Question 1

The present population of a town is 26,010 and it increases annually at a rate of 2%. What was the population of the town two years ago?

Accepted Answer

Introduction / Context: This question involves reverse calculation with compound growth. The population of a town grows every year by a fixed percentage, and we are given the present population. We must find the population two years ago by reversing the growth process. This is a typical example of reverse compound interest or population growth problems in quantitative aptitude. Given Data / Assumptions: - Present population of the town is 26,010. - The population increases by 2 percent per year. - We need to find the population two years ago, assuming the same growth rate each year and no migration effects. Concept / Approach: If P0 is the population two years ago and it grows at 2 percent per year, then after one year the population is P0 * 1.02, and after two years it is P0 * 1.02^2. We are given the final population after two years as 26,010. Therefore, we use the relation: 26,010 = P0 * (1.02)^2. We then solve for P0 by dividing 26,010 by (1.02)^2. Step-by-Step Solution: Let the population two years ago be P0. Growth rate per year = 2%, so growth factor per year = 1.02. After two years, population = P0 * 1.02 * 1.02 = P0 * (1.02)^2. Given that P0 * (1.02)^2 = 26,010. Compute (1.02)^2 = 1.0404. So, P0 = 26,010 / 1.0404. Evaluating this gives P0 = 25,000. Verification / Alternative check: Check forward. If the population two years ago was 25,000 and it grows at 2 percent per year, then after one year population becomes 25,000 * 1.02 = 25,500. After the second year, it becomes 25,500 * 1.02 = 26,010. This exactly matches the given present population, confirming that 25,000 is the correct population two years ago. Why Other Options Are Wrong: 25,100, 25,200, 25,350, and 25,500: If any of these were used as the starting population, applying 2 percent growth for two years would not result exactly in 26,010. They lead to slightly higher or lower final values, so they are not correct. Common Pitfalls: A common mistake is to subtract 2 percent twice from the current population rather than reversing compound growth properly. Another error is to assume linear growth instead of multiplicative growth. When the growth rate is per year and depends on the current population, it always behaves like compound interest and must be handled with powers of the growth factor, not simple subtraction. Final Answer: The population of the town two years ago was 25,000 people.

Question 2

A tyre is sold at three successive discounts of 20%, 30% and 20% on its marked price. What is the net single equivalent discount (in percentage) on the marked price?

Accepted Answer

Introduction / Context: This problem deals with successive discounts. When a shopkeeper offers several discounts one after another on the same marked price, the overall reduction cannot be found by simply adding the individual percentages. Instead, each discount acts on the already reduced price. The question asks for the net single discount that would give the same final selling price as the three successive discounts of 20%, 30% and 20%. Given Data / Assumptions: - Marked price (MP) of a tyre is some amount, say M rupees. - First discount = 20 percent. - Second discount = 30 percent on the reduced price after the first discount. - Third discount = 20 percent on the price after the second discount. - We must find the equivalent single discount percentage on M that gives the same final price. Concept / Approach: Each discount is implemented by multiplying the current price by a factor (1 - discount / 100). The final price factor after all discounts is the product of the individual factors. Once we know the final factor relative to the marked price, the net discount is 1 minus this factor, converted into a percentage. This method is analogous to successive percentage change problems in general. Step-by-Step Solution: Let the marked price be M. First discount 20% means factor = 1 - 20 / 100 = 0.80. Second discount 30% means factor = 1 - 30 / 100 = 0.70. Third discount 20% again means factor = 1 - 20 / 100 = 0.80. Final selling price factor = 0.80 * 0.70 * 0.80. Compute 0.80 * 0.70 = 0.56. Then 0.56 * 0.80 = 0.448. So final selling price = 0.448M. Net discount fraction = 1 - 0.448 = 0.552. Net discount percentage = 0.552 * 100 = 55.2 percent. Verification / Alternative check: Assume a marked price of Rs. 1000 for an easy numerical example. After the first 20% discount, price becomes 1000 * 0.80 = 800. After the second 30% discount, the price becomes 800 * 0.70 = 560. After the third 20% discount, the price becomes 560 * 0.80 = 448. The total discount is 1000 - 448 = 552, which is 55.2% of 1000. This confirms the net discount we calculated. Why Other Options Are Wrong: 59.6, 62.8 and 51.4: These values result from incorrect multiplication of factors or from naive addition of percentages without considering the reduced bases. They do not match the correct final price factor 0.448. 48.0: This is much too small and could come from averaging the three discounts rather than computing their combined effect. Common Pitfalls: Students often add discounts directly (20 + 30 + 20 = 70) or average them, both of which are wrong. Every discount applies to a new base, which is the reduced price after the previous discount. Forgetting this leads to large errors. The correct method is always to convert each discount to its multiplicative factor and multiply them to get the final factor. Final Answer: The net single equivalent discount on the tyre is 55.2 percent of the marked price.

Question 3

In a city, 40% of the people are illiterate and 60% are poor. Among the rich people, 10% are illiterate. What percentage of the total population are both poor and illiterate?

Accepted Answer

Introduction / Context: This question is a percentage and set intersection problem involving two categories, poverty and literacy. We are given the percentages of illiterates and poor people and partial information about illiteracy among the rich. We must determine what fraction of the total population is both poor and illiterate. Such questions are very common in data interpretation and reasoning sections. Given Data / Assumptions: - 40 percent of the total population is illiterate. - 60 percent of the total population is poor, so 40 percent is rich. - Among the rich, 10 percent are illiterate. - We are asked to find the percentage of the total population that is both poor and illiterate. Concept / Approach: We can treat the total population as 100 units for simplicity. Then the percentages can be interpreted directly as counts. First, we calculate how many rich people are illiterate using the given information. Then we subtract this number from the total illiterate population to obtain the number of poor illiterates. Finally, we express that number as a percentage of the total population. Step-by-Step Solution: Assume total population = 100 units. Illiterates = 40% of 100 = 40 people. Poor people = 60% of 100 = 60 people. Therefore, rich people = 100 - 60 = 40 people. Among the rich, 10% are illiterate. Illiterate rich = 10% of 40 = 4 people. Total illiterates = 40 people, of which 4 are rich and illiterate. Hence, poor illiterates = total illiterates - illiterate rich = 40 - 4 = 36 people. As a percentage of the total population, this is 36 out of 100, that is 36%. Verification / Alternative check: We can also think in terms of a table: rows for poor and rich, columns for literate and illiterate. Rich illiterate are 4, rich literate are 36. Total illiterates are 40, so poor illiterates must be 36. Poor literates then are 60 - 36 = 24. All numbers are consistent and add up: rich (4 + 36) = 40, poor (36 + 24) = 60, illiterates (36 + 4) = 40, literates (24 + 36) = 60. This confirms that 36 out of 100 are poor and illiterate. Why Other Options Are Wrong: 60: This would mean all poor people are illiterate, which contradicts the information about rich illiterates and the total illiterates. 50 and 40: These do not satisfy the breakdown of illiterates between rich and poor when checked with the given data. 64: This would exceed the total number of poor people and is impossible. Common Pitfalls: A frequent mistake is to add or subtract percentages without a clear reference to the total population, or to think that 40% illiterate and 60% poor means something like 40% of 60% without understanding the overlap. The safest way is to assume a convenient total population, such as 100, and then work out the actual counts in each intersecting group. Drawing a small table often helps to avoid confusion. Final Answer: The percentage of the total population that is both poor and illiterate is 36 percent.

Question 4

The list price (marked price) of an article is Rs. 900. It is sold after allowing two successive discounts of 20% and 10%. What is the final selling price of the article in rupees?

Accepted Answer

Introduction / Context: This question is another example of successive discounts on a marked price. A discount is applied, and then another discount is applied on the already reduced price. We must compute the final selling price starting from the marked price of Rs. 900. It illustrates how multiple discounts combine multiplicatively rather than additively. Given Data / Assumptions: - Marked price (MP) of the article = Rs. 900. - First discount = 20 percent on MP. - Second discount = 10 percent on the price after the first discount. - We must find the final selling price in rupees. Concept / Approach: Each discount reduces the current price by a certain percentage. A 20 percent discount corresponds to multiplying the price by 0.80. A 10 percent discount corresponds to multiplying the current price by 0.90. The final selling price is MP multiplied by both discount factors. It is important not to add the percentages directly (20 + 10) since the second discount is applied to a smaller base price. Step-by-Step Solution: Marked price MP = Rs. 900. First discount 20% gives factor = 1 - 20 / 100 = 0.80. Price after first discount = 900 * 0.80 = Rs. 720. Second discount 10% gives factor = 1 - 10 / 100 = 0.90. Final selling price = price after first discount * 0.90 = 720 * 0.90. Compute 720 * 0.90 = Rs. 648. Verification / Alternative check: We can also compute directly from the marked price: final factor = 0.80 * 0.90 = 0.72. Then selling price = 900 * 0.72 = 648. This matches the previous calculation, confirming that the final selling price is Rs. 648. Why Other Options Are Wrong: 640 and 700: These amounts do not correspond to any simple factor from 900 after two correct discounts of 20% and 10%. 720: This is the price after only the first 20% discount, before the 10% discount is applied. 738: This might arise from an incorrect multiplication or from applying one discount incorrectly. Common Pitfalls: One common mistake is to add discounts and take a single discount of 30%, which would give 900 * 0.70 = 630, not matching any option. Another error is to apply the second discount on the original price instead of the already discounted price. Always remember that successive discounts act one after another, each on the current reduced price. Final Answer: The final selling price of the article after two successive discounts is Rs. 648.

Question 5

A merchant marks cloth at Rs. 50 per metre. He allows two successive discounts of 15% and 20% on this marked price. What is the net selling price per metre of the cloth?

Accepted Answer

Introduction / Context: This is another example of successive discounts, this time on cloth priced per metre. The merchant first gives a 15 percent discount and then a further 20 percent discount on the reduced price. We are to compute the final price per metre that the customer pays. It reinforces the concept of multiplying discount factors rather than adding percentages. Given Data / Assumptions: - Marked price of cloth = Rs. 50 per metre. - First discount = 15 percent on the marked price. - Second discount = 20 percent on the price after the first discount. - We must find the net selling price per metre after both discounts. Concept / Approach: A 15 percent discount means the customer pays 85 percent of the current price. Mathematically, this is a factor of 0.85. A 20 percent discount means paying 80 percent of the current price, a factor of 0.80. The final price is obtained by multiplying the marked price by both factors, in order. This is the standard method for handling multiple discounts. Step-by-Step Solution: Marked price per metre = Rs. 50. First discount 15% gives factor = 1 - 15 / 100 = 0.85. Price after first discount = 50 * 0.85 = Rs. 42.50. Second discount 20% gives factor = 1 - 20 / 100 = 0.80. Final selling price per metre = 42.50 * 0.80. Compute 42.50 * 0.80 = 34.00. Therefore, net price per metre = Rs. 34.00. Verification / Alternative check: We can also compute directly from the marked price: overall factor = 0.85 * 0.80 = 0.68. So selling price = 50 * 0.68 = 34.00. This matches the earlier stepwise calculation and confirms the answer. Why Other Options Are Wrong: Rs. 32.50: This is the price after a single discount of 35% or a miscalculation and does not match the correct successive discount factors. Rs. 42.50: This is only the price after the first 15% discount, before the 20% discount is applied. Rs. 40.00 and Rs. 36.50: These numbers do not correspond to the correct final factor of 0.68 when applied to 50. Common Pitfalls: Candidates sometimes add the discounts (15 + 20) to incorrectly conclude a single 35% discount. That would give 50 * 0.65 = 32.50, which is not the same as the correct answer. Another pitfall is to calculate the second discount on the original price instead of on the reduced price after the first discount. Always apply each discount to the current price in sequence. Final Answer: The net selling price per metre of the cloth after both discounts is Rs. 34.00.

Question 6

Ramesh marks an article at Rs. 6000. After allowing a discount of 20% to the customer, he still earns a profit of 60%. What is the cost price of the article in rupees?

Accepted Answer

Introduction / Context: This question connects marked price, discount, selling price and cost price in a profit and loss situation. We are told the marked price and discount offered, plus the profit percentage on cost price. From this, we must work backwards to find the cost price. This type of question is typical in shopkeeper and trader profit problems. Given Data / Assumptions: - Marked price (MP) of the article = Rs. 6000. - Discount given = 20 percent on the marked price. - Profit earned = 60 percent of cost price. - We must determine the cost price (CP) of the article. Concept / Approach: The selling price (SP) is found from marked price and discount. Profit is calculated using SP and CP. Since we know the required profit percentage on CP, we can relate SP and CP using the formula: SP = CP * (1 + profit percent / 100). We first compute SP from the discount, then equate it to CP * 1.60 and solve for CP. Step-by-Step Solution: Marked price MP = Rs. 6000. Discount = 20%, so discount factor = 1 - 20 / 100 = 0.80. Selling price SP = MP * 0.80 = 6000 * 0.80 = Rs. 4800. Profit percentage = 60%, so SP = CP * (1 + 60 / 100) = CP * 1.60. Therefore, CP * 1.60 = 4800. So CP = 4800 / 1.60. Compute CP = 3000. Verification / Alternative check: Check the result: If CP is 3000 and profit is 60%, then SP should be 3000 * 1.60 = 4800. This matches the SP we computed from discount on the marked price (6000 * 0.80 = 4800). Therefore, both approaches agree, confirming that the cost price must be Rs. 3000. Why Other Options Are Wrong: 3600, 4200 and 4800: These values do not satisfy both conditions simultaneously when used as CP. They lead to incorrect profit percentages or wrong selling prices when rechecked. 3500: This value would result in a profit different from exactly 60%, so it does not match the given information. Common Pitfalls: Some learners mistakenly treat the 60 percent profit as being calculated on the marked price or selling price instead of cost price. Others may directly subtract the discount from the marked price and treat that result as cost price, ignoring the profit information. Always remember that profit percentage is defined with respect to cost price, while discount percentage is defined with respect to marked price. Final Answer: The cost price of the article is Rs. 3000.

Question 7

In a television program called Jabardast, 1/8 of the audience were children, 2/5 of the audience were men and the rest were women. If the number of women in the audience was 380, what was the total audience for the program?

Accepted Answer

Introduction / Context: This question is a fractions and percentage type word problem about composition of an audience. The audience is divided into three groups: children, men and women. The fractions for children and men are given, and the absolute number of women is given. From this information, we must find the total audience. It is a classic example of using fractional parts of a whole. Given Data / Assumptions: - Children constitute 1/8 of the total audience. - Men constitute 2/5 of the total audience. - The remaining audience members are women. - Number of women in the audience = 380. - We must find the total audience size. Concept / Approach: Let the total audience be A. Then children = A / 8, men = (2 / 5) * A, and women = A minus children minus men. Using the given number of women, we can write an equation in A and solve for A. Working with fractions requires careful handling of denominators, but the process is straightforward. Step-by-Step Solution: Let total audience be A. Children = 1 / 8 of A = A / 8. Men = 2 / 5 of A = (2A) / 5. Women = A - (A / 8) - (2A / 5). Find a common denominator for 8 and 5, which is 40. Convert fractions: A / 8 = 5A / 40 and 2A / 5 = 16A / 40. So women = A - (5A / 40) - (16A / 40). Rewrite A as 40A / 40. Thus women = (40A / 40) - (5A / 40) - (16A / 40) = (40A - 5A - 16A) / 40. Simplify numerator: 40A - 5A - 16A = 19A. So women = 19A / 40. Given that women = 380, so 19A / 40 = 380. Therefore, A = 380 * 40 / 19. Compute 380 / 19 = 20, so A = 20 * 40 = 800. Verification / Alternative check: With A = 800, compute each group: children = 1/8 of 800 = 100, men = 2/5 of 800 = 320. Women = 800 - 100 - 320 = 380, which matches the given number. All fractions and totals are consistent, so 800 is indeed the correct total audience size. Why Other Options Are Wrong: 600, 500, 400, and 700: None of these values produce 380 women when the given fractional proportions for children and men are applied and subtracted from the total. Checking any of them quickly shows that the resulting number of women is different from 380. Common Pitfalls: Students sometimes misinterpret "remaining" and forget that both children and men must be subtracted from the total to find women. Another common error is incorrect fraction arithmetic, such as adding denominators or failing to find a common denominator. Writing all steps clearly, especially when combining fractions, helps avoid mistakes. Final Answer: The total audience in the Jabardast program was 800 people.

Question 8

If 4/5 of 6/7 of a number is equal to 216, then what is 8/9 of that number?

Accepted Answer

Introduction / Context: This question checks understanding of fractions of a number and the ability to work backward from a given fractional value. You are given that a composite fraction of a number equals 216, and you must find another fraction of the same number. It is a typical algebraic fractions problem appearing in aptitude tests. Given Data / Assumptions: - 4/5 of 6/7 of a number N is equal to 216. - We must find the value of 8/9 of N. Concept / Approach: The statement 4/5 of 6/7 of N equals 216 can be written as: (4 / 5) * (6 / 7) * N = 216. From this, we solve for N. After obtaining N, we compute (8 / 9) * N. The key is careful manipulation of fractions and simplification to avoid arithmetic mistakes. Step-by-Step Solution: We have (4 / 5) * (6 / 7) * N = 216. First simplify the fraction (4 / 5) * (6 / 7) = 24 / 35. So (24 / 35) * N = 216. To solve for N, multiply both sides by 35 / 24. N = 216 * (35 / 24). Simplify 216 / 24 = 9. Thus N = 9 * 35 = 315. Now compute 8 / 9 of N. 8 / 9 of N = (8 / 9) * 315. Compute 315 / 9 = 35. So 8 / 9 of N = 8 * 35 = 280. Verification / Alternative check: We can verify by plugging N = 315 back into the original condition. Compute 6/7 of 315: since 315 / 7 = 45, 6/7 of 315 = 6 * 45 = 270. Then 4/5 of 270 is (4 / 5) * 270 = 4 * 54 = 216, which matches the given value. This confirms that N = 315 is correct, and therefore 8/9 of N = 280 is also correct. Why Other Options Are Wrong: 179, 160, 149, and 200: None of these values matches 8/9 of 315. They result from incorrect handling of fractions or miscalculation while solving for N. Only 280 is consistent with both the given condition and the derived value of N. Common Pitfalls: A common mistake is to misinterpret "4/5 of 6/7 of a number" and either add or average the fractions instead of multiplying them. Another error is to make arithmetic mistakes when simplifying 216 * 35 / 24. It helps to cancel common factors early to keep numbers small and manageable, such as reducing 216 / 24 to 9 before multiplying by 35. Final Answer: The value of 8/9 of the number is 280.

Question 9

If the price of a pen decreases by 20%, a man can buy 10 more pens for Rs. 100. What is the new price of each pen in rupees?

Accepted Answer

Introduction / Context: This question combines percentage decrease in price with a change in quantity purchased for a fixed total amount of money. We are told that after a 20 percent decrease in price, the same amount of money buys 10 more pens. Using this information, we must find the new price per pen. It is a standard type of question in percentage and unitary method topics. Given Data / Assumptions: - Let the original price of one pen be P rupees. - New price after 20% decrease is 0.80P rupees per pen. - Total money spent in both cases is Rs. 100. - After the price reduction, the man can buy 10 more pens than before with the same Rs. 100. - We must find the new price per pen in rupees. Concept / Approach: The number of pens purchasable with a fixed amount is equal to total money divided by price per pen. Let the original number of pens be n. Then after the price decrease, the number becomes n + 10. By setting up equations for these quantities using Rs. 100, we can solve for P and then compute the new price 0.80P. The key is to handle the ratio 100 / price carefully and express the change in quantity correctly. Step-by-Step Solution: Let original price per pen be P. Original number of pens that can be bought with Rs. 100 = 100 / P. New price per pen after 20% decrease = 0.80P. New number of pens that can be bought with Rs. 100 = 100 / (0.80P). We are told that the man can buy 10 more pens after the price decrease. So, 100 / (0.80P) = 100 / P + 10. Note that 100 / (0.80P) = 100 / (4P / 5) = 100 * (5 / 4P) = 125 / P. Thus the equation becomes 125 / P = 100 / P + 10. Subtract 100 / P from both sides: (125 / P) - (100 / P) = 10. This simplifies to 25 / P = 10. Therefore, P = 25 / 10 = 2.5. So the original price per pen was Rs. 2.50. New price per pen after 20% decrease = 0.80 * 2.5 = 2.0. Hence, the new price per pen is Rs. 2. Verification / Alternative check: At original price P = 2.50, the man can buy 100 / 2.50 = 40 pens. At the new price Rs. 2, he can buy 100 / 2 = 50 pens. The difference is 50 - 40 = 10 pens, which matches the condition in the problem. This confirms that the new price of each pen is Rs. 2. Why Other Options Are Wrong: 1: At this price, 100 / 1 = 100 pens, which is far more than 10 extra pens compared with the original quantity. 4 and 5: These values are larger than the original price 2.5 and would decrease the number of pens the man can buy, not increase it. 3: This price is still higher than 2.5 and would result in fewer pens purchased, not more. Common Pitfalls: Some students mistakenly assume that the new price is 20 percent directly of the old price instead of 20 percent less. Others mix up the change in the number of pens and treat 10 as the original number of pens. The correct approach is to express the numbers of pens before and after the price change in terms of P and then use the given difference to form an equation. Careful algebra leads to the correct answer. Final Answer: The new price of each pen after the 20 percent decrease is Rs. 2 per pen.

Question 10

After deducting 60% from a number and then 15% from the remainder, 1428 is left. What was the original number?

Accepted Answer

Introduction / Context: This question tests the concept of successive percentage deductions from an original quantity. Such problems are common in aptitude exams and real life when a value is reduced in more than one step, for example through successive taxes, discounts, or losses. The key idea is that each percentage is applied on the remaining amount, not on the original number, and therefore we must treat them as multiplicative factors. Here, we know the final remaining value after two reductions and we are asked to recover the original number. Given Data / Assumptions: The original number is unknown and we can call it N. First deduction is 60 percent of N, so only 40 percent of N remains. Second deduction is 15 percent of the remaining amount, so 85 percent of that remainder is left. After both deductions, the final amount is 1428. Concept / Approach: When a quantity is successively reduced by some percentages, we multiply the original quantity by the remaining percentage factors written in decimal form. Here the remaining fraction after a 60 percent deduction is 40 percent, which is 0.40. After that, a further 15 percent deduction leaves 85 percent, which is 0.85. Thus the final amount can be written as N * 0.40 * 0.85. We then set this expression equal to the given final value, 1428, and solve for N using simple algebra and division. Step-by-Step Solution: Let the original number be N. After deducting 60 percent, the remaining amount is 40 percent of N, which is 0.40 * N. Next, 15 percent is deducted from this remainder, so 85 percent is left: remaining amount = 0.85 * (0.40 * N). Therefore final amount = 0.40 * 0.85 * N = 0.34 * N. We are told the final amount is 1428, so 0.34 * N = 1428. Hence N = 1428 / 0.34. Compute N: 1428 / 0.34 = 4200. Therefore the original number is 4200. Verification / Alternative check: We can verify the answer by working forward from 4200. First deduction: 60 percent of 4200 is 0.60 * 4200 = 2520. Remaining amount is 4200 - 2520 = 1680, which matches 40 percent of 4200. Second deduction: 15 percent of 1680 is 0.15 * 1680 = 252. Remaining amount is 1680 - 252 = 1428. This final value matches the number given in the question, confirming that 4200 is correct. The arithmetic is consistent at each step, so no contradiction appears. Why Other Options Are Wrong: Option 3962 is wrong because 0.34 * 3962 is not equal to 1428; it gives a different final value. Option 4150 is also incorrect because applying 60 percent then 15 percent to 4150 does not produce 1428. Option 4300 is too large; 0.34 * 4300 gives a final amount greater than 1428. Only 4200 exactly reproduces the final value when both reductions are applied correctly. Common Pitfalls: A common mistake is to simply add 60 percent and 15 percent and assume a single 75 percent deduction, which is incorrect because the second percentage is applied after the first deduction. Another frequent error is to take 60 percent and 15 percent of the same original number rather than using the remainder. Some candidates also incorrectly round intermediate values, which can change the final result. Always convert percentages to decimal factors and apply them successively to avoid these issues. Final Answer: The original number before both deductions were applied was 4200.

Question 11

A shopkeeper allows a 20% discount on the marked price. What is the marked price of an article whose selling price after discount is Rs. 740?

Accepted Answer

Introduction / Context: This percentage problem involves finding the marked price of an article when the discount percentage and the selling price after discount are given. Situations like this are common in shopping and in competitive examinations. Understanding the relationship between marked price, discount percentage, and selling price helps to reverse engineer the original price before discount from the final selling price. Given Data / Assumptions: The shopkeeper gives a discount of 20 percent on the marked price (MP). The selling price (SP) after discount is Rs. 740. Discount is calculated on the marked price, not on cost price. We are asked to find the marked price of the article. Concept / Approach: A discount of 20 percent means that the customer pays 80 percent of the marked price. In percentage form this is 100% - 20% = 80%. In decimal form, 80 percent is 0.80. Therefore, the selling price can be expressed as SP = 0.80 * MP. Since we know SP, we rearrange this relationship to find MP by dividing the selling price by 0.80. This approach avoids confusion and directly links discount percentage, marked price, and selling price in a simple algebraic equation. Step-by-Step Solution: Let the marked price be MP. Discount = 20 percent of MP, so the customer pays 80 percent of MP. Thus, SP = 80 percent of MP = 0.80 * MP. We are given SP = Rs. 740. So, 0.80 * MP = 740. Therefore, MP = 740 / 0.80. Compute MP: 740 / 0.80 = 925. Therefore, the marked price of the article is Rs. 925. Verification / Alternative check: We can verify by applying the discount to Rs. 925. Discount amount at 20 percent is 0.20 * 925 = 185. Subtract this from the marked price: 925 - 185 = 740. This matches the given selling price exactly, confirming that our calculated marked price is correct. This forward check is a simple way to ensure that no arithmetic mistake was made while rearranging and solving the equation. Why Other Options Are Wrong: If the marked price were Rs. 725, 80 percent of 725 gives 580, not 740, so that option is incorrect. If the marked price were Rs. 875, 20 percent discount leads to a selling price of 700, which again does not match 740. If the marked price were Rs. 1040, then at 20 percent discount the selling price would be 832, not 740. Only the marked price of Rs. 925 correctly produces the given selling price after applying the 20 percent discount. Common Pitfalls: Many learners mistakenly subtract 20 percent directly from the selling price instead of from the marked price, which is conceptually wrong. Others divide 740 by 0.20 rather than by 0.80, confusing the discount rate with the payable rate. Another typical mistake is to use 80 instead of 0.80 in the equation, which leads to drastically wrong results. Always remember that selling price after a discount of D percent is (100 - D) percent of the marked price and convert that percentage properly into a decimal factor before calculation. Final Answer: The marked price of the article before discount was Rs. 925.

Question 12

The number of trees in a town is 17,640 after a 5% annual increase for two years. How many trees were there 2 years ago?

Accepted Answer

Introduction / Context: This question examines percentage growth over multiple years, which is a classic example of compound increase. Instead of simple addition of percentages, each year the increase is applied to the updated value. In population growth, interest calculations, or value appreciation, we must treat these as multiplicative, not additive, processes. Here we are given the current number of trees and the annual growth rate, and we must find the original number of trees two years earlier. Given Data / Assumptions: Present number of trees in the town = 17,640. Annual percentage increase in the number of trees = 5 percent per year. The increase has taken place for 2 consecutive years at the same rate. We need to find the number of trees 2 years ago, before these increases. Concept / Approach: A 5 percent increase means the quantity becomes 105 percent of its previous value, which is 1.05 times the earlier amount. After two years with the same annual increase, the overall multiplication factor is 1.05 * 1.05, that is 1.05^2. If the initial number of trees is N, then after two years the number becomes N * 1.05^2. We know this final number is 17,640. Therefore, N = 17,640 / (1.05^2). Evaluating this gives the original population of trees. This is the standard compound growth formula rearranged to recover the initial value. Step-by-Step Solution: Let the number of trees 2 years ago be N. After 1 year at 5 percent increase, trees become N * 1.05. After another year at 5 percent increase, trees become N * 1.05 * 1.05 = N * 1.05^2. We are given that after 2 years the number of trees is 17,640. So N * 1.05^2 = 17640. Compute 1.05^2: 1.05 * 1.05 = 1.1025. Therefore N = 17640 / 1.1025. 17640 / 1.1025 = 16000. Hence, there were 16,000 trees in the town 2 years ago. Verification / Alternative check: Verify using forward calculation. Start with 16,000 trees. After 1 year with 5 percent increase: 16,000 * 1.05 = 16,800. After the second year: 16,800 * 1.05 = 17,640. This matches the final value given in the question, so our initial value is correct. Any other starting value would not produce exactly 17,640 after two successive increases of 5 percent each. Why Other Options Are Wrong: If the initial number were 14,000, then after 2 years at 5 percent per year the final value would be 14,000 * 1.05^2, which is less than 17,640. Similarly, with 15,000 as the starting value, the final population would still be smaller than 17,640. If we take 18,000 as the starting value, the final population becomes 18,000 * 1.05^2, which exceeds 17,640. Only 16,000 gives the exact required final count. Common Pitfalls: A frequent mistake is to double the 5 percent and assume the total growth over two years is 10 percent, then divide 17,640 by 1.10. This ignores compounding and leads to an incorrect answer. Another error is to subtract twice 5 percent from 17,640 instead of reversing the growth properly. Always remember that repeated percentage increases or decreases must be handled with multiplication by growth factors, not simple addition or subtraction of percentages, especially when reversing the process to find an original amount. Final Answer: The town had 16,000 trees 2 years ago before the two successive 5 percent increases.

Question 13

If A scored 20% more marks than B, by what percentage are the marks of B less than the marks of A?

Accepted Answer

Introduction / Context: This question is about the relationship between two quantities when one is a given percentage more than the other. It highlights an important concept that percentage increase and percentage decrease between two values are not symmetric. If A is 20 percent more than B, the percentage by which B is less than A is not 20 percent, but a different value that must be computed carefully. Given Data / Assumptions: Marks of B are taken as a base reference value. A scored 20 percent more marks than B. We need the percentage by which B is less than A. We assume marks are positive numbers and percentages are based on standard definitions. Concept / Approach: Let the marks of B be 100 units for simplicity. Then a 20 percent increase means A has 120 marks. The percentage by which B is less than A is calculated using A as the base, because the question asks relative to A. So we compute (A - B) / A * 100 percent. With A = 120 and B = 100, the difference is 20. Therefore the required percentage is 20 / 120 * 100, which simplifies to 16.66 percent (approximately). This illustrates that a 20 percent increase from B to A corresponds to about a 16.66 percent decrease when going from A back to B. Step-by-Step Solution: Assume marks of B = 100 units. A scored 20 percent more than B, so marks of A = 100 + 20 percent of 100. 20 percent of 100 = 20, so marks of A = 120. Now we find by what percent B is less than A. Difference in marks = A - B = 120 - 100 = 20. Required percentage = (difference / A) * 100 = (20 / 120) * 100. Compute 20 / 120 = 1 / 6, and 1 / 6 * 100 = 16.66 percent (approximately). Therefore, the marks of B are 16.66 percent less than the marks of A. Verification / Alternative check: We can choose any convenient number for B, not just 100, and the percentage will be the same. For example, let B = 50. Then A is 20 percent more, so A = 50 + 0.20 * 50 = 60. The difference is 10. Now the percentage by which B is less than A is (10 / 60) * 100 = 16.66 percent. This confirms that the result does not depend on the particular starting value chosen, as the ratio remains constant. Why Other Options Are Wrong: 20 percent would be correct only if we compared relative to B instead of A, which is not what the question asks. 33.33 percent corresponds to a situation where one value is half of the other, which is not the case here. The value 14.28 percent corresponds roughly to 1 / 7 and does not fit the actual ratio 100 to 120. Only 16.66 percent correctly represents how much smaller B is compared to A when A is 20 percent more than B. Common Pitfalls: The most common mistake is to assume that if A is 20 percent more than B, then B is automatically 20 percent less than A. This is incorrect because the bases for the two percentages are different. Another error is to subtract 20 from 100 and say the answer is 80 percent, completely misinterpreting the question. Always identify clearly which value is being taken as the base for the percentage comparison and set up the ratio accordingly. Final Answer: The marks of B are approximately 16.66 percent less than the marks of A.

Question 14

The price of onion rises from Rs. 24 per kg to Rs. 36 per kg. By what percentage should a household reduce consumption so that its expenditure on onions remains the same?

Accepted Answer

Introduction / Context: This question deals with the relationship between price, quantity, and expenditure. When the price of a commodity increases, a consumer must either increase expenditure or reduce consumption to keep the total spending constant. Here, we want to maintain the same expenditure on onions even though the price per kilogram has increased, so we must find the required percentage reduction in quantity purchased. Given Data / Assumptions: Original price of onion = Rs. 24 per kg. New price of onion = Rs. 36 per kg. Total expenditure before and after the price rise should remain constant. We must find the percentage decrease in consumption (quantity) of onions. Concept / Approach: Expenditure is given by price * quantity. Let the original quantity be Q kilograms. The original expenditure is 24 * Q. After the price rise, new price is 36 per kg. Let the new quantity be q kilograms. To keep expenditure the same, 24 * Q must equal 36 * q. From this equation, we can express q in terms of Q and find the ratio q / Q. Once we have the ratio of new quantity to old quantity, we can compute the percentage decrease in quantity as (Q - q) / Q * 100. Because price increased by a factor of 1.5 (from 24 to 36), quantity must adjust inversely by a factor of 2 / 3. Step-by-Step Solution: Let the original quantity purchased be Q kg. Original price per kg = Rs. 24, so original expenditure = 24 * Q. New price per kg = Rs. 36, let new quantity be q kg. Since expenditure is unchanged: 24 * Q = 36 * q. Solve for q: q = (24 / 36) * Q = (2 / 3) * Q. Thus, new quantity is two thirds of the original quantity. Decrease in quantity = Q - q = Q - (2 / 3) * Q = (1 / 3) * Q. Percentage decrease = (decrease / original quantity) * 100 = (1 / 3) * 100 = 33.33 percent (approximately). Verification / Alternative check: Assume a concrete original quantity, for example Q = 3 kg. Original expenditure = 24 * 3 = 72 rupees. After the price rise, the new price is 36 per kg. If the quantity is reduced to q = 2 kg (which is two thirds of 3), then new expenditure = 36 * 2 = 72 rupees, exactly the same as before. The reduction from 3 kg to 2 kg is 1 kg, which is 1 / 3 of 3 kg, that is 33.33 percent. This numerical example confirms our algebraic result. Why Other Options Are Wrong: A 25 percent reduction would make the new quantity 75 percent of the old one, which would not balance the 50 percent increase in price. A 50 percent reduction in quantity would lower expenditure far below the original, since price increased only by 50 percent. A 20 percent reduction is too small and would cause total expenditure to increase. Only a 33.33 percent reduction exactly offsets the price increase from 24 to 36 per kg. Common Pitfalls: Some learners try to subtract the percentages directly, which is not valid. Others incorrectly assume that a 50 percent increase in price demands a 50 percent decrease in quantity, not realizing the inverse proportionality in the expenditure formula. It is also easy to mix up the roles of price and quantity when setting up the equation. Always start from the equation price1 * quantity1 = price2 * quantity2 when expenditure must stay the same, then solve systematically for the unknown quantity and compute the required percentage change. Final Answer: The household should reduce its onion consumption by approximately 33.33 percent to keep expenditure the same.

Question 15

A book has a marked price of Rs. 2150 and is sold for Rs. 1892. What is the discount percentage offered on the marked price?

Accepted Answer

Introduction / Context: This question is a straightforward percentage discount problem. The marked price of a product is given, along with the selling price after discount. The task is to find what percentage discount the buyer received. Such problems are very common in commerce, shopping, and aptitude tests, and they reinforce the basic idea of percentage decrease from an original reference value. Given Data / Assumptions: Marked price (MP) of the book = Rs. 2150. Selling price (SP) after discount = Rs. 1892. Discount is calculated as the difference between MP and SP. We need to find the discount percentage with respect to the marked price. Concept / Approach: The discount amount is MP minus SP. Once we know the discount amount, the discount percentage is given by (discount amount / marked price) * 100. Here, this translates to ((2150 - 1892) / 2150) * 100. It is important to remember that this percentage is always calculated on the marked price, since that is the reference value before any deduction. This simple formula provides the required percentage reduction. Step-by-Step Solution: Marked price MP = Rs. 2150. Selling price SP = Rs. 1892. Discount amount = MP - SP = 2150 - 1892. Compute 2150 - 1892 = 258. Discount percentage = (discount amount / MP) * 100. So, discount percentage = (258 / 2150) * 100. Compute 258 / 2150 = 0.12. Therefore discount percentage = 0.12 * 100 = 12 percent. Verification / Alternative check: We can verify by checking what 12 percent of 2150 is. Calculate 12 percent of 2150: (12 / 100) * 2150 = 0.12 * 2150 = 258. Subtract this discount from the marked price: 2150 - 258 = 1892, which matches the given selling price. This confirms that the discount percentage is indeed 12 percent and that there are no arithmetic errors in our calculation. Why Other Options Are Wrong: If the discount were 13 percent, then discount amount would be 13 percent of 2150 = 279.5, leading to a selling price of 1870.5, not 1892. For a 14 percent discount, the discount amount would be 301, and the selling price would become 1849, again incorrect. For a 15 percent discount, the selling price would reduce even further to 1827.5. None of these match the given selling price, so only 12 percent is consistent with the data. Common Pitfalls: Some candidates mistakenly use the selling price as the base and compute discount percentage relative to SP, which is not standard in discount problems. Others confuse the concept of profit and discount and try to relate the values to cost price instead of marked price. A few may also commit arithmetic errors by mixing up subtraction or by misplacing decimal points. To avoid such mistakes, always clearly identify the base (here, marked price), compute the difference, and then apply the formula percentage = (difference / base) * 100. Final Answer: The discount percentage offered on the book is 12 percent.

Question 16

Three successive discounts of 20%, 10% and 30% are allowed on an article. What is the net single discount percentage equivalent to these three discounts?

Accepted Answer

Introduction / Context: This question explores the idea of successive discounts and asks for a single equivalent discount that gives the same final price as applying three different discounts one after another. In practical life, retailers sometimes offer multiple discounts, and customers or exam candidates must understand that these discounts do not simply add up. Instead, each discount is applied to the price remaining after the previous discount. Given Data / Assumptions: An article is given three successive discounts of 20 percent, 10 percent, and 30 percent. We are asked to find one equivalent discount that produces the same final price. No information about the marked price is required because we can assume any convenient initial value, for example 100 units. Concept / Approach: If the marked price is M, then after a discount of D percent the price becomes M * (1 - D / 100). For successive discounts, we multiply all the remaining percentage factors. Here, the remaining factors are 80 percent (0.80) after 20 percent, then 90 percent (0.90) after 10 percent, and 70 percent (0.70) after 30 percent. Thus, the final price factor is 0.80 * 0.90 * 0.70. The equivalent single discount D_eq is found from the equation final factor = 1 - D_eq / 100. Solving for D_eq gives the net discount percentage. Step-by-Step Solution: Assume marked price = 100 units for simplicity. After first discount of 20 percent, remaining factor = 1 - 0.20 = 0.80. After second discount of 10 percent, new factor = 1 - 0.10 = 0.90. After third discount of 30 percent, new factor = 1 - 0.30 = 0.70. Overall multiplication factor = 0.80 * 0.90 * 0.70. Compute 0.80 * 0.90 = 0.72. Then 0.72 * 0.70 = 0.504. So final price = 100 * 0.504 = 50.4 units. This means the equivalent discount is 100 - 50.4 = 49.6 units on a base of 100 units. Hence net discount percentage = 49.6 percent. Verification / Alternative check: We can also express the equivalent discount as D_eq using the formula 1 - D_eq / 100 = product of remaining factors. So 1 - D_eq / 100 = 0.504. Thus D_eq / 100 = 1 - 0.504 = 0.496 and D_eq = 0.496 * 100 = 49.6 percent. This matches the result obtained by applying the assumed marked price of 100 units. Since the calculation uses pure percentages, the result does not depend on the actual marked price. Why Other Options Are Wrong: A net discount of 45.2 percent would leave a final factor of 0.548, which is larger than 0.504, so the final price would be higher than it should be. A net discount of 54.6 percent would leave only 45.4 percent of the price, which is too low compared to 50.4 percent. A net discount of 50.4 percent would leave 49.6 percent of the price, which is smaller than the computed 50.4 percent. Only a 49.6 percent discount correctly reflects the combined effect of the three given discounts. Common Pitfalls: A very common error is to simply add the three discounts: 20 percent + 10 percent + 30 percent = 60 percent. This ignores the fact that each discount successively reduces the base on which the next discount is calculated. Another trap is rounding intermediate values too aggressively, which can lead to a slightly off final discount. To avoid mistakes, always convert each discount to a remaining factor, multiply these factors, and then convert back to a percentage discount from the original price. Final Answer: The net single discount equivalent to the three successive discounts is 49.6 percent.

Question 17

If A is six times more than B, by what percentage are the value of B and the value of A different? Specifically, by what percentage is B less than A?

Accepted Answer

Introduction / Context: This question assesses the relationship between two quantities when one is described as being several times more than the other. The phrase "six times more than" is usually interpreted in aptitude questions as A = B + 6B = 7B. The task is to compute the percentage by which the smaller quantity is less than the larger one. This emphasizes that percentage comparisons depend on which value is taken as the base in the calculation. Given Data / Assumptions: A is stated to be six times more than B, interpreted as A = 7B. B is the smaller quantity, A is the larger quantity. We must find the percentage by which B is less than A, using A as the base for percentage comparison. Values are positive real numbers, and percentage calculations use standard definitions. Concept / Approach: If we let B = 1 unit, then A = 7 units based on the interpretation A = B + 6B. The difference A - B is 6 units. The percentage by which B is less than A equals (A - B) / A * 100. Substituting A = 7 and B = 1, the required percentage is 6 / 7 * 100. This fraction evaluates to approximately 85.71 percent. Thus B is 85.71 percent less than A. The exact numbers chosen for B do not matter because the ratio between A and B is fixed. Step-by-Step Solution: Let B = 1 unit for simplicity. A is six times more than B, so A = B + 6B = 7B. Therefore A = 7 * 1 = 7 units. Difference between A and B = A - B = 7 - 1 = 6 units. We need the percentage by which B is less than A. Required percentage = (difference / A) * 100 = (6 / 7) * 100. Compute (6 / 7) * 100 = 85.71 percent (approximately). Hence, the value of B is 85.71 percent less than the value of A. Verification / Alternative check: We can choose a different convenient value for B, for example B = 10. Then A = 10 + 6 * 10 = 70. The difference is 70 - 10 = 60. Percentage by which B is less than A is (60 / 70) * 100 = 6 / 7 * 100 = 85.71 percent. This reproduces the same percentage result, confirming that the chosen base value for B does not affect the final percentage as long as the ratio A : B remains 7 : 1. Why Other Options Are Wrong: 64.82 percent does not align with the ratio 7 : 1 and typically arises from a different, incorrect interpretation. 83.33 percent corresponds to a difference ratio of 5 / 6, which is not the case here. 28.56 percent is far too small and does not reflect how much larger A is compared to B. Only 85.71 percent matches the exact ratio of the difference to the larger value A in this situation. Common Pitfalls: A common error is misinterpreting "six times more than" as merely A = 6B instead of A = 7B, which would change the percentage completely. Another typical mistake is to calculate the percentage difference using B as the base, which would give 600 percent, describing how much more A is than B rather than how much less B is than A. Always read the wording carefully and identify which quantity is the reference base for the percentage calculation. Final Answer: The value of B is 85.71 percent less than the value of A.

Question 18

The selling price of a table is Rs. 2808 and the profit percentage is 17%. What is the cost price of the table?

Accepted Answer

Introduction / Context: This problem relates selling price, cost price, and profit percentage. In commercial arithmetic, when the profit percentage and selling price are known, we can find the cost price by reversing the percentage increase. Profit is always calculated on the cost price, so the selling price is cost price plus profit. Understanding the direct formula linking these values is essential for quickly solving such questions. Given Data / Assumptions: Selling price (SP) of the table = Rs. 2808. Profit percentage = 17 percent. Profit is calculated on the cost price (CP). We are required to find the cost price of the table. Concept / Approach: If the profit on cost price is P percent, then selling price = cost price * (1 + P / 100). Here, with a 17 percent profit, we have SP = CP * 1.17. Hence, cost price CP can be obtained by CP = SP / 1.17. This is a direct algebraic rearrangement and avoids any confusion between the various price components. Once we compute CP, we can check by adding 17 percent of CP to see whether we indeed get back the selling price. Step-by-Step Solution: Let cost price be CP. Profit percentage = 17 percent, so profit = 17 percent of CP. Thus selling price SP = CP + 0.17 * CP = 1.17 * CP. We are given SP = Rs. 2808. So 1.17 * CP = 2808. Therefore CP = 2808 / 1.17. Compute 2808 / 1.17 = 2400. Hence, the cost price of the table is Rs. 2400. Verification / Alternative check: Check by computing profit on the found cost price. Profit = 17 percent of 2400 = 0.17 * 2400 = 408. Add this profit to the cost price: 2400 + 408 = 2808. This matches the given selling price, confirming that our calculated cost price is correct. If we had chosen any other cost price, the resulting selling price after adding 17 percent would not equal 2808. Why Other Options Are Wrong: If the cost price were 2288, then SP at 17 percent profit would be 2288 * 1.17, which does not equal 2808. Similarly, 2300 as cost price would generate a selling price different from 2808 when profit is 17 percent. A cost price of 3285 is actually higher than the selling price and cannot yield a 17 percent profit. Only a cost price of 2400 yields the correct selling price when we apply a 17 percent profit margin. Common Pitfalls: One common mistake is to compute 17 percent of the selling price instead of the cost price and then subtract, which produces incorrect results because the base for percentage is wrong. Another error involves treating selling price as cost price times (100 / 117) without careful calculation, leading to arithmetic slips. To avoid such problems, always write the fundamental relation SP = CP * (1 + profit percent / 100) and solve explicitly for the unknown variable. Final Answer: The cost price of the table is Rs. 2400.

Question 19

At a 24% discount, the selling price of a washing machine is Rs. 38,000. What will be its selling price if the discount is increased to 40%?

Accepted Answer

Introduction / Context: This problem combines the concepts of discount and marked price. We are given the selling price at one discount rate and then asked to find the selling price at a different discount rate. The key is to first recover the marked price of the washing machine, which is the original price before any discount, and then apply the new discount percentage on this marked price to get the new selling price. Given Data / Assumptions: Discount rate 1 = 24 percent. Selling price at 24 percent discount = Rs. 38,000. Discount rate 2 (new discount) = 40 percent. We assume the marked price remains the same in both cases. Concept / Approach: A discount of 24 percent means that the customer pays 76 percent of the marked price. In decimal form this is 0.76. Therefore, selling price SP1 = marked price MP * 0.76. From this relation we find MP = SP1 / 0.76. Once we know the marked price, we apply the new discount of 40 percent, which leaves 60 percent of the marked price payable. Hence, new selling price SP2 = MP * 0.60. This two step approach is standard whenever we move from one discount rate to another using the same marked price. Step-by-Step Solution: Let marked price be MP. At 24 percent discount, payable percentage = 100 percent - 24 percent = 76 percent. So SP1 = 0.76 * MP. We are given SP1 = Rs. 38,000, so 0.76 * MP = 38000. Therefore MP = 38000 / 0.76. Compute 38000 / 0.76 = 50000. Now apply the new discount of 40 percent, so payable percentage = 60 percent, or 0.60. New selling price SP2 = MP * 0.60 = 50000 * 0.60 = 30000. Thus, the selling price at a 40 percent discount will be Rs. 30,000. Verification / Alternative check: We can verify by checking the relationship between the two selling prices. When discount rises from 24 percent to 40 percent, the payable fraction reduces from 0.76 to 0.60. The ratio of the new selling price to the old selling price should be 0.60 / 0.76. Compute 0.60 / 0.76 and multiply by 38,000; the result is 30,000. This confirms that the new selling price is consistent with the change in discount percentage and the constant marked price. Why Other Options Are Wrong: Rs. 18,000 would correspond to a payable fraction of only 0.36 of the marked price, which matches a 64 percent discount, not 40 percent. Rs. 22,320 and Rs. 31,920 do not align with the exact ratio 0.60 of the computed marked price 50,000. Only Rs. 30,000 is exactly 60 percent of the marked price, which is the correct pay amount at a 40 percent discount. Common Pitfalls: Some candidates mistakenly try to adjust the 38,000 directly by 16 percent (the difference between 24 percent and 40 percent), which is incorrect because discount percentages apply to the marked price, not to the selling price. Others may wrongly add or subtract discount percentages as if they were absolute rupee amounts. The safe method is always to first retrieve the marked price using the first discount information and then apply the second discount to that marked price. Final Answer: With a 40 percent discount, the washing machine will sell for Rs. 30,000.

Question 20

In a town the population increases by 44% in the first year and by 75% in the second year. What is the approximate average annual rate of increase in population over these two years?

Accepted Answer

Introduction / Context: This question asks for an average rate of increase when a population grows by different percentages in two consecutive years. In many aptitude exams, such questions expect a simple arithmetic average of the two yearly percentage increases, unless explicitly asking for a compound equivalent rate. Here, the options provided match the simple average approach, which is widely used as an approximate average rate of increase per year when exact compounding is not required. Given Data / Assumptions: Population increases by 44 percent in the first year. Population increases by 75 percent in the second year. We must find the average annual rate of increase in percentage over two years. We assume that the question refers to a simple average of the yearly percentage increases, as suggested by the answer choices. Concept / Approach: When a question about average percentage increase over multiple years is framed in school level aptitude style and no explicit compounding requirement is mentioned, it usually refers to the arithmetic mean of the individual yearly rates. That is, average rate per year = (rate in year 1 + rate in year 2) / 2. Therefore, we simply add 44 percent and 75 percent, then divide the sum by 2 to obtain the approximate average annual increase. This method is straightforward and matches one of the given options. Step-by-Step Solution: Rate of increase in year 1 = 44 percent. Rate of increase in year 2 = 75 percent. Sum of the two percentage increases = 44 + 75. Compute 44 + 75 = 119. Average annual rate of increase = total of rates / number of years. So, average rate per year = 119 / 2. Compute 119 / 2 = 59.5. Therefore, the approximate average annual rate of increase in population is 59.5 percent. Verification / Alternative check: If we wanted the exact compound equivalent rate, we could consider a starting population of 100 units. After year 1, population = 100 * 1.44 = 144. After year 2, population = 144 * 1.75 = 252. The overall factor is 100 to 252, or 2.52 times. The compound equivalent annual rate r would satisfy (1 + r)^2 = 2.52, so r is approximately 58.75 percent. This is close to 59.5 percent, and since 58.75 percent is not among the options, the closest and intended answer is the simple average of 59.5 percent. Why Other Options Are Wrong: 45 percent is far smaller than either of the yearly rates and cannot represent a realistic average of 44 percent and 75 percent. 74 percent and 76 percent are larger than the higher annual increase of 75 percent and cannot be average values. Only 59.5 percent lies between 44 percent and 75 percent and matches the arithmetic mean, which fits the typical exam convention here. Common Pitfalls: One common confusion is mixing up simple average rate and compound equivalent rate. Some candidates may try to average the final values rather than the rates. Others may incorrectly attempt to subtract the smaller rate from the larger one or divide the sum by the wrong number of years. To avoid mistakes, clearly identify that this question targets the arithmetic mean of two given percentages and confirm that the result lies between the two original rates. Final Answer: The approximate average annual rate of population increase over the two years is 59.5%.

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