350% of D.The resulting fraction after these changes is equal to 25/51.We are asked to find the original fraction N/D.Concept / Approach:Percent increases are best handled by converting them into multiplicative factors. Increasing a quantity by 150% means the final value is 250% of the original, or 2.5 times the original. Similarly, increasing a quantity by 350% makes it 450% of the original, or 4.5 times the original. Once we express the new numerator and denominator in terms of N and D, we equate the resulting fraction to 25/51, simplify, and solve for the ratio N/D. This is an algebraic approach based on proportions and is more reliable than trial and error.Step-by-Step Solution:Step 1: Let the original fraction be N/D.Step 2: Increasing the numerator by 150% makes the new numerator N + 1.5N = 2.5N.Step 3: Increasing the denominator by 350% makes the new denominator D + 3.5D = 4.5D.Step 4: The new fraction is therefore (2.5N) / (4.5D).Step 5: According to the question, (2.5N) / (4.5D) = 25 / 51.Step 6: Multiply numerator and denominator by 2 to remove decimals: (5N) / (9D) = 25 / 51.Step 7: Cross-multiply to get 5N * 51 = 25 * 9D.Step 8: This simplifies to 255N = 225D.Step 9: Divide both sides by 15 to get 17N = 15D.Step 10: Therefore N / D = 15 / 17, which is the original fraction.Verification / Alternative check:To verify, start from the original fraction 15/17. Increase the numerator 15 by 150%. That makes the new numerator 15 + 1.5 * 15 = 15 + 22.5 = 37.5. Increase the denominator 17 by 350%. That makes the new denominator 17 + 3.5 * 17 = 17 + 59.5 = 76.5. The new fraction is 37.5 / 76.5. Multiply numerator and denominator by 2 to clear halves, giving 75 / 153. Dividing numerator and denominator by 3, we get 25 / 51, which matches the fraction given in the question. This confirms that the original fraction is indeed 15/17.Why Other Options Are Wrong:Option 31/25 would not yield 25/51 when the described percentage changes are applied; the resulting fraction would be very different. Option 14/25, 11/16, and 9/11 each give transformed fractions that do not simplify to 25/51 when the numerator is multiplied by 2.5 and the denominator by 4.5. Only 15/17 gives the exact required resulting fraction. Therefore, the other options are mathematically inconsistent with the given condition.Common Pitfalls:Many learners misinterpret "increased by 150%" as becoming 150% of the original, instead of 250% of the original, leading to incorrect factors. Another common error is to add percentage values for numerator and denominator separately without converting to proper multipliers. Some also try to guess the original fraction from the options without doing the algebra, which can be risky when options are close in value. Always convert percentage changes into multiplication factors and solve the proportional equation carefully.Final Answer:The original fraction is 15/17.

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