Question 1

Raman spends 80% of his income. If his income increases by 25% and his expenditure increases by 10%, then by what percentage do his savings increase?

Accepted Answer

Introduction / Context: This question tests understanding of the relationship between income, expenditure and savings when both income and expenditure change by different percentages. It is not enough to look only at the changes in percentages; we must also consider how these changes affect the absolute amounts and the resulting savings. This type of question is common in topics on profit, loss and savings in aptitude tests. Given Data / Assumptions: Initially, Raman spends 80% of his income. Therefore, initially he saves 20% of his income. His income then increases by 25%. His expenditure increases by 10%. We must find the percentage increase in his savings. Concept / Approach: Let the initial income be I. Initial expenditure is 0.80 * I, and initial saving is 0.20 * I. When income increases by 25%, new income becomes 1.25 * I. Expenditure increases by 10%, so new expenditure becomes 1.10 times the original expenditure (0.80 * I), that is 0.88 * I. New savings then equal new income minus new expenditure. After finding initial and new savings in terms of I, we compare them to determine the percentage increase in savings. Step-by-Step Solution: Step 1: Let initial income = I. Step 2: Initial expenditure = 80% of I = 0.80 * I. Step 3: Initial savings = income - expenditure = I - 0.80 * I = 0.20 * I. Step 4: After a 25% increase, new income = 1.25 * I. Step 5: After a 10% increase in expenditure, new expenditure = 1.10 * (0.80 * I) = 0.88 * I. Step 6: New savings = new income - new expenditure = 1.25 * I - 0.88 * I = 0.37 * I. Step 7: Initial savings were 0.20 * I and new savings are 0.37 * I. Step 8: Increase in savings = 0.37 * I - 0.20 * I = 0.17 * I. Step 9: Percentage increase in savings = (increase / original savings) * 100 = (0.17 * I / 0.20 * I) * 100. Step 10: The I cancels out, leaving (0.17 / 0.20) * 100 = 0.85 * 100 = 85%. Verification / Alternative check: To verify, choose a concrete value, for example I = Rs. 100. Initially, Raman spends Rs. 80 and saves Rs. 20. After the changes, new income = 125, new expenditure = 1.10 * 80 = 88, and new savings = 125 - 88 = 37. The increase in savings is 37 - 20 = 17. Percentage increase = (17 / 20) * 100 = 85%. This confirms the earlier algebraic result. Why Other Options Are Wrong: 17 is the absolute increase (in our I = 100 example), not the percentage increase. 70% and 77% are smaller than the actual ratio of new savings to old savings. The calculations clearly show that the increase is 85% of the original saving, so only 85 is correct. Common Pitfalls: A frequent error is to think that since income increased by 25% and expenditure increased by 10%, savings must increase by 15%. This is wrong because savings are the difference between two quantities that are both changing. Another pitfall is to forget that the 10% increase in expenditure is on the original expenditure, not on income. Always translate the situation into explicit algebraic expressions before computing percentage changes in derived quantities like savings. Final Answer: Raman's savings increase by 85%.

Question 2

In a school there are 1400 students. Of these, 25% wear spectacles and 2/7 of those wearing spectacles are boys. How many girls in the school wear spectacles?

Accepted Answer

Introduction / Context: This question is about percentages and fractions applied one after another. First, a percentage of the total students are identified as wearing spectacles. Then, among those, a fraction are boys, and the rest are girls. We are asked to find how many girls wear spectacles. This combines percentage calculation with simple ratio reasoning. Given Data / Assumptions: Total number of students in the school = 1400. 25% of the students wear spectacles. Among those who wear spectacles, 2/7 are boys. The remaining spectacle wearers are girls. Concept / Approach: First we compute the total number of students who wear spectacles by taking 25% of 1400. Then we apply the fraction 2/7 to that number to find how many of those are boys. Finally the number of girls who wear spectacles equals total spectacle wearers minus spectacle wearing boys. This stepwise approach ensures we use both the percentage and fractional information correctly. Step-by-Step Solution: Step 1: Total students = 1400. Step 2: Percentage of students wearing spectacles = 25%. Step 3: Number of students wearing spectacles = 25% of 1400 = (25 / 100) * 1400. Compute 25% of 1400 = 0.25 * 1400 = 350. Step 4: Of these 350 spectacle wearers, 2/7 are boys. Number of boys wearing spectacles = (2 / 7) * 350. (2 / 7) * 350 = 2 * 50 = 100. Step 5: Number of girls wearing spectacles = total spectacle wearers - boys wearing spectacles. Girls wearing spectacles = 350 - 100 = 250. Step 6: Therefore, 250 girls in the school wear spectacles. Verification / Alternative check: We can also compute directly the fraction of total students who are girls wearing spectacles. Spectacle wearers are 25% of 1400, that is 350. Of these, 2/7 (or about 28.57%) are boys, so 5/7 (or about 71.43%) must be girls. 5/7 of 350 = 5 * 50 = 250. This agrees with our earlier result, confirming that 250 is correct. Why Other Options Are Wrong: 100 corresponds to the number of boys wearing spectacles, not girls. 200 and 300 do not satisfy the fraction 2/7 for boys when combined with 1400 total students and 25% spectacle wearers. Only 250 fits consistently with all pieces of information given in the problem. Common Pitfalls: A common mistake is to take 2/7 of the total number of students, instead of 2/7 of only those who wear spectacles. Another mistake is to miscalculate 25% of 1400 or to wrongly subtract when computing the girls. Carefully tracking which group each percentage or fraction applies to is essential in such layered percentage problems. Final Answer: The number of girls in the school who wear spectacles is 250.

Question 3

The marked price of an article is Rs. 5000. Due to a special festive offer a certain percentage discount is allowed, so that a customer buys the article at the reduced price and then sells it at Rs. 5000, making a profit of 11 1/9%. What was the per…

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Introduction / Context: This problem combines discount and profit concepts. The article has a marked price, is purchased at a discounted price, and then sold at the full marked price, generating a given percentage profit for the customer. We need to work backwards from the selling price and profit percentage to find the cost price for the customer, and then compare that cost price with the marked price to determine the discount percentage. Given Data / Assumptions: Marked price of the article = Rs. 5000. Customer buys at a discounted price (unknown) and then sells at Rs. 5000. The customer makes a profit of 11 1/9% on cost price. We assume normal definition of profit: profit = selling price minus cost price. We must determine the percentage discount allowed on the marked price. Concept / Approach: Let the cost price (discounted price) for the customer be C. The selling price is 5000. Given profit percentage is 11 1/9%, which is 100 / 9 percent, or 1 / 9 in fractional form when expressed relative to cost price. So selling price = C * (1 + 1 / 9) = C * (10 / 9). Using this, we can find C from the known selling price. After obtaining C, we compare it with the original marked price 5000. The discount is the difference between the marked price and C, expressed as a percentage of the marked price. Step-by-Step Solution: Step 1: Write the profit percentage as a fraction. Given profit = 11 1/9% = 100 / 9%. As a fraction of cost price, this is 1 / 9. Step 2: Let the customer's cost price (discounted price) = C. Step 3: Selling price SP = C * (1 + 1 / 9) = C * (10 / 9). Step 4: SP is given as Rs. 5000, so C * (10 / 9) = 5000. Step 5: Solve for C: C = 5000 * (9 / 10) = 4500. Step 6: Thus, the discounted price (cost price for the customer) is Rs. 4500. Step 7: Discount amount = marked price - discounted price = 5000 - 4500 = 500. Step 8: Discount percentage = (discount amount / marked price) * 100 = (500 / 5000) * 100 = 10%. Verification / Alternative check: Starting from the discounted price C = 4500, compute the profit if the article is sold at 5000. Profit = 5000 - 4500 = 500. Profit percentage on cost = 500 / 4500 * 100 = (1 / 9) * 100 = 11 1/9%. This matches the given profit condition. Also, a discount of 10% on 5000 gives 500 as the discount, making the discounted price 4500, consistent with the calculated cost price. Thus the solution is confirmed. Why Other Options Are Wrong: 10/3%, 15/2% and 100/9% are much smaller or larger discount percentages and would yield discounted prices on 5000 that do not allow a resale at 5000 with profit exactly equal to 11 1/9%. They would either produce a different profit percentage or fail to match the given relationship between profit and selling price. Common Pitfalls: A common mistake is to confuse the profit percentage with the discount percentage and try to set them equal. Another error is misinterpreting the mixed fraction 11 1/9% and incorrectly converting it to an improper fraction or decimal. It is also easy to accidentally divide by the selling price instead of the cost price when computing profit percentage. Always base the percentage profit on cost price and carefully convert mixed percentages to fractions. Final Answer: The percentage discount allowed on the marked price was 10%.

Question 4

P, Q and R secured 45%, 50% and 60% marks respectively in Biology. M scored 12.5 marks more than P and 4 marks less than R. You are given three statements about their marks. Which statements are actually required to determine the individual marks of…

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Introduction / Context: This is a data sufficiency style question framed in the context of Biology marks. The problem describes the percentages obtained by three students, P, Q and R, and then relates the marks of a fourth student M to the marks of P and R. It then lists three additional statements about totals and specific marks, and asks which of these statements are required to determine the individual marks of all four students. The challenge is to see whether the original information alone is enough to uniquely determine the marks, or whether some of the extra statements are necessary. Given Data / Assumptions: P secures 45% marks in Biology. Q secures 50% marks in Biology. R secures 60% marks in Biology. Let the maximum marks in Biology be Mmax. M secures 12.5 more marks than P. M secures 4 fewer marks than R. Statements given: A: The total of the marks obtained by the four students is 311.5. B: The total of M and P marks is 147.5. C: R has obtained 84 marks. We must decide which of A, B, C are actually required to find individual marks. Concept / Approach: Let the maximum marks be Mmax. Then P = 0.45 * Mmax, Q = 0.50 * Mmax and R = 0.60 * Mmax. M is defined both as P + 12.5 and as R - 4. These relationships already form a system of equations involving P, Q, R, M and Mmax. If this system has a unique solution, then we do not need any of the additional statements A, B or C to determine the marks. So our main task is to check whether the base equations from the question alone are sufficient to uniquely determine all values. Step-by-Step Solution: Step 1: Express P, Q and R in terms of Mmax. P = 0.45 * Mmax, Q = 0.50 * Mmax, R = 0.60 * Mmax. Step 2: Use the relations for M. M = P + 12.5 and M = R - 4. Step 3: Equate P + 12.5 and R - 4. P + 12.5 = R - 4. Step 4: Substitute P and R in terms of Mmax: 0.45 * Mmax + 12.5 = 0.60 * Mmax - 4. Step 5: Rearrange: 12.5 + 4 = 0.60 * Mmax - 0.45 * Mmax. Step 6: 16.5 = 0.15 * Mmax, so Mmax = 16.5 / 0.15 = 110. Step 7: Now compute P, Q and R: P = 0.45 * 110 = 49.5, Q = 0.50 * 110 = 55, R = 0.60 * 110 = 66. Step 8: From M = P + 12.5, M = 49.5 + 12.5 = 62. From M = R - 4, M = 66 - 4 = 62, which is consistent. Step 9: Thus P, Q, R, M and Mmax are uniquely determined without using statements A, B or C. Verification / Alternative check: We can quickly check statement C: it says R has 84 marks, but from our direct derivation, R has 66 marks when maximum marks are 110. This means statement C actually contradicts the values derived from the original data. Therefore, relying on statement C could mislead us. Since the question is about which statements are required, and the core data already gives a unique solution, the correct conclusion is that none of the extra statements are required. The base relationships are sufficient by themselves. Why Other Options Are Wrong: Saying “Only C” or “Only A and B” or “All are required” assumes that the base data is not sufficient to compute the marks. However, we have already determined all marks using only the original percentages and the relationships between M, P and R. Therefore, all such choices are incorrect. The additional statements may either be redundant or even inconsistent with the derived values. Common Pitfalls: In data sufficiency style questions, it is easy to assume that the extra statements must be used, especially when they look informative. Many learners do not verify whether the base problem is already solvable. Another common mistake is to take any one of the given statements as a fact without checking for consistency with the original data. A systematic approach is to first test whether the original conditions alone yield a unique solution, and only then consider if further statements are necessary. Final Answer: None of the additional statements A, B or C is required to determine the individual marks; the original data is sufficient.

Question 5

Dot Cabs has 8 Swift Dzire cars, 6 Scorpios, 7 Innovas and some small cars. If the number of Swift Dzire cars is one-fourth of the total fleet, how many small cars does the company have?

Accepted Answer

Introduction / Context: This question is a simple application of ratios and fractions in a real world setting. The total fleet of cars in a cab company includes different types of vehicles, and we know that one particular type (Swift Dzire) constitutes one-fourth of the entire fleet. By treating the total as an unknown and using the given counts of specific vehicles, we can determine how many small cars must be present so that Swift Dzire cars make up exactly one-fourth of the total fleet. Given Data / Assumptions: Number of Swift Dzire cars = 8. Number of Scorpios = 6. Number of Innovas = 7. Number of small cars = S (unknown). The number of Swift Dzire cars is one-fourth of the total number of cars in the fleet. Total fleet size = Swift Dzire + Scorpios + Innovas + small cars. Concept / Approach: Let the total number of cars be T. Then T = 8 + 6 + 7 + S. The condition “Swift Dzire makes one-fourth of the total fleet” means 8 = (1 / 4) * T. We can use this relation to express T and then find S. This is a straightforward algebraic problem involving solving a linear equation in one variable. Step-by-Step Solution: Step 1: Let the total number of cars be T. Step 2: We know that T = 8 (Swift Dzire) + 6 (Scorpios) + 7 (Innovas) + S (small cars). So T = 21 + S. Step 3: The number of Swift Dzire cars is one-fourth of the total fleet. So 8 = (1 / 4) * T. Step 4: Multiply both sides by 4: T = 4 * 8 = 32. Step 5: Now use T = 21 + S: 21 + S = 32. Step 6: Solve for S: S = 32 - 21 = 11. Step 7: Hence the company has 11 small cars. Verification / Alternative check: We can verify by recomputing the total fleet with S = 11. Total = 8 + 6 + 7 + 11 = 32 cars. One-fourth of 32 is 32 / 4 = 8, which matches the given number of Swift Dzire cars. Therefore, our solution is consistent and correct. Why Other Options Are Wrong: If S were 9, total would be 30, and one-fourth of 30 is 7.5, not 8. If S were 10, total would be 31, and one-fourth of 31 is not an integer and certainly not 8. If S were 12, total would be 33, and one-fourth of 33 is 8.25. Only S = 11 produces a total where Swift Dzire cars are exactly one-fourth of the fleet. Common Pitfalls: Learners sometimes misinterpret “one-fourth of the total fleet” as “Swift Dzire is one-fourth more than some other category” or confuse which quantity is supposed to be a fraction of which. Another mistake is to forget to include all car types in the total. It is important to define variables clearly, write down the total explicitly and translate the fractional relationship into an equation before solving. Final Answer: Dot Cabs has 11 small cars in its fleet.

Question 6

The payment per hour for a daily-wage employee is increased by 33.33% and at the same time the working hours are reduced by 33.33%. What is the net percentage change in the employee's total income?

Accepted Answer

Introduction / Context: This question tests the combined effect of two opposite percentage changes on total earnings. The wage rate per hour increases, but the number of hours worked decreases. Total income is the product of wage per hour and hours worked, so we must apply the changes multiplicatively, not add or subtract percentage changes directly. Such problems are classic examples of successive percentage changes in quantitative aptitude. Given Data / Assumptions: Hourly wage rate is increased by 33.33% (approximately one-third). Working time is reduced by 33.33%. Total income = wage per hour * number of hours worked. We assume there are no other components of income. Concept / Approach: Let the initial hourly wage be W and the initial number of hours be H. Then initial income = W * H. After the changes, the new wage becomes W * (1 + 1 / 3) = (4 / 3) * W, and the new hours become H * (1 - 1 / 3) = (2 / 3) * H. New income is then (4 / 3) * W * (2 / 3) * H. Comparing new income with initial income allows us to find the net percentage change. This is a standard successive change problem where the combined factor is the product of individual change factors. Step-by-Step Solution: Step 1: Let hourly wage initially be W and hours worked be H. Step 2: Initial income = W * H. Step 3: Wage increases by 33.33%, which is approximately 1 / 3. So new wage = W * (1 + 1 / 3) = (4 / 3) * W. Step 4: Working time decreases by 33.33%, so new hours = H * (1 - 1 / 3) = (2 / 3) * H. Step 5: New income = (4 / 3) * W * (2 / 3) * H. Step 6: Simplify the product: (4 / 3) * (2 / 3) = 8 / 9. Step 7: So new income = (8 / 9) * W * H. Step 8: Therefore, new income is 8 / 9 of the old income, or approximately 0.8889 times the original. Step 9: Net percentage change in income = (new - original) / original * 100 = (8 / 9 - 1) * 100. Step 10: 8 / 9 - 1 = -1 / 9, so percentage change = (-1 / 9) * 100 ≈ -11.11%. Step 11: Hence the income decreases by about 11.11%. Verification / Alternative check: Take specific numbers for clarity. Suppose initially W = Rs. 30 per hour and H = 9 hours, so income = 30 * 9 = Rs. 270. After a 33.33% increase, new wage ≈ 40 (since one-third more of 30 is 10). After a 33.33% decrease, new hours ≈ 6 (one-third fewer than 9). New income = 40 * 6 = Rs. 240. The decrease in income is 270 - 240 = 30. Percentage change = 30 / 270 * 100 ≈ 11.11% decrease. This matches the earlier calculation. Why Other Options Are Wrong: 11.11% increase would require the combined factor to be 10 / 9 instead of 8 / 9. 21.11% increase or decrease do not correspond to any natural product of the given change factors. The only correct combined effect of a one-third increase and a one-third decrease in two multiplicative components is an 11.11% decrease in their product. Common Pitfalls: A common error is to simply subtract 33.33% from 33.33% and conclude there is no net change, which is incorrect because the rate and time multiply, not add. Another mistake is to compute 33.33% of hours using the new wage as a base or vice versa. Each change must be applied to its own quantity, and only then should their combined effect on income be calculated. Final Answer: The employee's total income decreases by approximately 11.11%.

Question 7

The population of a colony of mosquitoes increases by 20% per day. If the population on Monday is 3000, on which day of the same week will the population first reach 5184?

Accepted Answer

Introduction / Context: This problem is an example of exponential growth, where a population increases by a fixed percentage each day. We are given the population on Monday and the daily growth rate, and we are asked to find on which subsequent day the population equals a specific target value. Such questions help develop comfort with repeated percentage growth and geometric sequences. Given Data / Assumptions: Initial population on Monday = 3000 mosquitoes. Daily growth rate = 20% increase. Population each day is 120% (that is 1.20 times) of the previous day. We need to determine on which day of the week the population becomes 5184. We assume the growth continues consistently with no external losses. Concept / Approach: Let the population on Monday be P0 = 3000. Each day multiplies the previous population by 1.20. Thus, population after n days is Pn = P0 * (1.20^n). We need to find the integer n such that Pn = 5184. Then we interpret n in terms of days of the week, counting from Monday. Since these are small numbers, we can simply compute populations for successive days until we reach 5184. Step-by-Step Solution: Step 1: Population on Monday (Day 0) = 3000. Step 2: Population on Tuesday (Day 1) = 3000 * 1.20 = 3600. Step 3: Population on Wednesday (Day 2) = 3600 * 1.20 = 4320. Step 4: Population on Thursday (Day 3) = 4320 * 1.20. 4320 * 1.20 = 4320 * (6 / 5) = 5184. Step 5: Thus, on Thursday the population is exactly 5184. Verification / Alternative check: We can also solve using exponents. We want 3000 * (1.20^n) = 5184. So 1.20^n = 5184 / 3000 = 1.728. It is known that 1.20^3 = 1.2 * 1.2 * 1.2 = 1.728. Therefore n = 3. Counting three days after Monday leads to Tuesday as 1, Wednesday as 2 and Thursday as 3. This confirms that Thursday is the correct day. Why Other Options Are Wrong: Tuesday corresponds to one 20% increase, giving 3600, not 5184. Wednesday corresponds to two increases, giving 4320. Friday would give a fourth increase, resulting in a value greater than 5184. Monday is the starting day with 3000. Only Thursday matches the target population of 5184. Common Pitfalls: Learners sometimes treat the 20% daily increase additively, adding 20% of the original population each day instead of multiplying by 1.20 each time. This yields a linear pattern rather than exponential growth. Another error is miscounting days when interpreting the exponent n as the step from Monday. Careful enumeration of days and consistent use of the multiplier 1.20 prevent such mistakes. Final Answer: The population reaches 5184 on Thursday.

Question 8

Three independent cost-cutting strategies P, Q and R are applied in a company. Strategy P reduces costs by 20%, Q reduces the (already reduced) cost by 30%, and R further reduces it by 10%. Assuming all three strategies operate independently and seq…

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Introduction / Context: This problem explores the combined effect of multiple successive percentage reductions on cost. Each strategy independently cuts costs by a given percentage and acts on the currently reduced amount, not on the original. The question asks for the net saving after all strategies have been applied. This is a classic example of successive percentage change where multiplicative factors must be used instead of simple addition of percentages. Given Data / Assumptions: Strategy P saves 20%, so it leaves 80% of the cost. Strategy Q saves 30%, so it leaves 70% of the cost it acts on. Strategy R saves 10%, so it leaves 90% of the cost it acts on. The strategies are applied one after another, each acting on the cost remaining after the previous strategy. We must find the overall percentage saving compared to the original cost. Concept / Approach: Let the original cost be C. After strategy P, the cost is 0.80 * C. After strategy Q, it becomes 0.70 of that, and after strategy R, it becomes 0.90 of the already reduced cost. So final cost = C * 0.80 * 0.70 * 0.90. The net saving is then original cost minus final cost, and the net percentage saving is that difference divided by original cost, multiplied by 100. Because each strategy acts multiplicatively, we must multiply their remaining fractions to get the final factor. Step-by-Step Solution: Step 1: Let original cost = C. Step 2: After strategy P (20% saving), remaining cost = 80% of C = 0.80 * C. Step 3: After strategy Q (30% saving), remaining cost = 70% of the current amount = 0.70 * (0.80 * C). Step 4: After strategy R (10% saving), remaining cost = 90% of the current amount = 0.90 * [0.70 * (0.80 * C)]. Step 5: Combine the factors: final cost = C * 0.80 * 0.70 * 0.90. Step 6: Compute the product: 0.80 * 0.70 = 0.56. Step 7: Now 0.56 * 0.90 = 0.504. Step 8: Thus, final cost = 0.504 * C. Step 9: Net saving fraction = 1 - 0.504 = 0.496. Step 10: Net percentage saving = 0.496 * 100 = 49.6%. Verification / Alternative check: Take an example with C = 100 units. After strategy P, cost is 80. After applying Q, cost is 70% of 80, which is 56. After R, cost is 90% of 56, which is 50.4. Net saving = 100 - 50.4 = 49.6. Expressed as a percentage of the original 100, this is 49.6%. The example confirms the algebraic result and matches option 49.6%. Why Other Options Are Wrong: 50.4% would be the percentage of cost remaining, not the percentage saved. 33.67% or 66.66% do not correspond to the calculated net savings and likely stem from adding or subtracting percentages instead of multiplying remaining fractions. The correct combined effect of 20%, 30% and 10% savings is a 49.6% overall saving. Common Pitfalls: A very common mistake is to simply add the three savings and say 20 + 30 + 10 = 60% saving. This ignores the fact that each subsequent strategy acts on an already reduced cost. Another mistake is to add the remaining percentages (80, 70, 90) and attempt to average them, which does not reflect the multiplicative effect. Always convert each saving into a remaining fraction and multiply them to get the final fraction of cost left. Final Answer: The net percentage saving achieved is 49.6%.

Question 9

An individual deposits 20% of 25% of 30% of the money he has into his bank account. If the deposited amount is Rs. 750, how much money did he initially have before depositing?

Accepted Answer

Introduction / Context: This question deals with successive percentages applied to an initial amount, but in the form of a nested expression: 20% of 25% of 30% of the original amount. We are given the final deposited amount and asked to determine the original amount. This is a reverse percentage problem where we work backwards from a compound fraction of an unknown total. Given Data / Assumptions: Let the original amount of money be X. The individual takes 30% of X. From that, he takes 25%. From that, he takes 20% and deposits it into the bank. The amount deposited = Rs. 750. We must find the value of X. Concept / Approach: We interpret “20% of 25% of 30% of X” as a chain of multiplications with decimal or fractional forms of the percentages. Specifically, 30% of X is 0.30 * X, 25% of that is 0.25 * 0.30 * X, and 20% of that is 0.20 * 0.25 * 0.30 * X. This final expression equals 750. We simplify the combined factor 0.20 * 0.25 * 0.30, set up an equation and solve for X by dividing 750 by the combined fraction. Step-by-Step Solution: Step 1: Let the original amount be X. Step 2: Compute the combined percentage factor. 30% of X = 0.30 * X. 25% of that amount = 0.25 * (0.30 * X) = 0.25 * 0.30 * X. 20% of that result = 0.20 * (0.25 * 0.30 * X) = 0.20 * 0.25 * 0.30 * X. Step 3: Multiply the decimal factors: 0.20 * 0.25 = 0.05. Step 4: Then 0.05 * 0.30 = 0.015. Step 5: Hence, deposited amount = 0.015 * X. Step 6: The deposited amount is given as Rs. 750. So 0.015 * X = 750. Step 7: Solve for X: X = 750 / 0.015. Step 8: 750 / 0.015 = 750 * (1 / 0.015) = 750 * (1000 / 15) = (750000 / 15) = 50,000. Step 9: Therefore, the individual initially had Rs. 50,000. Verification / Alternative check: Check by moving forward from Rs. 50,000. First, 30% of 50,000 is 0.30 * 50,000 = 15,000. Next, 25% of 15,000 is 0.25 * 15,000 = 3750. Finally, 20% of 3750 is 0.20 * 3750 = 750. This agrees exactly with the given deposited amount. Hence, the calculated original amount of Rs. 50,000 is correct. Why Other Options Are Wrong: If the original amount were Rs. 38,000, 20% of 25% of 30% of that would be 0.015 * 38,000 = 570, not 750. Similarly, for Rs. 47,000 we get 0.015 * 47,000 = 705, and for Rs. 35,000 we get 0.015 * 35,000 = 525. None of these match the required deposit of 750. Only Rs. 50,000 yields the correct deposit amount. Common Pitfalls: One mistake is to add the percentages (20 + 25 + 30) and treat the total 75% as a single direct percentage of X, which is incorrect. Another common error is to misplace the decimal when multiplying percentages, for example using 0.2 instead of 0.20 * 0.25 * 0.30. Always convert each percentage properly, multiply them stepwise, and remember that chaining percentages multiplies their fractions, it does not add them. Final Answer: The individual initially had Rs. 50,000.

Question 10

Tanks A and B are right circular cylinders with different heights and circumferences. Tank A has interior height 10 m and interior circumference 8 m, and tank B has interior height 8 m and interior circumference 10 m. If both tanks are completely fu…

Accepted Answer

Introduction / Context: This question compares the capacities of two cylindrical tanks when their heights and circumferences are interchanged. It tests understanding of how the volume of a cylinder depends on its radius and height, and how to express one volume as a percentage of another. Given Data / Assumptions: • Tank A is a right circular cylinder with height 10 m and interior circumference 8 m. • Tank B is a right circular cylinder with height 8 m and interior circumference 10 m. • Both tanks are considered completely full when comparing volumes. • Volumes are compared using exact geometric formulas, and the percentage is rounded as a simple ratio. Concept / Approach: The volume of a right circular cylinder is given by V = π * r^2 * h, where r is the radius and h is the height. The circumference of a circle is C = 2 * π * r. From the circumference we can find r, and then substitute into the volume formula. Because π will cancel when forming the ratio, we can work directly with C and h to compare the two capacities. Step-by-Step Solution: Step 1: Use C = 2 * π * r so r = C / (2 * π). Step 2: Substitute r into V = π * r^2 * h to get V = (C^2 * h) / (4 * π). Step 3: For tank A: C = 8 m, h = 10 m, so V_A is proportional to 8^2 * 10 = 64 * 10 = 640. Step 4: For tank B: C = 10 m, h = 8 m, so V_B is proportional to 10^2 * 8 = 100 * 8 = 800. Step 5: Ratio V_A : V_B = 640 : 800 = 64 : 80 = 4 : 5. Step 6: Therefore V_A is (4 / 5) * 100% = 80% of V_B. Verification / Alternative check: Instead of writing the full formula, one can directly use the proportional relation V is proportional to C^2 * h. Plugging the same values again gives 640 for tank A and 800 for tank B, confirming that the fraction is 640 / 800 = 0.8, which is 80%. Why Other Options Are Wrong: • 76% and 72% are lower than the correct ratio 80% and do not match the computed fraction 640 / 800. • 82% is close to 80% but still does not equal the exact ratio 4 / 5. Common Pitfalls: A frequent mistake is to compare only heights or only circumferences instead of using both together. Another common error is forgetting that volume depends on the square of the radius, which means it depends on the square of the circumference. Ignoring this square leads to wrong ratios. Being systematic with the formulas avoids such errors. Final Answer: The capacity of tank A is 80% of the capacity of tank B.

Question 11

Kalyan purchased a Maruti van for Rs. 1,96,000, and the value of the van depreciates at a fixed rate of 14 2/7 percent per annum. What will be the approximate value of the van after two full years of depreciation at this constant rate?

Accepted Answer

Introduction / Context: This is a standard percentage depreciation question. It checks whether you understand how to apply a fixed percentage decrease year after year using the idea of successive percentage changes and compound reduction rather than subtracting a flat amount each year. Given Data / Assumptions: • Initial cost price of the Maruti van = Rs. 1,96,000. • Annual depreciation rate = 14 2/7 percent per year. • Depreciation is applied every year on the reduced value, not on the original cost. • The number of years of depreciation to be considered is exactly two years. Concept / Approach: A depreciation rate of 14 2/7 percent can be converted into an exact fraction. Since 14 2/7 percent equals 100 / 7 percent, this is 1 / 7 of the value each year. If a quantity loses 1 / 7 of its value, it retains 6 / 7 of its value. For successive years, we multiply the original value by (6 / 7) for each year. Thus the final value after n years is initial value * (retained fraction)^n. Step-by-Step Solution: Step 1: Convert 14 2/7 percent to a fraction: 14 2/7 = 100 / 7 percent. Step 2: As a fraction of the value, 100 / 7 percent is (100 / 7) / 100 = 1 / 7. Step 3: Depreciation of 1 / 7 means the van retains 1 − 1 / 7 = 6 / 7 of its value each year. Step 4: After two years, value = 1,96,000 * (6 / 7) * (6 / 7) = 1,96,000 * 36 / 49. Step 5: Compute 1,96,000 / 49 = 4,000, then 4,000 * 36 = 1,44,000. Verification / Alternative check: You can approximate 14 2/7 percent as about 14.28 percent and use a calculator or mental estimation. A single year reduces the value from 1,96,000 to around 1,68,000. A second similar reduction leads to a figure near 1,44,000. This supports the exact calculation result and shows the answer is reasonable. Why Other Options Are Wrong: • Rs. 1,24,000 represents an excessively large reduction over two years compared to 14 2/7 percent annually. • Rs. 1,32,000 is still lower than the correctly computed amount and does not match the fraction 36 / 49 of the original price. • Rs. 1,62,000 is too high and would correspond to a much smaller depreciation rate than given in the question. Common Pitfalls: Many learners mistakenly subtract 14 2/7 percent of the original amount twice instead of applying the percentage on the reduced value each year. Others convert the mixed fraction incorrectly. Always convert a mixed percentage such as 14 2/7 percent carefully and use the compound multiplier method to avoid errors. Final Answer: The value of the van after two years is Rs. 1,44,000.

Question 12

A user types a long piece of content on 20 sheets, where each sheet has 55 lines and each line can contain up to 66 characters. The same content is then retyped on a new set of sheets, where each sheet has 65 lines and each line can take 70 characte…

Accepted Answer

Introduction / Context: This problem deals with capacity and percentage reduction. It asks you to compare how many sheets are required to hold the same amount of text when the number of lines per sheet and characters per line change. This type of question tests multiplication, division, and percentage calculation skills in a practical formatting context. Given Data / Assumptions: • Original typing format: 20 sheets, 55 lines per sheet, 66 characters per line. • New typing format: 65 lines per sheet, 70 characters per line. • The total amount of content (characters) remains exactly the same in both formats. • Every line is treated as fully filled for capacity calculations. Concept / Approach: The total text capacity can be measured in characters. First find the total number of characters in the original arrangement. Then find how many characters one new sheet can hold. Dividing the total character count by the capacity of one new sheet gives the number of new sheets needed. Finally, compare old and new numbers of sheets to get the reduction and the percentage decrease. Step-by-Step Solution: Step 1: Original total characters = 20 * 55 * 66. Step 2: Compute 55 * 66 = 3,630, so total characters = 20 * 3,630 = 72,600. Step 3: Capacity of one new sheet = 65 * 70 = 4,550 characters. Step 4: Number of new sheets required = 72,600 / 4,550 ≈ 15.95, so 16 sheets are needed. Step 5: Original sheets = 20, new sheets = 16, so reduction = 20 − 16 = 4 sheets. Step 6: Percentage reduction = (4 / 20) * 100% = 20%. Verification / Alternative check: We can also approximate by observing that the capacity per sheet increases from 55 * 66 = 3,630 characters to 4,550 characters. This is clearly more, so fewer sheets are needed. The exact calculations above confirm that only 16 sheets are required and that the reduction of 4 sheets corresponds to a 20 percent decrease, matching the computed fraction 4 / 20. Why Other Options Are Wrong: • 2 sheets or 10% reduction underestimates the actual saving; the math shows a bigger reduction. • 6 sheets or 30% reduction would require only 14 sheets in the new format, which does not fit the total characters correctly. • 8 sheets or 40% reduction would mean using only 12 sheets, which again contradicts the necessary capacity for 72,600 characters. Common Pitfalls: Common mistakes include rounding the intermediate values too early or confusing lines and characters. Some learners also try to compare only lines per sheet without considering characters per line. Always work with total characters to avoid miscounts and use proper rounding when computing the final number of sheets. Final Answer: The reduction is 4 sheets or 20% reduction.

Question 13

If the value of L is 30 percent greater than the value of K, then by what percentage is K less than L? Choose the correct percentage decrease for K relative to L.

Accepted Answer

Introduction / Context: This question explores the difference between percentage increase and percentage decrease. If one quantity is a certain percent more than another, the reverse relation is not the same percentage in the opposite direction. Understanding this asymmetry is very important in percentage calculations and many comparative problems in aptitude exams. Given Data / Assumptions: • L exceeds K by 30 percent. • That means L is 30 percent more than K. • We are required to find by what percent K is less than L. • No other conditions or units are involved, so we can work with simple numbers. Concept / Approach: If L is 30 percent more than K, then L = K * (1 + 30 / 100) = K * 1.3. To find how much K is less than L in percentage terms, we consider the change from L down to K. The formula for percentage decrease is (difference / original) * 100, where the original here is L, since we compare K relative to L. Step-by-Step Solution: Step 1: Express L in terms of K: L = 1.3 * K. Step 2: The difference between L and K is L − K = 1.3K − K = 0.3K. Step 3: Percentage by which K is less than L = (difference / L) * 100. Step 4: Substitute the values: percentage = (0.3K / 1.3K) * 100 = (0.3 / 1.3) * 100. Step 5: Simplify 0.3 / 1.3 = 3 / 13 ≈ 0.23077. Step 6: Multiply by 100 to get approximately 23.08 percent, which is close to 23 1/3 percent. Verification / Alternative check: Take a simple assumed value for K. For example, let K = 100. Then L is 30 percent more, so L = 130. The difference is 30. Now compute 30 as a percentage of 130: (30 / 130) * 100 ≈ 23.08 percent, confirming our earlier result and matching the closest option of 23 1/3 percent. Why Other Options Are Wrong: • 14 2/5% is far smaller than the correct decrease and does not match the ratio 30 : 130. • 20% would be correct only if L were 25 percent more than K, which is not the case here. • 16% is also inconsistent with the actual numerical comparison 30 out of 130. Common Pitfalls: A common mistake is to assume that if L is 30 percent more than K, then K is 30 percent less than L, which is incorrect. The base changes from K to L, so the percentages are different. Always identify clearly which value is treated as the base when calculating percentage increase or decrease. Final Answer: K is approximately 23 1/3% less than L.

Question 14

Calculate 12 percent of 70,000. Choose the correct numerical value for this percentage of the given quantity.

Accepted Answer

Introduction / Context: This is a direct percentage calculation that tests your ability to convert a percentage into a fraction and multiply it by a given number. Such problems appear frequently in profit and loss, interest, and data interpretation questions, so it is helpful to solve them quickly and accurately. Given Data / Assumptions: • The given number is 70,000. • We are asked to find 12 percent of this number. • No rounding or approximation is required; we want the exact value. Concept / Approach: To find p percent of a number N, we use the relation p percent of N = (p / 100) * N. In this question, p = 12 and N = 70,000. Simplifying the fraction 12 / 100 before multiplication makes the arithmetic faster and reduces errors in mental calculations. Step-by-Step Solution: Step 1: Express 12 percent as a fraction: 12 percent = 12 / 100. Step 2: Apply the formula: 12 percent of 70,000 = (12 / 100) * 70,000. Step 3: Simplify 12 / 100 to 3 / 25 by dividing numerator and denominator by 4. Step 4: Now compute (3 / 25) * 70,000. Step 5: First divide 70,000 by 25. Since 25 * 2,800 = 70,000, we get 70,000 / 25 = 2,800. Step 6: Multiply 2,800 by 3 to obtain 8,400. Verification / Alternative check: We can also think in two steps: 10 percent of 70,000 is 7,000, and 1 percent is 700. Therefore 2 percent is 1,400. Adding 10 percent and 2 percent gives 7,000 + 1,400 = 8,400. This alternate reasoning confirms that the computed value is correct. Why Other Options Are Wrong: • 7,200 corresponds to about 10.29 percent of 70,000, which is not equal to 12 percent. • 6,000 is even smaller and would represent less than 9 percent of 70,000. • 5,600 equals 8 percent of 70,000, which clearly does not match the required 12 percent. Common Pitfalls: Learners sometimes mistakenly divide by 12 instead of multiplying by 12 / 100, or they misplace the decimal point when converting a percentage to a decimal. Keeping the formula percent of N = (percent / 100) * N clear helps avoid these errors. Final Answer: 12 percent of 70,000 is 8,400.

Question 15

In the expression 75 percent of 450 plus an unknown percentage of 560 equals 600, find the unknown percentage of 560 that makes the statement true.

Accepted Answer

Introduction / Context: This question tests your ability to handle an equation involving percentages of different numbers. You are given one known percentage term, one unknown percentage term, and their sum. You must isolate the unknown percentage and compute its value carefully using basic algebra and percentage skills. Given Data / Assumptions: • 75 percent of 450 is known. • Let the unknown percentage of 560 be p percent. • The equation is: 75 percent of 450 + p percent of 560 = 600. • We solve for p and then match it to one of the given options. Concept / Approach: First compute 75 percent of 450. Then subtract this from 600 to find the remaining value that must be contributed by p percent of 560. Once you know the absolute contribution from 560, calculate p by treating it as (p / 100) * 560. Rearranging the equation gives p = (required value / 560) * 100. The key idea is setting up and solving a straightforward linear equation in p. Step-by-Step Solution: Step 1: Compute 75 percent of 450 = (75 / 100) * 450 = 0.75 * 450 = 337.5. Step 2: The total required is 600, so the remaining part from the second term is 600 − 337.5 = 262.5. Step 3: Let p be the unknown percentage. Then (p / 100) * 560 = 262.5. Step 4: Solve for p: p = (262.5 * 100) / 560. Step 5: Compute 262.5 / 560 = 0.46875, so p = 0.46875 * 100 = 46.875. Step 6: Therefore the unknown percentage is 46.875 percent. Verification / Alternative check: Check by substitution: 46.875 percent of 560 is (46.875 / 100) * 560 = 0.46875 * 560 = 262.5. Adding 337.5 and 262.5 gives 600, confirming the correctness of the computed percentage. Why Other Options Are Wrong: • 52.385% would make the second term too large and the total more than 600. • 42.155% and 40.555% would yield smaller contributions from 560, so the sum would be less than 600. Common Pitfalls: Errors often occur if you round too early or mis-handle decimal multiplication. Another common mistake is to divide 560 by 262.5 incorrectly. Writing each algebraic step clearly and simplifying with care helps avoid these issues. Final Answer: The unknown percentage of 560 is 46.875%.

Question 16

Evaluate the expression 40 percent of 75 plus 80 percent of 25, and then express this total as K percent of 250. What is the value of K?

Accepted Answer

Introduction / Context: This question checks your ability to compute basic percentages and then interpret the result as another percentage of a different base number. It reinforces the idea that percentage values must always be tied to the number they are taken from, and that different bases change the meaning of the same numerical result. Given Data / Assumptions: • First term: 40 percent of 75. • Second term: 80 percent of 25. • The sum of these two terms equals K percent of 250. • We must find the numeric value of K. Concept / Approach: The strategy is to evaluate each percentage term separately, add the results to obtain a total, and then compare this total with 250. If the total equals K percent of 250, then K = (total / 250) * 100. This is a straightforward application of the basic percent-of-a-number formula and solving a simple proportion. Step-by-Step Solution: Step 1: Compute 40 percent of 75 = (40 / 100) * 75 = 0.40 * 75 = 30. Step 2: Compute 80 percent of 25 = (80 / 100) * 25 = 0.80 * 25 = 20. Step 3: Add both results to get the total: 30 + 20 = 50. Step 4: Let this total equal K percent of 250, so (K / 100) * 250 = 50. Step 5: Solve for K: K = (50 * 100) / 250. Step 6: Simplify: 50 * 100 = 5,000 and 5,000 / 250 = 20, so K = 20. Verification / Alternative check: We can reason that 10 percent of 250 is 25. If the total is 50, that is double 25, so it must be 20 percent of 250. This mental check confirms that K equals 20 without doing the full algebra again. Why Other Options Are Wrong: • 10 corresponds to 10 percent of 250, which is 25, not 50. • 30 would give 75 as 30 percent of 250, larger than the total 50. • 35 would yield 87.5 as 35 percent of 250, which is far above the actual sum. Common Pitfalls: Some learners mistakenly add the percentages first (40 percent plus 80 percent) and then apply that combined 120 percent to a single number, which is not what the question describes. Always compute each percentage on its own base number and then combine the results as instructed. Final Answer: The required value of K is 20.

Question 17

Find 20 percent of 120 using a simple and clear method. Choose the correct numerical value for this percentage of the given number.

Accepted Answer

Introduction / Context: This question is a straightforward test of percentage calculation. Problems like this are common in many competitive exams and daily life, such as when computing discounts, tax, or simple proportions. Being able to quickly find 10 percent, 20 percent, and similar values is a useful mental math skill. Given Data / Assumptions: • The base number is 120. • The required percentage is 20 percent of this base. • We assume no rounding is needed; the answer is an exact integer. Concept / Approach: To find p percent of a number N, use the formula p percent of N = (p / 100) * N. Here p = 20 and N = 120. An even faster method for this particular case is to note that 20 percent is the same as 1 / 5, since 20 / 100 reduces to 1 / 5. Multiplying 120 by 1 / 5 directly gives the answer with minimal effort. Step-by-Step Solution: Step 1: Write 20 percent as a fraction: 20 percent = 20 / 100. Step 2: Simplify 20 / 100 by dividing numerator and denominator by 20, giving 1 / 5. Step 3: Apply the fraction to the number: (1 / 5) * 120. Step 4: Divide 120 by 5 to get 24. Step 5: Therefore, 20 percent of 120 is 24. Verification / Alternative check: We can also go through 10 percent steps. Ten percent of 120 is 12. Since 20 percent is double 10 percent, 20 percent must be 12 * 2 = 24. This matches the earlier calculation, confirming the correctness of the result. Why Other Options Are Wrong: • 20 corresponds to about 16.67 percent of 120, not 20 percent. • 32 is larger and equals about 26.67 percent of 120. • 36 corresponds to 30 percent of 120, not 20 percent. Common Pitfalls: Some learners confuse 20 percent with one fifth but then mistakenly divide by 20 instead of 5. Others might multiply 120 by 20 and forget to divide by 100. Always remember that percent means “per hundred” and that you can simplify the fraction before multiplying for easier computation. Final Answer: 20 percent of 120 is 24.

Question 18

A person wants to find 6 percent of Rs. 3,50,000. Using basic percentage calculation, determine the exact value that corresponds to 6 percent of 3,50,000.

Accepted Answer

Introduction / Context: This is a simple but very practical percentage question involving a relatively large amount of money. It reflects common real life calculations such as computing tax, commissions, or discounts expressed as a percentage of a large value. Given Data / Assumptions: • Base amount = Rs. 3,50,000. • Required percentage = 6 percent of this amount. • No rounding or approximation is necessary; the answer is exact. Concept / Approach: To find p percent of an amount N, use p percent of N = (p / 100) * N. For p = 6 and N = 3,50,000, we compute (6 / 100) * 3,50,000. It is often convenient to first find 1 percent of the amount and then multiply appropriately, especially for small percentages like 6 percent. Step-by-Step Solution: Step 1: Express 6 percent as a fraction: 6 percent = 6 / 100. Step 2: Compute 1 percent of 3,50,000. Since 1 percent is 1 / 100, 1 percent of 3,50,000 is 3,500. Step 3: Multiply this by 6 to get 6 percent: 3,500 * 6 = 21,000. Step 4: Therefore, 6 percent of 3,50,000 equals 21,000. Verification / Alternative check: Using the direct formula confirms the same result: (6 / 100) * 3,50,000 = 0.06 * 3,50,000. Multiplying 3,50,000 by 0.06 gives 21,000, which matches the stepwise calculation using 1 percent. Why Other Options Are Wrong: • 19,500 corresponds to 5.57 percent of 3,50,000, not 6 percent. • 23,600 would be more than 6 percent and does not match the exact formula output. • 24,000 represents about 6.86 percent of 3,50,000 and is therefore too large. Common Pitfalls: A frequent error is mishandling the zeros when multiplying large numbers by decimals. Another issue arises when learners confuse 6 percent with 0.6 or 0.06. Remember that 6 percent means 6 / 100, which equals 0.06, not 0.6. Explicitly writing the fraction before multiplying can help avoid mistakes. Final Answer: 6 percent of Rs. 3,50,000 is 21,000.

Question 19

In a zinc ore mine, 60 percent of the total mass consists of metals. Of this metallic portion, 3/4 percent is silver and the rest is zinc. If the total mass of ore extracted is 8,000 kg, what is the mass of zinc in kilograms?

Accepted Answer

Introduction / Context: This question combines percentage composition with stepwise deduction. It asks you to work from the percentage of metals in the ore, then from that metallic portion identify how much is silver and finally how much is zinc. It is typical of mixture and composition questions often seen in aptitude exams. Given Data / Assumptions: • Total mass of ore = 8,000 kg. • Metals form 60 percent of the ore by mass. • Of the metallic portion, 3/4 percent is silver. • The remainder of the metallic portion is zinc. Concept / Approach: First determine the total mass of metals by applying 60 percent to the total ore mass. Then find the amount of silver by applying 3/4 percent to the metallic mass. The zinc mass is simply the metallic mass minus the silver mass. This requires working with a very small percentage of a subset quantity, so careful calculation is important. Step-by-Step Solution: Step 1: Compute the mass of metals: 60 percent of 8,000 kg = (60 / 100) * 8,000 = 4,800 kg. Step 2: Silver is 3/4 percent of the metallic mass. Convert 3/4 percent to a fraction: 3/4 percent = (3 / 4) / 100 = 0.0075. Step 3: Silver mass = 0.0075 * 4,800 kg. Step 4: Compute 4,800 * 0.0075 = 36 kg of silver. Step 5: Zinc mass = total metallic mass − silver mass = 4,800 − 36 = 4,764 kg. Verification / Alternative check: We can confirm by checking that silver is a very small fraction. Since 1 percent of 4,800 kg is 48 kg, 0.75 percent (which is 3/4 percent) would be 0.75 * 48 = 36 kg, matching our computation. Subtracting 36 kg from 4,800 kg again yields 4,764 kg of zinc, so the result is consistent. Why Other Options Are Wrong: • 5,147 kg is greater than the total metallic mass of 4,800 kg, so it is impossible for zinc alone to weigh that much. • 3,587 kg is too small and would imply an unrealistically large silver component. • 2,125 kg is far below the correct value and does not arise from any reasonable calculation of the given percentages. Common Pitfalls: Learners may confuse 3/4 percent with 75 percent or with 3/4 of 60 percent, both of which are incorrect. Another common error is applying 3/4 percent directly to the total ore mass instead of the metallic portion. Always pay attention to which quantity a percentage is being applied to in multi-step problems. Final Answer: The mass of zinc in the ore is 4,764 kg.

Question 20

Twenty one is equal to 35 percent of a certain number. Based on this information, find that original number.

Accepted Answer

Introduction / Context: This problem reverses the usual direction of percentage questions. Instead of asking for a percentage of a known number, it gives the result of a percentage and asks you to find the original number. This is a common pattern in aptitude tests and helps you practice solving simple equations involving percentages. Given Data / Assumptions: • 21 is given as 35 percent of some unknown number N. • Mathematically, 21 = 35 percent of N. • We need to find the value of N that satisfies this relation. Concept / Approach: If p percent of N equals a known value V, then (p / 100) * N = V. To find N, rearrange this to N = (V * 100) / p. Here, p = 35 and V = 21. Careful handling of the fraction and cancellation makes the calculation easy and exact. Step-by-Step Solution: Step 1: Write the percentage equation: (35 / 100) * N = 21. Step 2: Rearrange to solve for N: N = (21 * 100) / 35. Step 3: Simplify the fraction 100 / 35. Divide numerator and denominator by 5 to get 20 / 7. Step 4: Now N = 21 * (20 / 7). Step 5: Simplify 21 / 7 = 3, so N = 3 * 20 = 60. Verification / Alternative check: Check the answer by computing 35 percent of 60: (35 / 100) * 60 = 0.35 * 60 = 21. Since this matches the given value, the solution is verified as correct. Why Other Options Are Wrong: • 85 would give 35 percent of 85 as 29.75, not 21. • 300 would give 105 as 35 percent, which is far too large. • 21 would imply 35 percent of 21 equals 21, which is impossible since any proper percentage less than 100 is smaller than the original number. Common Pitfalls: A common error is to multiply 21 by 0.35 instead of dividing by 0.35. Remember that 21 is the percentage result, not the original base. Always set up the equation carefully and solve for the unknown base when reversing a percentage calculation. Final Answer: The original number is 60.

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