Question 1

If 35% of a number is 24 less than 50% of that number, what is the number?

Accepted Answer

Introduction / Context:This is an algebraic percentage problem that involves comparing two different percentages of the same unknown number. The statement says that 35% of a number is 24 less than 50% of that number. From this information we need to determine the original number. Such questions train the ability to translate verbal percentage relations into equations and solve them efficiently.Given Data / Assumptions:Let the unknown number be N.35% of N is 24 less than 50% of N.In equation form, 35% of N = 50% of N minus 24.We are required to find the value of N.Concept / Approach:Convert the percentages into their decimal forms and then set up an equation. Specifically, 35% of N is 0.35N and 50% of N is 0.50N. The statement that one is 24 less than the other can be written as 0.35N = 0.50N - 24. Solving this linear equation in N involves bringing like terms together and isolating N. This straightforward method is standard in many percentage comparison questions.Step-by-Step Solution:Let the number be N.35% of N = 0.35N.50% of N = 0.50N.Given that 35% of N is 24 less than 50% of N.So, 0.35N = 0.50N - 24.Bring terms involving N to one side: 0.50N - 0.35N = 24.0.15N = 24.Hence N = 24 / 0.15.Compute 24 / 0.15 = 2400 / 15 = 160.Therefore, the number is 160.Verification / Alternative check:Check by substituting N = 160 back into the original condition. 35% of 160 = 0.35 * 160 = 56. 50% of 160 = 0.50 * 160 = 80. Now compute the difference: 80 - 56 = 24. This matches the condition that 35% of the number is 24 less than 50% of the number. Thus, 160 is correct.Why Other Options Are Wrong:80 would give 35% of 80 = 28 and 50% of 80 = 40, with a difference of 12, not 24.100 would give 35 and 50, with a difference of 15, not 24.120 would give 42 and 60, with a difference of 18, not 24.140 would give 49 and 70, with a difference of 21, also not equal to 24.Common Pitfalls:One common mistake is to interpret 35% as 0.35 and 50% as 0.5 correctly, but then to add or subtract the percentages themselves instead of forming the equation with N. Another error is in handling the algebra, for example writing 0.35N - 0.50N = 24, which leads to a negative coefficient. Misplacing the decimal points when dividing 24 by 0.15 can also cause errors. Using fraction forms, such as 35 / 100 and 50 / 100, can help if decimals feel confusing.Final Answer:The number that satisfies the given percentage condition is 160.

Question 2

A number, when increased by 30%, becomes 390. What is the original number before the increase?

Accepted Answer

Introduction / Context:This question is a direct application of percentage increase. We are told that after a 30% increase, a number becomes 390. The goal is to work backwards to find the original number. This type of reverse percentage calculation is very common in topics like profit and loss, population growth, and price changes.Given Data / Assumptions:Let the original number be N.The number is increased by 30%.After the increase, the new value is 390.We need to determine the value of N.Concept / Approach:When a quantity is increased by 30%, the new value becomes 1.30 times the original value. In symbols, new value = N * (1 + 30 / 100) = 1.30N. Since we know the new value is 390, we set 1.30N = 390 and solve for N by dividing 390 by 1.30. This is a straightforward linear equation in one variable.Step-by-Step Solution:Let the original number be N.After a 30% increase, new number = N * (1 + 30 / 100) = 1.30N.We are given that 1.30N = 390.So N = 390 / 1.30.Compute N: 390 / 1.30 = 39000 / 130 = 300.Therefore, the original number is 300.Verification / Alternative check:Check by applying the 30% increase to 300. Thirty percent of 300 is (30 / 100) * 300 = 90. Adding this increase to the original number gives 300 + 90 = 390. This matches the given final value, so the original number 300 is confirmed to be correct.Why Other Options Are Wrong:180 increased by 30% becomes 234, not 390.480 decreased by 30% or increased by 30% does not yield 390.260 increased by 30% becomes 338, not 390.240 increased by 30% becomes 312, also not equal to 390.Common Pitfalls:Some learners mistakenly treat 30% as 0.30 and subtract it from 390 instead of dividing by 1.30. Others try to compute 70% of 390, assuming that is the original number, which is incorrect in this context. The original number is the base on which the 30% was applied, so we must use division by the factor 1.30 rather than simple subtraction. Thinking in terms of multiplication factors like 1.30 for a 30% increase helps avoid confusion.Final Answer:The original number, before it was increased by 30%, is 300.

Question 3

The salary of Mr. X is 30% more than the salary of Mr. Y. By what percentage is Mr. Y’s salary less than Mr. X’s salary?

Accepted Answer

Introduction / Context:This question tests the important concept that percentage increase in one direction does not equal the percentage decrease in the reverse direction. We are told that Mr. X earns 30% more than Mr. Y, and we must find by what percentage Mr. Y earns less than Mr. X. These relative percentage problems are common in salary, price comparison and profit questions, and they require careful use of ratios rather than simple subtraction of percentages.Given Data / Assumptions:Let Mr. Y's salary be Y.Mr. X's salary is 30% more than Y.Therefore, Mr. X's salary is X = 1.30Y.We need to find what percent Mr. Y's salary is less than Mr. X's salary.Concept / Approach:The percentage by which Y is less than X is computed as (difference between X and Y) divided by X, multiplied by 100. The difference between their salaries is X - Y. Since X = 1.30Y, the difference becomes 0.30Y. The required percentage is (0.30Y / X) * 100. Substituting X = 1.30Y gives (0.30Y / 1.30Y) * 100. The Y cancels, and we get a percentage that is less than 30%, illustrating the asymmetry between increase and decrease percentages.Step-by-Step Solution:Let Mr. Y's salary be Y.Mr. X's salary is 30% more, so X = 1.30Y.Difference in salaries = X - Y = 1.30Y - Y = 0.30Y.We need the percentage by which Y is less than X.Required percentage = (difference / X) * 100.So, required percentage = (0.30Y / 1.30Y) * 100.Simplify the ratio: 0.30Y / 1.30Y = 0.30 / 1.30 = 30 / 130.30 / 130 reduces to 3 / 13.Now convert to percentage: (3 / 13) * 100 ≈ 23.08%.This is approximately equal to 23 1/3%.Verification / Alternative check:Assume a convenient value for Y, for example Y = 100 units. Then X = 30% more than 100, which is 130. The difference is 30. Now compute what percent 30 is of X (130). The percentage is (30 / 130) * 100 = 23.08%. This again is approximately 23 1/3%. So Mr. Y's salary is about 23 1/3% less than Mr. X's salary. The exact fraction is 3 / 13, which matches this approximate mixed number.Why Other Options Are Wrong:24% and 25 1/2% are larger than the true value and come from rounding errors or using the wrong base for comparison.22 8/9 % is slightly smaller than the true value and does not match the exact ratio 3 / 13.30% is the percentage increase from Y to X, not the percentage by which Y is less than X. Using 30% here ignores the change of base.Common Pitfalls:The most frequent mistake is to answer 30% directly, forgetting that the base has changed from Y to X. Another pitfall is computing (difference / Y) * 100 instead of (difference / X) * 100, which again yields 30%. Some students also mishandle the fraction 30 / 130 when simplifying. Remember that when the question asks “how much less is Y compared to X,” the base must be X, the larger quantity.Final Answer:Mr. Y's salary is approximately 23 1/3% less than Mr. X's salary.

Question 4

A girl wants to finish reading a book. She reads 2/5 of the book on the first day, 34% of the book on the second day, and the remaining pages over the next two days in equal parts. If she reads 52 pages on the last day, how many pages are there in t…

Accepted Answer

Introduction / Context:This is a mixed fraction and percentage word problem that appears often in aptitude exams. The girl reads different portions of the book over several days, with the first portion expressed as a fraction, the second as a percentage, and the remaining portion split equally between two more days. We are told how many pages she reads on the last day, and from this we must determine the total number of pages in the book. This tests the ability to convert fractions and percentages into consistent forms and to track remaining quantities step by step.Given Data / Assumptions:Let the total number of pages in the book be P.On day one she reads 2 / 5 of the entire book.On day two she reads 34% of the entire book.The remaining pages are read over the next two days in equal amounts.On the last day she reads 52 pages, which equals half of the remaining pages after day two.Concept / Approach:We will first express all parts as fractions of the total number of pages. Two fifths is 0.4P and 34% is 0.34P. After these two days, the total proportion read is 0.4P + 0.34P, so the remaining proportion is 1 - (0.4 + 0.34). That remaining part is split equally into two equal portions, one for each of the last two days. Since the number of pages on the last day is given, we can set the last day pages equal to half of the remaining portion and solve for P.Step-by-Step Solution:Let the total number of pages be P.Pages read on day one = (2 / 5)P = 0.4P.Pages read on day two = 34% of P = 0.34P.Total pages read in first two days = 0.4P + 0.34P = 0.74P.Remaining pages after two days = P - 0.74P = 0.26P.This remaining 0.26P is read equally over the next two days, so each day she reads (0.26P) / 2 = 0.13P.Given that the pages read on the last day equal 52.So, 0.13P = 52.Therefore, P = 52 / 0.13 = 400.Thus, the book contains 400 pages.Verification / Alternative check:Verify by reconstructing the reading schedule using P = 400. Day one: 2 / 5 of 400 = 160 pages. Day two: 34% of 400 = 0.34 * 400 = 136 pages. Total after two days = 160 + 136 = 296 pages. Remaining pages = 400 - 296 = 104 pages. These 104 pages are split equally over days three and four, so each day the girl reads 52 pages. This matches the information that she reads 52 pages on the last day. Therefore, the calculated total of 400 pages is consistent.Why Other Options Are Wrong:525, 325, 475 and 450 do not satisfy the conditions when you repeat the calculation steps; the last day pages do not come out to 52.For example, with 450 pages, the remaining portion after first two days would not divide into equal parts of 52 pages each.Common Pitfalls:Some learners mistakenly apply 34% to the remaining pages instead of the whole book. Others confuse the fraction 2 / 5 with 2 / 50 or misinterpret 34% as 0.034. Another common error is to forget that the last day pages represent exactly half of the remaining pages after day two. Writing every step in terms of P and carefully simplifying helps avoid these mistakes.Final Answer:The total number of pages in the book is 400.

Question 5

In an examination with maximum marks of 500, K gets 10% less marks than L, L gets 25% more marks than M, and M gets 20% less marks than N. If K scores 360 marks, what percentage of the maximum marks does N obtain?

Accepted Answer

Introduction / Context:This question involves a chain of percentage relations between the marks of four students K, L, M and N in an examination with a known maximum score. We are given that K scores 10% less than L, L scores 25% more than M, and M scores 20% less than N. The score of K is known, and we must ultimately find the percentage of maximum marks obtained by N. This tests the ability to move through multiple percentage relationships systematically.Given Data / Assumptions:Maximum marks in the examination are 500.K gets 10% less than L.L gets 25% more than M.M gets 20% less than N.K's marks are given to be 360.We must find N's marks as a percentage of 500.Concept / Approach:We work backward from the known value. Starting with K = 360, we express each relation in terms of the previous one. If K is 10% less than L, then K = 0.90L. If L is 25% more than M, then L = 1.25M. If M is 20% less than N, then M = 0.80N. Combining these, we can express K directly in terms of N, solve for N, and then compute N's percentage of 500. This chain method keeps the structure clear and avoids confusion.Step-by-Step Solution:Let N's marks be N.M gets 20% less than N, so M = 0.80N.L gets 25% more than M, so L = 1.25M = 1.25 * 0.80N = 1.00N.Thus L = N.K gets 10% less than L, so K = 0.90L = 0.90N.Given K = 360, so 0.90N = 360.Therefore N = 360 / 0.90 = 400.N's percentage of the maximum 500 marks is (400 / 500) * 100.Compute: (400 / 500) * 100 = 80%.Verification / Alternative check:Check all relationships using N = 400. M = 20% less than N, so M = 0.80 * 400 = 320. L is 25% more than M, so L = 1.25 * 320 = 400. Thus L and N both have 400 marks, consistent with the earlier result. K is 10% less than L, so K = 0.90 * 400 = 360, which matches the given score. This confirms that the chain of calculations is correct and N's marks are 400, corresponding to 80% of 500.Why Other Options Are Wrong:87, 78 and 76 percent would correspond to different total marks and would break the relationships among K, L, M and N.82% of 500 is 410 marks, which does not satisfy the given conditions when propagated back through M, L and K.Common Pitfalls:Students often confuse phrases like 10% less than and 10% of, or mistakenly apply increments and decrements in the wrong direction. Another error is trying to work from N downward without anchoring values, which can be messy. Instead, starting from the known score, expressing each relation as a multiplier and solving systematically keeps the work organized. Remember that less than means multiplying by (1 - percentage / 100), while more than means multiplying by (1 + percentage / 100).Final Answer:Mr. N obtains 80 percent of the maximum marks in the examination.

Question 6

Sneha’s weight is 140% of Veena’s weight. Rina’s weight is 90% of Sana’s weight, and Sana weighs twice as much as Veena. If Sneha’s weight is x% of Rina’s weight, what is the value of x?

Accepted Answer

Introduction / Context:This problem links four people's weights with several percentage relationships. Sneha's weight is described relative to Veena's, Rina's relative to Sana's, and Sana's relative to Veena's. From these, we must find the percentage that Sneha's weight represents of Rina's weight. Such chained percentage ratio questions are typical in quantitative aptitude tests and require clear variable assignment and algebraic manipulation.Given Data / Assumptions:Let Veena's weight be V.Sneha's weight is 140% of Veena's weight.Sana's weight is twice Veena's weight.Rina's weight is 90% of Sana's weight.We must find x such that Sneha's weight is x% of Rina's weight.Concept / Approach:We express all weights in terms of the common base V (Veena's weight). Percentages are converted to multiplication factors. Then we write Sneha's and Rina's weights in these terms, form the ratio Sneha / Rina, and convert this ratio into a percentage. This method avoids dealing with specific numbers and highlights the proportional structure of the problem.Step-by-Step Solution:Let Veena's weight be V.Sneha's weight is 140% of V, so Sneha = 1.40V.Sana's weight is twice Veena's weight, so Sana = 2V.Rina's weight is 90% of Sana's weight, so Rina = 0.90 * Sana = 0.90 * 2V = 1.80V.We need x such that Sneha = (x / 100) * Rina.That is, 1.40V = (x / 100) * 1.80V.Divide both sides by V: 1.40 = (x / 100) * 1.80.So x / 100 = 1.40 / 1.80.Simplify 1.40 / 1.80 = 14 / 18 = 7 / 9.Thus x / 100 = 7 / 9, so x = (7 / 9) * 100 = 700 / 9.Verification / Alternative check:We can choose a convenient value for V to check. Let V = 9 kg. Then Sneha's weight = 1.40 * 9 = 12.6 kg. Sana's weight = 2 * 9 = 18 kg. Rina's weight = 0.90 * 18 = 16.2 kg. Now find what percent Sneha is of Rina: (12.6 / 16.2) * 100. Compute 12.6 / 16.2 = 7 / 9. Hence the percentage is (7 / 9) * 100 = 700 / 9. This confirms the algebraic result.Why Other Options Are Wrong:596/9, 605/9, 645/9 and 650/9 are all different from 700/9 and arise from incorrect algebra, wrong percentage conversions, or misinterpreting which weight is being compared to which.Only 700/9 exactly equals (Sneha / Rina) * 100 when the correct relationships are applied.Common Pitfalls:Students sometimes mix up the direction of percentage relationships, for example thinking Sana is 90% of Rina instead of Rina being 90% of Sana. Others may fail to use a common base like Veena's weight and instead attempt to guess numbers, which can lead to arithmetic confusion. Another mistake is to divide the wrong way when forming the ratio Sneha / Rina, which inverts the percentage. Carefully expressing all quantities in terms of V and proceeding step by step prevents these errors.Final Answer:The value of x, the percentage that Sneha's weight is of Rina's weight, is 700/9 percent.

Question 7

In the expression 3.2% of 500 multiplied by 2.4% of an unknown number is equal to 312. Find the value of the unknown number.

Accepted Answer

Introduction / Context:This question involves a product of two percentage expressions equated to a fixed number. Specifically, 3.2% of 500 multiplied by 2.4% of some unknown quantity results in 312. The task is to find that unknown quantity. This type of problem tests your ability to handle small decimal percentages and to set up a clean algebraic equation involving products.Given Data / Assumptions:First factor: 3.2% of 500.Second factor: 2.4% of an unknown number, say N.The product of these two factors equals 312.We must find the value of N.Concept / Approach:We use the percentage formula value = (percentage / 100) * base for both percentages. So 3.2% of 500 is (3.2 / 100) * 500, and 2.4% of N is (2.4 / 100) * N. Their product is then set equal to 312. This gives us a single equation in the variable N, which we can solve by isolating N. Careful handling of decimal percentages is important to avoid computational errors.Step-by-Step Solution:Let the unknown number be N.3.2% of 500 = (3.2 / 100) * 500.Compute: (3.2 / 100) * 500 = 0.032 * 500 = 16.2.4% of N = (2.4 / 100) * N = 0.024N.We are told that (3.2% of 500) * (2.4% of N) = 312.So, 16 * 0.024N = 312.Compute the coefficient: 16 * 0.024 = 0.384.Therefore, 0.384N = 312.Solve for N: N = 312 / 0.384.Compute: 312 / 0.384 = 812.5.So the unknown number N is 812.5.Verification / Alternative check:Substitute N = 812.5 back into the original expression. First factor: 3.2% of 500 = 16 as before. Second factor: 2.4% of 812.5 = 0.024 * 812.5. Compute 0.024 * 812.5 = 19.5. Now the product is 16 * 19.5 = 312. This matches the given product exactly, so N = 812.5 is correct.Why Other Options Are Wrong:829.75, 749.85, 799.5 and 780 all fail when substituted back into the equation, because the product 16 * (2.4% of N) does not equal 312.Only N = 812.5 gives exactly 312 as the product, satisfying the given condition.Common Pitfalls:Some students misinterpret the phrase 3.2% of 500 multiplied by 2.4% of N and add the percentages instead of multiplying the resulting values. Others mistakenly divide 312 by 3.2% or 2.4% directly without taking into account the factor of 500. Another common issue is handling decimal multiplication incorrectly. Writing each percentage as a decimal and carefully forming the equation avoids these mistakes.Final Answer:The value of the unknown number that satisfies the given percentage product equation is 812.5.

Question 8

On its anniversary, a showroom offers an 18% discount on the marked price of bikes. Raghuram buys a bike for Rs. 54,540 after discount. What is the marked price of the bike?

Accepted Answer

Introduction / Context:This is a typical discount and marked price problem from the topic of percentage and profit and loss. The showroom gives a fixed discount rate on the marked price, and the buyer pays the discounted price. We are given the final price paid and the discount percentage, and we must work backwards to find the original marked price. Such questions are very common in retail and commerce related aptitude tests.Given Data / Assumptions:Let the marked price of the bike be M rupees.A discount of 18% is offered on this marked price.So, the selling price after discount is 82% of M, since 100% - 18% = 82%.Raghuram pays Rs. 54,540 for the bike.Therefore, 82% of M equals 54,540.Concept / Approach:We express the discounted price as a percentage of the marked price. If a discount of d% is given, the buyer pays (100 - d)% of the marked price. In this case, the buyer pays 82% of M. Thus we set up the equation (82 / 100) * M = 54,540 and solve for M by dividing 54,540 by 0.82. This is a simple reverse percentage calculation.Step-by-Step Solution:Let the marked price be M.Discount = 18% of M.Selling price after discount = M - 0.18M = 0.82M.We are given that the selling price is Rs. 54,540.So, 0.82M = 54,540.Therefore, M = 54,540 / 0.82.Compute: 54,540 / 0.82 ≈ 66,512.20.Rounded to the nearest rupee, M ≈ Rs. 66,512.Verification / Alternative check:Verify using the option Rs. 66,512 as the marked price. Discount at 18% of 66,512 is 0.18 * 66,512 ≈ 11,972.16. Subtracting this from the marked price gives 66,512 - 11,972.16 ≈ 54,539.84, which rounds to 54,540. This matches the given selling price. The slight difference is due to rounding to the nearest rupee, and the option Rs. 66,512 is therefore the correct marked price.Why Other Options Are Wrong:Rs. 63,548, Rs. 65,000, Rs. 67,857 and Rs. 70,500, when reduced by 18%, do not give a selling price of approximately Rs. 54,540.Only Rs. 66,512, when reduced by 18%, yields a discounted price that matches the amount paid by Raghuram.Common Pitfalls:Some students mistakenly add 18% to the selling price instead of reversing from the marked price. Others compute 18% of 54,540 instead of 18% of the unknown marked price. Another common mistake is to divide 54,540 by 18 or by 0.18 directly, which does not correspond to the relationship between discount and selling price. Always remember that selling price = (100 - discount percentage) percent of marked price and set up the equation accordingly.Final Answer:The marked price of the bike before the 18% discount was approximately Rs. 66512.

Question 9

36 is what percent of 80?

Accepted Answer

Introduction / Context:This question is the reverse of the usual percentage calculation. Instead of finding a percentage of a given number, we are asked to determine what percentage one number (36) is of another number (80). This is a fundamental concept in percentage comparisons, frequently used in exam score analysis, growth calculations and ratio interpretation.Given Data / Assumptions:The part value is 36.The base or reference value is 80.We must find the percentage p such that 36 = p% of 80.No rounding is necessary because the result is an exact terminating decimal.Concept / Approach:To find what percent one number is of another, we use the formula p% = (part / whole) * 100. Here, the part is 36 and the whole is 80. So we compute (36 / 80) * 100. Simplifying the fraction first often makes the calculation easier and reduces the chance of mistakes. After simplifying, we convert the result into a percentage value.Step-by-Step Solution:We want p such that 36 is p% of 80.Use the formula p% = (36 / 80) * 100.First simplify the fraction 36 / 80.Divide numerator and denominator by 4: 36 / 80 = 9 / 20.Now compute (9 / 20) * 100.100 / 20 = 5, so (9 / 20) * 100 = 9 * 5 = 45.Thus, 36 is 45% of 80.Verification / Alternative check:We can verify by checking if 45% of 80 equals 36. Compute 45% of 80 as (45 / 100) * 80 = 0.45 * 80. Now 0.45 * 80 = 36. Since the original number is recovered, our calculated percentage is confirmed to be correct.Why Other Options Are Wrong:54% of 80 is (54 / 100) * 80 = 43.2, not 36.60% of 80 is 48, which is larger than 36.64% of 80 is 51.2, still not 36.50% of 80 is 40, which is also different from 36.Common Pitfalls:Some learners mistakenly compute 80 as a percent of 36 instead of 36 as a percent of 80, which reverses the ratio. Others incorrectly set up the formula as (80 / 36) * 100. Another error is to forget to multiply by 100 at the end, thus leaving the result as a decimal rather than a percentage. Always remember that the part goes in the numerator, the whole in the denominator, and then multiply by 100 to convert to percent.Final Answer:Thirty six is 45 percent of 80.

Question 10

Find the value of the missing term in the equation: 56% of 870 plus 82% of 180 equals 32% of 90 plus an unknown number. What is this unknown number?

Accepted Answer

Introduction / Context:This question presents a percentage equation where an unknown number appears on one side along with another percentage expression. The goal is to compute each percentage term, simplify both sides, and then solve for the unknown number. Such questions check accuracy with percentage calculations and algebraic manipulation in a single step.Given Data / Assumptions:We have 56% of 870.We also have 82% of 180.The sum of these two values equals 32% of 90 plus an unknown number, say X.We need to find the value of X that satisfies this equation.Concept / Approach:The method is to evaluate each percentage separately using value = (percentage / 100) * base. First compute 56% of 870 and 82% of 180, and add them to get the left hand side. Then compute 32% of 90 on the right hand side. Finally, express the equation as left side = 32% of 90 + X and solve for X by subtracting the known percentage value from the left side total. Careful arithmetic is essential due to multiple percentage operations.Step-by-Step Solution:Let the unknown number be X.Compute 56% of 870: (56 / 100) * 870 = 0.56 * 870.0.56 * 870 = 487.2.Compute 82% of 180: (82 / 100) * 180 = 0.82 * 180.0.82 * 180 = 147.6.Left hand side total = 487.2 + 147.6 = 634.8.Compute 32% of 90: (32 / 100) * 90 = 0.32 * 90 = 28.8.According to the question, 634.8 = 28.8 + X.So X = 634.8 - 28.8 = 606.Therefore, the unknown number is 606.Verification / Alternative check:Substitute X = 606 back into the equation. Right hand side becomes 32% of 90 plus 606. We already have 32% of 90 = 28.8. So right hand side = 28.8 + 606 = 634.8. Left hand side was computed as 56% of 870 plus 82% of 180 = 634.8. Since both sides are equal, the equation is satisfied and X = 606 is correct.Why Other Options Are Wrong:777, 666, 789 and 696 all give right hand side totals that do not equal 634.8 when added to 28.8.Only X = 606 makes the right hand side equal to the left hand side sum of 634.8.Common Pitfalls:Some students accidentally add 56 and 82 as percentages before applying them, or they compute 32% of 870 by mistake. Another common error is to subtract incorrectly when isolating X, for example adding 28.8 instead of subtracting it. A few learners also misplace decimal points when multiplying by percentages such as 0.56 and 0.82. Writing each step clearly and rechecking arithmetic helps avoid these mistakes.Final Answer:The unknown number that satisfies the given percentage equation is 606.

Question 11

If the price of a commodity is decreased by 20% and, as a result, its consumption is increased by 10%, what will be the overall percentage increase or decrease in the total expenditure on that commodity?

Accepted Answer

Introduction / Context: This percentage question tests the effect of successive percentage changes on total expenditure for a commodity. Expenditure on a commodity is the product of its price and the quantity consumed. When one factor decreases and the other increases, many students are unsure whether the overall expenditure increases or decreases and by how much. This problem helps to build intuition about combined percentage changes in daily life situations like shopping, budgeting, and cost analysis. Given Data / Assumptions: The original price of the commodity is taken as some base value, say P. The original consumption (quantity bought) is taken as some base value, say Q. Price decreases by 20%, so the new price is 80% of P. Consumption increases by 10%, so the new quantity is 110% of Q. Expenditure is calculated as Price * Quantity. Concept / Approach: Expenditure is given by the formula: Expenditure = Price * Quantity. If the price changes by a certain percentage and the quantity changes by another percentage, then the new expenditure is the product of the new price and the new quantity. We compare this new expenditure with the original expenditure and express the difference as a percentage of the original expenditure to find the net percentage increase or decrease. Using assumed convenient numbers like 100 for easy calculation is a common and valid technique in percentage questions. Step-by-Step Solution: Step 1: Assume the original price P = 100 units of money and original quantity Q = 100 units. Step 2: Original expenditure = P * Q = 100 * 100 = 10000. Step 3: Price decreases by 20%, so new price = 80% of 100 = 80. Step 4: Quantity increases by 10%, so new quantity = 110% of 100 = 110. Step 5: New expenditure = 80 * 110 = 8800. Step 6: Change in expenditure = New expenditure - Original expenditure = 8800 - 10000 = -1200. Step 7: Percentage change in expenditure = (Change / Original) * 100 = (-1200 / 10000) * 100 = -12%. Step 8: A negative percentage indicates a decrease, so the expenditure has decreased by 12%. Verification / Alternative check: We can verify using percentage multiplication factors. A decrease of 20% in price means multiplying by 0.8. An increase of 10% in quantity means multiplying by 1.1. So the overall factor on expenditure is 0.8 * 1.1 = 0.88. This means the new expenditure is 88% of the original, which is 12% less than 100%. Therefore, the net effect is a 12% decrease in total expenditure, confirming our previous calculation. Why Other Options Are Wrong: 12% increase is wrong because the combined factor 0.8 * 1.1 is less than 1, so expenditure cannot increase. 16% decrease and 16% increase are incorrect because they do not match the derived factor of 0.88. No change in expenditure is also incorrect since there is a clear reduction from 100% to 88% of the original expenditure. Common Pitfalls: A frequent mistake is to simply add the percentages, for example treating the change as 10% - 20% = -10% or sometimes 20% - 10% = 10% without calculating the product. Percentage changes on different factors (price and quantity) must be applied multiplicatively, not additively. Another common error is to use inconsistent base values or to forget that the price change acts on price only and the consumption change acts on quantity only. Final Answer: The total expenditure on the commodity experiences a 12% decrease compared to the original expenditure.

Question 12

King Ajay distributes his gold among his three queens by giving 50% of his wealth to the first queen, 50% of the remaining gold to the second queen, and again 50% of the remaining gold to the third queen. If the combined share of the three queens is…

Accepted Answer

Introduction / Context: This is a percentage and fraction based distribution problem involving successive sharing of wealth. The question describes how King Ajay distributes his gold among three queens by repeatedly giving half of the remaining amount. Such problems are common in aptitude exams and test understanding of fractions, percentages, and reverse calculations when the final distributed amount is known but the initial amount is unknown. Given Data / Assumptions: Let the initial amount of gold be G kilograms. First queen receives 50% of G, which is G / 2. Second queen receives 50% of the remaining gold. Third queen receives 50% of what remains after the second queen receives her share. The total quantity received by all three queens together is 1,30,900 kilograms of gold. Concept / Approach: The main idea is to model the distribution process as fractions of the original amount G. Because each queen receives half of what is available at her turn, we can express each queen's share in terms of G and then sum those expressions. After that, we set the sum equal to the given combined share of 1,30,900 kilograms and solve for G. This method is often easier and safer than trying to work backward step by step without algebra. Step-by-Step Solution: Step 1: Let the total initial gold be G kilograms. Step 2: First queen gets 50% of G, so her share = G / 2. Remaining gold = G - G / 2 = G / 2. Step 3: Second queen gets 50% of the remaining G / 2, so her share = (1 / 2) * (G / 2) = G / 4. Remaining gold = G / 2 - G / 4 = G / 4. Step 4: Third queen gets 50% of the remaining G / 4, so her share = (1 / 2) * (G / 4) = G / 8. Remaining gold after this is G / 4 - G / 8 = G / 8. Step 5: Total gold given to the queens = G / 2 + G / 4 + G / 8 = (4G / 8 + 2G / 8 + G / 8) = 7G / 8. Step 6: We are told that 7G / 8 = 130900. Step 7: Solve for G: G = 130900 * (8 / 7) = 149600. Step 8: Therefore, King Ajay originally had 1,49,600 kilograms of gold. Verification / Alternative check: We can forward check using G = 149600. First queen: 50% of 149600 = 74800. Remaining = 74800. Second queen: 50% of 74800 = 37400. Remaining = 37400. Third queen: 50% of 37400 = 18700. Total given = 74800 + 37400 + 18700 = 130900, which matches the given combined share. This confirms that the calculated initial wealth is correct. Why Other Options Are Wrong: Values like 153145 kgs, 147890 kgs, and 152400 kgs do not produce a combined share of exactly 130900 kilograms when the same pattern of distribution is applied. The option 140000 kgs is a round figure but also fails to yield the correct total. Only 149600 kgs reproduces the given final combined share, so it is the correct answer. Common Pitfalls: A common mistake is to assume that the total given is simply 50% + 50% + 50% of the original amount, which would wrongly suggest that 150% of the gold was distributed. The key is that each 50% is taken from the current remainder, not from the original amount every time. Another error is to miscalculate the fraction sum 1 / 2 + 1 / 4 + 1 / 8, which must be carefully handled as 7 / 8. Final Answer: The total quantity of gold King Ajay originally had was 149600 kgs.

Question 13

Two numbers are respectively 40% and 30% more than a third number. The second number, when expressed as a percentage of the first number, is equal to:

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Introduction / Context: This question tests understanding of percentage increase and comparison between numbers that are related to a common base. The idea is to express two numbers in terms of a third reference number and then compare them as a percentage. Such problems are common in percentage and ratio based aptitude questions and help develop comfort with relative comparisons. Given Data / Assumptions: Let the third number be T. The first number is 40% more than T. The second number is 30% more than T. We need to find the second number as a percentage of the first number. Concept / Approach: A standard technique is to assume a convenient value for the base number T, such as T = 100, to simplify the percentage calculations. Then the first and second numbers are expressed in terms of T. Once both numbers are written numerically, we compute the ratio of the second to the first and convert this ratio into a percentage. This is more reliable than trying to directly work with percentage expressions without a numeric base. Step-by-Step Solution: Step 1: Assume T = 100 for simplicity. Step 2: First number = T increased by 40% = 100 + 40 = 140. Step 3: Second number = T increased by 30% = 100 + 30 = 130. Step 4: We need second as a percentage of first, that is (Second / First) * 100. Step 5: Compute ratio: 130 / 140. Step 6: 130 / 140 = 13 / 14. Step 7: Multiply by 100 to get percentage: (13 / 14) * 100 ≈ 92.857%. Step 8: Rounded to one decimal place, this is approximately 92.8%. Verification / Alternative check: We can cross check by approximating. Since both numbers are close to each other, the second number is slightly less than the first. If the first is 140 and the second is 130, the difference is 10, which is about 7.14% of 140. So the second is roughly 100% minus 7.14%, which is about 92.86%. This matches our earlier calculation of about 92.8%, confirming the result. Why Other Options Are Wrong: 84.23%, 87.14%, and 94.1% are not consistent with the exact ratio 13 / 14. These values would correspond to different numerical relationships between the first and second numbers. The option 100% would imply both numbers are equal, which is not true because one is 140 and the other is 130. Only 92.8% correctly represents the ratio of the second number to the first based on the given conditions. Common Pitfalls: Many learners incorrectly compare 40% and 30% directly and assume a simple difference of 10% as the answer, which is wrong. The percentages are taken with respect to the third number T, not directly between the first and second numbers. The correct method is to convert both to concrete values using the same base and then compute their ratio. Forgetting to multiply by 100 when converting to a percentage is another common error. Final Answer: The second number is 92.8% of the first number.

Question 14

The value of a machine depreciates at the rate of 10% every year. It was purchased 3 years ago, and its present value is Rs. 8,748. What was the original purchase price of the machine?

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Introduction / Context: This is a depreciation percentage problem, where the value of an asset such as a machine decreases by a fixed percentage each year. The present value after a certain number of years is given, and we need to work backwards to find the original purchase price. Such questions are common in aptitude exams, finance, and accounting, and they test a clear understanding of successive percentage changes and reverse calculations. Given Data / Assumptions: The machine depreciates at 10% per year. Number of years since purchase = 3 years. Present value after 3 years = Rs. 8,748. Let the original purchase price be P rupees. Concept / Approach: Depreciation at a fixed percentage means the value is multiplied each year by (1 - depreciation rate). Here, the depreciation rate is 10%, so each year the value is multiplied by 0.9. Over 3 years, the value is multiplied by 0.9 three times, that is by 0.9^3. Therefore, Present value = Original value * (0.9^3). To find the original value, we divide the present value by 0.9^3. This is an example of using the compound depreciation formula, which is similar in structure to compound interest but with a negative rate. Step-by-Step Solution: Step 1: Let the original purchase price be P. Step 2: Depreciation rate per year = 10%, so each year the value is multiplied by 0.9. Step 3: After 3 years, value factor = 0.9^3 = 0.9 * 0.9 * 0.9. Step 4: Compute 0.9^3 = 0.729. Step 5: Present value after 3 years = P * 0.729 = 8748. Step 6: Solve for P: P = 8748 / 0.729. Step 7: Perform the division: 8748 / 0.729 = 12000. Step 8: Therefore, the machine was originally purchased for Rs. 12,000. Verification / Alternative check: Forward check: Start from Rs. 12,000 and apply 10% depreciation each year. End of year 1: 12000 * 0.9 = 10800. End of year 2: 10800 * 0.9 = 9720. End of year 3: 9720 * 0.9 = 8748. This matches the given present value exactly, which verifies that Rs. 12,000 is correct. Why Other Options Are Wrong: Rs. 14,000, Rs. 12,800, and Rs. 15,000, when depreciated by 10% over 3 years, do not reduce to Rs. 8,748. For example, 14000 * 0.729 = 10206, which is too high, and 10000 * 0.729 = 7290, which is too low. Only Rs. 12,000 produces the exact present value. Therefore the other options are incorrect. Common Pitfalls: A common mistake is to subtract 10% of the original value three times without compounding, which treats depreciation as simple rather than compound. Another error is to divide by 0.9 instead of 0.9^3 when reversing three years of depreciation. Learners must remember that repeated percentage changes require powers of the multiplication factor, not linear subtraction or division by the rate only once. Final Answer: The original purchase price of the machine was Rs. 12,000.

Question 15

In one year, the amount collected for social welfare in a village increases by 10%, and in the next year it decreases by 10%. If after these two consecutive changes the final amount is Rs. 7,920, what was the amount at the beginning, before any incr…

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Introduction / Context: This is a classic percentage change problem involving a successive increase and decrease on the same base quantity. The question describes the amount collected for social welfare in a village over two years, with a 10% increase followed by a 10% decrease. Learners must carefully understand that a 10% increase and a 10% decrease do not cancel out because the base values are different in each step. Given Data / Assumptions: In year 1, the amount increases by 10%. In year 2, the new amount decreases by 10%. The final amount after two years is Rs. 7,920. Let the initial amount be A rupees. Concept / Approach: Successive percentage changes are handled by multiplying the initial amount by the corresponding percentage factors. An increase of 10% means multiplying by 1.10, and a decrease of 10% means multiplying by 0.90. Therefore, the overall factor after both changes is 1.10 * 0.90. The final amount is the initial amount multiplied by this combined factor. To recover the initial amount, we divide the final amount by this factor. This method shows that equal percentage increase and decrease produce a net decrease, not zero change. Step-by-Step Solution: Step 1: Let the initial amount be A rupees. Step 2: After a 10% increase, amount becomes A * 1.10. Step 3: After a 10% decrease on this new value, amount becomes A * 1.10 * 0.90. Step 4: Compute combined factor: 1.10 * 0.90 = 0.99. Step 5: Final amount is given as 7920, so A * 0.99 = 7920. Step 6: Solve for A: A = 7920 / 0.99. Step 7: 7920 / 0.99 = 8000. Step 8: Therefore, the starting amount for social welfare was Rs. 8,000. Verification / Alternative check: Forward check: Start with 8000. After a 10% increase, amount = 8000 * 1.10 = 8800. After a 10% decrease in the next year, amount = 8800 * 0.90 = 7920. This matches the given final amount, so our initial estimate is correct. We also see that the net effect of +10% and -10% is a 1% decrease, since 0.99 is slightly less than 1. Why Other Options Are Wrong: Rs. 8,800 and Rs. 8,980 are too high; if we start with them and apply the same percentage changes, the final value will exceed 7,920. Rs. 8,650 and Rs. 7,500 do not reproduce the final amount of 7,920 when the same 10% increase and 10% decrease are applied. Only Rs. 8,000 yields exactly 7,920 after the two successive percentage changes. Common Pitfalls: Many learners incorrectly assume that a 10% increase followed by a 10% decrease returns the amount to its original value. This is incorrect because the decrease is calculated on a higher amount. Another pitfall is adding or subtracting the percentages directly instead of multiplying the factors. It is important to remember that percentage changes act multiplicatively, not additively, over time. Final Answer: The amount collected at the beginning was Rs. 8,000.

Question 16

In an election between two candidates K and L, 20% of the total 8,720 votes polled were invalid. The number of valid votes received by K exceeds those received by L by 15% of the total votes polled. How many valid votes did candidate L receive?

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Introduction / Context: This election based percentage problem tests your ability to handle valid and invalid votes, and to convert textual information about vote differences into algebraic equations. Such questions are common in quantitative aptitude and reasoning exams, because they combine percentage calculations with simple algebra and logical interpretation of real life scenarios. Given Data / Assumptions: Total votes polled = 8,720. 20% of the total votes are invalid. Let the valid votes received by K be V_K and by L be V_L. The valid votes of K exceed those of L by 15% of the total votes polled. We need to find V_L, the number of valid votes received by L. Concept / Approach: First, we compute the number of valid votes by removing the invalid votes from the total. Then we use the statement about K's lead to form two equations. One equation is V_K + V_L = total valid votes. The other comes from the fact that the difference V_K - V_L equals 15% of total votes polled. Solving these two simple linear equations gives values for V_K and V_L. This method is systematic and avoids confusion between percentages of total votes and percentages of valid votes. Step-by-Step Solution: Step 1: Total votes polled = 8720. Step 2: Invalid votes = 20% of 8720 = 0.20 * 8720 = 1744. Step 3: Valid votes = 8720 - 1744 = 6976. Step 4: Let V_K be valid votes for K and V_L be valid votes for L. Step 5: We know V_K + V_L = 6976 (all valid votes). Step 6: K exceeds L by 15% of total votes polled, so V_K - V_L = 15% of 8720. Step 7: Compute 15% of 8720 = 0.15 * 8720 = 1308. Step 8: So we have the system: V_K + V_L = 6976 and V_K - V_L = 1308. Step 9: Add both equations: 2V_K = 6976 + 1308 = 8284, so V_K = 4142. Step 10: Substitute into V_K + V_L = 6976 to get V_L = 6976 - 4142 = 2834. Step 11: Therefore, candidate L received 2,834 valid votes. Verification / Alternative check: Check the conditions using V_L = 2834 and V_K = 4142. First, V_K + V_L = 4142 + 2834 = 6976, which matches the valid votes count. Second, the difference is 4142 - 2834 = 1308, which equals 15% of 8720, confirming that K leads L by the correct margin. This confirms that our calculations are consistent and that 2834 is correct. Why Other Options Are Wrong: Values like 3241, 2457, 2642, and 3000 either do not yield the correct sum of valid votes when combined with K's total, or do not maintain the required difference of 1308. Only V_L = 2834 satisfies both the sum equation and the difference equation simultaneously. Therefore, the other options fail to meet the conditions given in the problem. Common Pitfalls: A common error is to treat 15% as a percentage of valid votes instead of total votes polled, which leads to incorrect equations. Another frequent mistake is forgetting to subtract invalid votes before distributing the remaining votes between the candidates. Some learners also make arithmetic errors when solving the pair of linear equations, especially while adding or subtracting them. Careful and systematic calculation avoids these issues. Final Answer: Candidate L received 2,834 valid votes.

Question 17

What is the value of 70% of 50?

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Introduction / Context: This is a very basic percentage calculation problem, asking for 70% of 50. Such questions check fundamental understanding of what a percentage means and how to convert a percentage into its numerical value relative to a given quantity. Mastering simple problems like this is essential before moving on to more complex percentage applications in profit and loss, discounts, and data interpretation. Given Data / Assumptions: We are asked to find 70% of 50. Percentage is a way to express a number as a fraction of 100. No hidden conditions or additional information are involved. Concept / Approach: To find X% of a number N, we convert X% to a fraction X / 100 and then multiply this fraction by N. Here X = 70 and N = 50. So we calculate (70 / 100) * 50. Another approach is to interpret 70% as 7 tenths (0.7) and then multiply 0.7 by 50. Both approaches lead to the same answer and reinforce the link between percentages, fractions, and decimals. Step-by-Step Solution: Step 1: Express 70% as a fraction: 70% = 70 / 100. Step 2: Simplify 70 / 100 = 7 / 10. Step 3: Multiply this fraction by 50: (7 / 10) * 50. Step 4: Compute (7 / 10) * 50 = 7 * 5 = 35. Step 5: Therefore, 70% of 50 is equal to 35. Verification / Alternative check: Use decimal form of the percentage. 70% can be written as 0.70. Multiply 0.70 by 50. That is 0.70 * 50 = 35. Both the fraction method and decimal method give the same answer, confirming that the calculation is correct. Why Other Options Are Wrong: 38, 42, and 44 are larger than 35 and would correspond to more than 70% of 50, which is not possible. For example, 42 would be 84% of 50. The option 28 is 56% of 50, which is too low. Only 35 exactly equals 70% of 50 and matches the proper calculation method. Common Pitfalls: Some learners mistakenly divide 50 by 70 instead of multiplying by 70 / 100, or they confuse 70% of 50 with 50% of 70. Another error is to misinterpret 70% as 0.07 rather than 0.70. Remember that to convert a percent to a decimal, you divide by 100, so 70 becomes 0.70, not 0.07. Careful handling of percentage conversion prevents these mistakes. Final Answer: The value of 70% of 50 is 35.

Question 18

4 4/5 of 259.99 is equal to what percentage of 1247.9?

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Introduction / Context: This question involves fractions, decimals, and percentages together. It asks you to find what percentage a multiplied value (4 4/5 of 259.99) is of another number (1247.9). Such problems are important in aptitude exams because they test your ability to handle mixed numbers, convert them into improper fractions or decimals, perform multiplication, and then express the result as a percentage of another quantity. Given Data / Assumptions: We need to compute 4 4/5 of 259.99. We then compare this value with 1247.9. We must express the computed value as a percentage of 1247.9. No rounding instructions are given, so standard reasonable rounding is acceptable. Concept / Approach: First, convert the mixed number 4 4/5 into an improper fraction or decimal. 4 4/5 equals 4.8. Then compute 4.8 * 259.99. After obtaining this product, to find what percentage it is of 1247.9, we use the formula: Percentage = (Value / Reference) * 100, where the value is 4.8 * 259.99 and the reference is 1247.9. This will show whether the result is close to 100% or some other option given in the choices. Step-by-Step Solution: Step 1: Convert 4 4/5 to decimal form: 4 4/5 = 4 + 4 / 5 = 4 + 0.8 = 4.8. Step 2: Compute 4.8 * 259.99. Step 3: 4.8 * 259.99 ≈ 1247.952 (since 4.8 * 260 ≈ 1248, the exact multiplication is 1247.952). Step 4: Now, we need the percentage of 1247.952 with respect to 1247.9. Step 5: Use Percentage = (Value / Reference) * 100 = (1247.952 / 1247.9) * 100. Step 6: Since 1247.952 is extremely close to 1247.9, their ratio is very close to 1. Step 7: Therefore, the percentage is approximately 100%. Step 8: Hence, 4 4/5 of 259.99 is essentially equal to 100% of 1247.9. Verification / Alternative check: We can approximate by rounding 259.99 to 260. Then 4.8 * 260 = 1248. The reference value is 1247.9, which is almost the same as 1248. So the percentage is (1248 / 1247.9) * 100, which is almost exactly 100%. This quick approximation confirms that 100% is the correct choice among the given options. Why Other Options Are Wrong: 96%, 78%, 88%, and 82% would represent values noticeably lower than the reference 1247.9. For example, 96% of 1247.9 is about 1197, and 88% is about 1098. These are far from 1247.952. Since our computed product is essentially equal to the reference value, only 100% correctly represents the relationship. Thus, all other percentage options are inconsistent with the calculation. Common Pitfalls: One common mistake is to misinterpret the mixed number 4 4/5 as 4.45 or 4.05 instead of 4.8. Another pitfall is to round too early or incorrectly, which may give a noticeably different percentage. Students sometimes also invert the percentage formula by computing (Reference / Value) * 100 instead of (Value / Reference) * 100, leading to the reciprocal percentage. Careful conversion and accurate use of the percentage formula prevents these errors. Final Answer: 4 4/5 of 259.99 is approximately 100% of 1247.9.

Question 19

What is 82.5% of 2,360?

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Introduction / Context: This question is a direct computation of a given percentage of a number. Calculating 82.5% of 2360 helps reinforce how to handle percentages that are not whole numbers, especially those involving halves such as 0.5%. This skill is useful in many real life contexts, including discounts, scores, and financial calculations. Given Data / Assumptions: We need to find 82.5% of 2360. 82.5% is a fractional percentage consisting of 82% plus 0.5%. No additional conditions or adjustments are given. Concept / Approach: To find X% of a number N, we can either convert X% to a decimal and multiply or break X% into simpler parts. Here, 82.5% can be written as 82.5 / 100 = 0.825. So we can compute 0.825 * 2360. Alternatively, we can split 82.5% into 80% + 2% + 0.5% of 2360, compute each part separately, and sum them. This second approach provides an easy mental calculation method and improves understanding of how percentages break down. Step-by-Step Solution: Step 1: Write 82.5% as a decimal: 82.5% = 82.5 / 100 = 0.825. Step 2: Multiply 0.825 by 2360: 0.825 * 2360. Step 3: Break it into components: 0.825 * 2360 = (0.8 + 0.02 + 0.005) * 2360. Step 4: Compute 80% of 2360: 0.8 * 2360 = 1888. Step 5: Compute 2% of 2360: 0.02 * 2360 = 47.2. Step 6: Compute 0.5% of 2360: 0.005 * 2360 = 11.8. Step 7: Add the parts: 1888 + 47.2 + 11.8 = 1947. Step 8: Therefore, 82.5% of 2360 is 1947. Verification / Alternative check: We can verify by direct decimal multiplication: 0.825 * 2360. Using standard multiplication, 825 * 2360 / 1000 yields 1947. This confirms the result obtained via percentage decomposition. Both methods support that the correct value is 1947. Why Other Options Are Wrong: 1960, 1954, and 1931 are close to 1947 but do not match the exact calculated value. They may correspond to incorrect approximations or rounding at intermediate steps. The option 2025 is significantly higher and clearly not correct. Only 1947 matches the precise computation of 82.5% of 2360. Common Pitfalls: Common mistakes include treating 82.5% as 0.8250 but then misaligning decimal places during multiplication, or mistakenly computing 82.5% as 0.8250 of 236 instead of 2360. Some learners also forget to include the 0.5% component or treat it as 5% by mistake. Ensuring careful decimal conversion and organized computation avoids these errors. Final Answer: The value of 82.5% of 2360 is 1,947.

Question 20

In a city there are 4,800 engineers in total. The number of female engineers is 3/8 of the total number of engineers. The number of female engineers is what percentage of the number of male engineers?

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Introduction / Context: This problem combines fractions and percentages in a demographic context. We are given the fraction of female engineers out of a total and asked to express the number of female engineers as a percentage of the number of male engineers. Such questions appear in data interpretation and percentage sections and help strengthen understanding of relative comparison between two groups. Given Data / Assumptions: Total number of engineers = 4800. Female engineers are 3/8 of the total. Male engineers are the remaining engineers after counting females. We must find female engineers as a percentage of male engineers. Concept / Approach: First, we use the fraction 3/8 to find the exact number of female engineers. Then we subtract this from the total to get the number of male engineers. Once we have both counts, we compute the percentage using the formula: Percentage = (Number of females / Number of males) * 100. It is important to note that we are not asked for the percentage of females out of the total, but specifically out of the male count. Step-by-Step Solution: Step 1: Total engineers = 4800. Step 2: Female engineers = (3 / 8) * 4800. Step 3: Compute (3 / 8) * 4800 = 3 * 600 = 1800. Step 4: Male engineers = Total - Female = 4800 - 1800 = 3000. Step 5: We need female engineers as a percentage of male engineers. Step 6: Percentage = (Number of females / Number of males) * 100. Step 7: Substitute values: Percentage = (1800 / 3000) * 100. Step 8: 1800 / 3000 = 0.6, and 0.6 * 100 = 60%. Step 9: So, the number of female engineers is 60% of the number of male engineers. Verification / Alternative check: We can reduce the ratio of female to male engineers first. Females : Males = 1800 : 3000. Divide both by 300 to get 6 : 10. This ratio says for every 10 male engineers there are 6 female engineers. To convert 6 out of 10 to a percentage, compute (6 / 10) * 100 = 60%. This matches the earlier result, confirming the calculation. Why Other Options Are Wrong: 76%, 56%, 52%, and 48% do not correspond to the ratio 1800 : 3000. For instance, 56% would imply a female to male ratio of 56 : 100, which is less than the actual 6 : 10, and 76% would imply a much higher ratio. Only 60% agrees with the simplified ratio 6 : 10. Therefore, the other options are inconsistent with the given data. Common Pitfalls: A common mistake is to compute the percentage of females out of the total (which would be 37.5%) rather than out of the male population. Another pitfall is miscalculating the fraction 3/8 of 4800 or forgetting to subtract from the total to find the male count. Carefully tracking what base quantity the percentage is relative to is crucial for correct answers in such questions. Final Answer: The number of female engineers is 60% of the number of male engineers.

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