Question 1

Compute the percent value: One-third of 1206 is what percent of 134?

Accepted Answer

Introduction / Context:This question checks quick percent conversion by ratio. We first evaluate one-third of a given number and then express the result as a percentage of another number.Given Data / Assumptions:Quantity A = one-third of 1206.Reference quantity B = 134.We need (A/B) * 100%.Concept / Approach:Compute A = 1206/3. Then divide A by 134 and convert the resulting ratio to a percentage. If A equals exactly 3 times B, the percentage is 300%.Step-by-Step Solution:A = 1206 / 3 = 402.Compute A/B = 402 / 134.Note 134 * 3 = 402 exactly.Therefore (A/B) * 100% = 3 * 100% = 300%.Verification / Alternative check:Since 402 is three times 134, the percent of 134 that is 402 is directly 300%, confirming the computation without a calculator.Why Other Options Are Wrong:3% and 30%: Off by factors of 100 or 10; they ignore the exact 3x relationship.None of these: Not needed because 300% is a listed option and correct.Common Pitfalls:Missing the exact multiple and doing unnecessary decimal division, which can lead to rounding mistakes.Final Answer:300%

Question 2

If 96% of the population of a village equals 23,040, find the total population of the village.

Accepted Answer

Introduction / Context:Finding the whole from a given percentage is a standard percentage application: if a known part corresponds to a percentage, divide by the decimal form of that percentage to get the total.Given Data / Assumptions:96% of total population = 23,040.We want 100% (the total).Concept / Approach:If 0.96 * Total = 23,040, then Total = 23,040 / 0.96. Another way: since 96% = 24/25, divide by 24 and multiply by 25 to avoid decimals.Step-by-Step Solution:Total = 23040 / 0.96.Using fraction: 96% = 24/25, so 23040 = (24/25)*Total → Total = 23040*(25/24).First compute 23040/24 = 960.Then Total = 960 * 25 = 24,000.Verification / Alternative check:Check: 96% of 24,000 = 0.96 * 24,000 = 23,040. Correct.Why Other Options Are Wrong:32,256; 24,936; 25,640: These do not satisfy 96% = 23,040 when multiplied back.Common Pitfalls:Dividing by 96 instead of 0.96, or mixing up the operations. Convert to fraction 24/25 to simplify and avoid decimal errors.Final Answer:24000

Question 3

After a 5% commission is deducted, a TV costs Rs. 9,595 to the buyer. What is its gross (before-commission) price?

Accepted Answer

Introduction / Context:The net price after deducting a commission is given. Since commission is 5%, the net equals 95% of the gross. We invert this to find the gross price (100%).Given Data / Assumptions:Net price = Rs. 9,595.Commission rate = 5%.Net = 95% of Gross.Concept / Approach:Let G be the gross price. Then 0.95G = 9,595 → G = 9,595 / 0.95. Use ratio 95/100 = 19/20 to convert division into simple multiplication.Step-by-Step Solution:0.95G = 9595 ⇒ G = 9595 / 0.95.Compute: 9595 / 0.95 = 9595 * (100/95) = 9595 * (20/19).First 9595/19 = 505. Then G = 505 * 20 = 10,100.Verification / Alternative check:Check: 5% of 10,100 is 505; gross − commission = 10,100 − 505 = 9,595, matching the given net price.Why Other Options Are Wrong:Rs. 10,000 or Rs. 10,074.75: Do not yield 9,595 after a 5% deduction.None of these: Not needed since Rs. 10,100 fits perfectly.Common Pitfalls:Subtracting 5% from 9,595 instead of reversing the percentage, or dividing by 5 instead of 0.95.Final Answer:Rs. 10100

Question 4

A person spends Rs. 3,500 per month and saves 12 1/2% of income. What is the monthly income?

Accepted Answer

Introduction / Context:This is a reverse calculation of income from given expenditure and savings rate. If savings are a fixed percentage of income, expenditure is the remainder. We use the complement of the saving rate to compute the income.Given Data / Assumptions:Monthly spend = Rs. 3,500.Savings rate = 12 1/2% = 12.5% = 1/8 of income.Income = Spend + Save.Concept / Approach:If savings are 1/8 of income, then expenditure is 7/8 of income. So 7I/8 = 3,500 → I = 3,500 * 8 / 7. This avoids trial-and-error and yields an exact figure.Step-by-Step Solution:Savings = (1/8)I, so expenditure = I − (1/8)I = (7/8)I.Given (7/8)I = 3500 ⇒ I = 3500 * 8 / 7.I = 3500 * 8 / 7 = 500 * 8 = Rs. 4,000.Verification / Alternative check:Savings at 12.5% of 4,000 = 500; expenditure = 4,000 − 500 = 3,500, matching the given data.Why Other Options Are Wrong:Rs. 4,400; Rs. 4,270; Rs. 3,937.50: These do not produce expenditure exactly Rs. 3,500 with a 12.5% saving rate.Common Pitfalls:Applying 12.5% on expenditure instead of income or mis-converting 12 1/2% to fraction.Final Answer:Rs. 4000

Question 5

A fruit seller sells 40% of his apples and still has 420 apples left. How many apples did he originally have?

Accepted Answer

Introduction / Context:This inventory question uses percentages of a whole. Selling 40% leaves 60% in stock. Knowing the remainder, we back-calculate the original stock using division by the remaining percentage.Given Data / Assumptions:Sold = 40% of original.Remaining = 420 apples = 60% of original.Concept / Approach:If R = remaining and O = original, then R = 0.60O ⇒ O = R / 0.60. Use fraction 3/5 for faster mental math: O = 420 ÷ (3/5) = 420 * (5/3).Step-by-Step Solution:Remaining fraction = 100% − 40% = 60% = 3/5.Original O = 420 ÷ (3/5) = 420 * (5/3).Compute 420/3 = 140; 140 * 5 = 700.Verification / Alternative check:Check: 40% of 700 = 280 sold; remaining = 700 − 280 = 420, which matches.Why Other Options Are Wrong:588, 600, 672: Do not yield 420 as 60% when tested.Common Pitfalls:Using 40% instead of 60% for the remaining, or incorrectly converting percentages to fractions.Final Answer:700 apples

Question 6

An ore contains 12% copper. How many kilograms of ore are needed to extract 69 kg of pure copper?

Accepted Answer

Introduction / Context:Percentage composition problems convert a desired pure quantity into the amount of mixture required. If ore has 12% copper, 12 parts out of 100 are copper. We scale up to obtain 69 kg of copper.Given Data / Assumptions:Ore copper content = 12% by mass.Required copper (pure) = 69 kg.Concept / Approach:Let the ore mass be M kg. Then copper from ore = 0.12M. Set 0.12M = 69 → M = 69 / 0.12. Working with fractions (12% = 12/100 = 3/25) speeds up the arithmetic: 69 ÷ (3/25) = 69 * (25/3).Step-by-Step Solution:M = 69 / 0.12.Write 0.12 as 3/25: M = 69 * (25/3).69/3 = 23; 23 * 25 = 575.Verification / Alternative check:Check: 12% of 575 = 0.12 * 575 = 69 exactly.Why Other Options Are Wrong:424, 828, 1736 2/3: These do not give 69 kg of copper at 12% concentration when tested.Common Pitfalls:Multiplying by 0.12 instead of dividing by it when recovering the whole from a part, or misreading 12% as 1/12.Final Answer:575 kg

Question 7

In an examination, 65% of the candidates passed. If the number of failures was 420, find the total number of examinees.

Accepted Answer

Introduction / Context:Pass percentage and number of failures combine to determine the total candidates. If 65% passed, then 35% failed. Given the count of failures, we scale up to the total population.Given Data / Assumptions:Pass rate = 65% → Fail rate = 35%.Failures = 420 candidates.Concept / Approach:If Total = T, then 0.35T = 420 ⇒ T = 420 / 0.35. Using fraction 35% = 7/20 makes mental math straightforward: T = 420 ÷ (7/20) = 420 * (20/7).Step-by-Step Solution:T = 420 / 0.35.0.35 = 7/20 ⇒ T = 420 * (20/7).420/7 = 60; 60 * 20 = 1200.Verification / Alternative check:Compute 65% of 1200 = 780 pass; 35% of 1200 = 420 fail, matching the given failure count.Why Other Options Are Wrong:567, 693, 1000: These totals do not yield exactly 420 failures at 35%.Common Pitfalls:Confusing pass and fail percentages, or dividing by 65 rather than 0.65/0.35. The failures correspond to 35% of the total.Final Answer:1200

Question 8

Seventy-five percent of a number, when added to 75, equals the number. Find the number.

Accepted Answer

Introduction / Context:This is a simple linear equation with a percentage term. Translating the text into an equation and solving for the unknown yields the number directly.Given Data / Assumptions:Let the number be x.Condition: 0.75x + 75 = x.Concept / Approach:Rearrange to isolate x: move the 0.75x term to the other side to get 25% of x equals 75, then solve by division. Recognize that 25% is one-quarter to speed up computation.Step-by-Step Solution:0.75x + 75 = x.x − 0.75x = 75 ⇒ 0.25x = 75.x = 75 / 0.25 = 75 * 4 = 300.Verification / Alternative check:75% of 300 = 225; 225 + 75 = 300, satisfying the condition.Why Other Options Are Wrong:150, 200, 225: Substitution shows the equality does not hold.Common Pitfalls:Adding 75 to 100% of x instead of 75% of x, or interpreting 75% incorrectly as 0.75 of 75.Final Answer:300

Question 9

What is 80% of a number if 200% of the same number equals 90?

Accepted Answer

Introduction / Context:This question tests proportional scaling among percentages of the same base number. Knowing one percentage value allows quick determination of another percentage by ratio.Given Data / Assumptions:Let the number be N.200% of N = 90.We want 80% of N.Concept / Approach:From 200% of N = 2N = 90, deduce N = 45. Then compute 80% of N = 0.8 * 45. Alternatively, scale directly: 80% is 0.8/2.0 = 40% of the given 200% value → 0.4 * 90 = 36.Step-by-Step Solution:2N = 90 ⇒ N = 45.80% of N = 0.8 * 45 = 36.Verification / Alternative check:Direct scaling: (80% / 200%) * 90 = (0.8 / 2.0) * 90 = 0.4 * 90 = 36. Matches.Why Other Options Are Wrong:72 and 144: These correspond to doubling or quadrupling errors; not consistent with N = 45.None of these: Not needed because 36 is correct and listed.Common Pitfalls:Confusing 200% with 20% or dividing 90 by 0.8 instead of by 2 first.Final Answer:36

Question 10

The price of rice rises by 60%. By what exact percentage should the new price be reduced to restore the original price?

Accepted Answer

Introduction / Context:This is a reverse change problem. After an increase of 60%, the compensating decrease to return to the original price is found relative to the increased base, not the original base, so it is less than 60% but not equal to 60% either.Given Data / Assumptions:Initial price = P.Increased price = P * 1.60.We need reduction r such that (1 − r) * 1.60P = P.Concept / Approach:Set up the equation and solve for r: (1 − r) = 1 / 1.60 = 0.625. Therefore r = 1 − 0.625 = 0.375 = 37.5%. This is the general reverse-percentage formula: required reduction r = (increase%) / (100% + increase%) * 100%.Step-by-Step Solution:New price after rise = 1.60P.(1 − r) * 1.60P = P ⇒ (1 − r) = 1/1.60 = 0.625.r = 1 − 0.625 = 0.375 = 37.5%.Verification / Alternative check:Let P = 100. After a 60% rise → 160. Reduce by 37.5%: 160 * (1 − 0.375) = 160 * 0.625 = 100, restored exactly.Why Other Options Are Wrong:33 1/3% and 40% and 45%: Do not satisfy (1 − r)*1.6 = 1 when tested.Common Pitfalls:Assuming the same 60% decrease would undo the increase; reverse changes are based on the new base.Final Answer:37.5%

Question 11

From 6 litres of a 5% salt solution, 1 litre of water evaporates. What is the new percentage of salt in the remaining solution?

Accepted Answer

Introduction / Context:This concentration problem involves evaporation of solvent (water) only. Since salt does not evaporate, the mass of solute remains constant while the volume of solution decreases, raising the concentration. We compute the new percentage after evaporation.Given Data / Assumptions:Initial volume = 6 L, salt concentration = 5%.Water loss due to evaporation = 1 L (solute unchanged).Final volume = 5 L.Concept / Approach:Amount of salt initially = 5% of 6 L = 0.05 * 6 = 0.30 L (in equivalent-volume terms). After evaporation, total solution volume is 5 L with the same 0.30 L salt. New percentage = (amount of salt / new volume) * 100%.Step-by-Step Solution:Initial salt = 0.05 * 6 = 0.30 L (unchanged).Final volume = 6 − 1 = 5 L.New % salt = (0.30 / 5) * 100% = 6%.Verification / Alternative check:If 5% of 6 L is 0.30 L, then in 5 L the same 0.30 L represents 6%, as 6% of 5 L equals 0.30 L. The logic is fully consistent.Why Other Options Are Wrong:5%: Would be true without evaporation; concentration must rise.44/19% or 55/7%: These are inconsistent fractions for this setup, not matching 0.30/5.Common Pitfalls:Subtracting salt along with water or recalculating salt amount incorrectly. Only water evaporates, so the solute quantity remains fixed.Final Answer:6%

Question 12

Evaluate: √(3.6% of 40). Choose the exact numerical value.

Accepted Answer

Introduction / Context:This evaluates a nested operation: first a percentage of a number, then a square root. Compute in the correct order to avoid mistakes, turning the percentage into a decimal multiplier and then taking the square root exactly.Given Data / Assumptions:Percent = 3.6% = 0.036.Number = 40.Concept / Approach:Compute 3.6% of 40 as 0.036 * 40. Then take √ of the result. Recognize perfect decimal squares to simplify mental calculation (1.44 has square root 1.2).Step-by-Step Solution:3.6% of 40 = 0.036 * 40 = 1.44.√(1.44) = 1.2.Verification / Alternative check:Note 12^2 = 144, so (1.2)^2 = 1.44. Thus the square root of 1.44 is precisely 1.2; no rounding necessary.Why Other Options Are Wrong:2.8 and 1.8: Do not square to 1.44; they are far from the correct square root.None of these: Not needed since 1.2 is exact and listed.Common Pitfalls:Treating 3.6% as 0.36 or 3.6 (misplacing the decimal), or taking the square root before computing the percentage.Final Answer:1.2

Question 13

A student must score at least 50% to pass. He scored 50 marks, which is short of the passing marks by 50. What are the maximum marks of the paper?

Accepted Answer

Introduction / Context:Translating passing criteria into an equation with maximum marks allows solving for the total marks. The statement indicates the student’s actual score is 50 below the passing threshold, which itself is 50% of the maximum marks.Given Data / Assumptions:Pass percentage = 50%.Actual score = 50 marks.Short by 50 marks → Passing marks = 50 + 50 = 100.Concept / Approach:If passing marks are 50% of the maximum, then 0.50 * Max = 100. Solve for Max. This directly yields the total marks of the paper.Step-by-Step Solution:Passing marks = 50 (actual) + 50 (shortfall) = 100.Let M be maximum marks: 0.50 * M = 100.M = 100 / 0.50 = 200.Verification / Alternative check:With maximum 200, passing marks are 100. The student scored 50, which is indeed 50 below passing, matching the prompt.Why Other Options Are Wrong:250, 275, 300: At 50% pass, these would imply passing marks of 125, 137.5, or 150, contradicting the stated shortfall.Common Pitfalls:Misreading “short by 50” as “scored 50%,” or incorrectly combining the data. The phrasing means the pass mark is exactly 100.Final Answer:200

Question 14

X is 30% less than Z and Y is 35% less than Z. By what percentage is Y less than X?

Accepted Answer

Introduction / Context:Comparative percentage questions require expressing both numbers relative to the same base and then comparing. Here both X and Y are defined as percentage reductions from Z; we must find the percent by which Y is below X.Given Data / Assumptions:X = 30% less than Z ⇒ X = 0.70Z.Y = 35% less than Z ⇒ Y = 0.65Z.Compute % difference relative to X: (X − Y)/X * 100%.Concept / Approach:Use the common base Z to write both values, then take the ratio. Plug the expressions and simplify the fraction to get an exact percentage. Recognize 0.05/0.70 = 1/14 for a clean fractional percentage 7 1/7%.Step-by-Step Solution:X − Y = 0.70Z − 0.65Z = 0.05Z.(X − Y)/X = (0.05Z)/(0.70Z) = 0.05/0.70 = 1/14.Percentage = (1/14) * 100% ≈ 7.142857% = 7 1/7%.Verification / Alternative check:Assume Z = 100. Then X = 70 and Y = 65. Difference = 5. Percent below X = 5/70 * 100% = 7 1/7%, confirming.Why Other Options Are Wrong:30%, 20%, 15%: These are common distractors when comparing to Z instead of to X, or mixing percentage points with percent differences.Common Pitfalls:Comparing Y to Z instead of to X, or computing (Y − X)/X which would produce a negative sign and confusion.Final Answer:7 1/7%

Question 15

A seller sells two tables at ₹1500 each, making 20% profit on one and 20% loss on the other. What is the net result on the overall deal?

Accepted Answer

Introduction / Context:Selling two items at the same selling price with equal profit and loss percentages leads to an overall loss, not break-even. This is a classic result due to different cost prices. We compute total cost and compare with total selling price.Given Data / Assumptions:SP of each table = ₹1500.One at 20% profit; one at 20% loss.Concept / Approach:Compute the individual cost prices (CP) from SP and profit/loss percentage. Then sum the CPs to get total cost. Sum SPs to get total SP. Compare to find overall profit or loss percentage relative to total CP.Step-by-Step Solution:For 20% profit: SP = 1.20 * CP ⇒ CP1 = 1500 / 1.20 = 1250.For 20% loss: SP = 0.80 * CP ⇒ CP2 = 1500 / 0.80 = 1875.Total CP = 1250 + 1875 = 3125.Total SP = 1500 + 1500 = 3000.Loss = 3125 − 3000 = 125.Loss % = 125 / 3125 * 100% = 4%.Verification / Alternative check:Equal and opposite percentages around the same SP do not cancel because the CPs are unequal; the product (1.20)*(0.80) = 0.96 shows a 4% net loss effect.Why Other Options Are Wrong:4% profit, 10% loss/profit: Contrary to the computation; the exact effect is 4% loss.Common Pitfalls:Assuming profit and loss cancel to zero, or averaging the percentages arithmetically without considering different bases.Final Answer:4% loss

Question 16

Compute a compound percentage: What is 5% of 50% of 500?

Accepted Answer

Introduction / Context:Nesting percentages is equivalent to multiplying the base by the product of the corresponding decimal fractions. This question checks order and fraction conversion accuracy.Given Data / Assumptions:Base number = 500.Percentages: first 50%, then 5% of that result.Concept / Approach:Compute 50% of 500 first, then take 5% of that answer. Alternatively, multiply directly by 0.50 * 0.05 = 0.025 to get the same result in one step.Step-by-Step Solution:50% of 500 = 0.50 * 500 = 250.5% of 250 = 0.05 * 250 = 12.5.Verification / Alternative check:Direct: 500 * (0.50 * 0.05) = 500 * 0.025 = 12.5. Both procedures agree.Why Other Options Are Wrong:25 and 6.25: These correspond to doubling or halving errors in one of the steps.1.25: Off by a factor of 10; likely from using 0.005 instead of 0.05.Common Pitfalls:Mixing up the order or misplacing the decimal point when converting percentages to decimals.Final Answer:12.5

Question 17

If 20% of N equals 90, find the value of N.

Accepted Answer

Introduction / Context:Recovering a whole from a known part and its percentage is a staple calculation. Converting 20% to decimal 0.20 allows direct division to obtain N.Given Data / Assumptions:0.20 * N = 90.Concept / Approach:Divide the known part by the decimal percentage: N = 90 / 0.20. Alternatively, since 20% is one-fifth, multiply by 5 to get N directly without decimals.Step-by-Step Solution:0.20N = 90 ⇒ N = 90 / 0.20.Compute: 90 / 0.20 = 90 * 5 = 450.Verification / Alternative check:20% of 450 = 0.20 * 450 = 90, confirming correctness.Why Other Options Are Wrong:350, 250, 550: Do not satisfy 20% equals 90 when tested.Common Pitfalls:Dividing 90 by 20 (getting 4.5) rather than by 0.20, or misreading 20% as 0.002.Final Answer:450

Question 18

Solve for the unknown: 89% of a number plus 365 equals 1075.22. What is the number?

Accepted Answer

Introduction / Context:This linear percentage equation involves isolating the unknown number from a mixed expression containing a percent of the number plus a constant. Converting to decimal and solving step-by-step yields the exact result.Given Data / Assumptions:Equation: 0.89x + 365 = 1075.22.x represents the unknown number.Concept / Approach:Subtract the constant to isolate the percentage term, then divide by the decimal percentage (0.89) to obtain x. This is a direct algebraic manipulation.Step-by-Step Solution:0.89x + 365 = 1075.22.0.89x = 1075.22 − 365 = 710.22.x = 710.22 / 0.89.Compute: 0.89 * 800 = 712; reduce by 0.89 * 2 = 1.78 to reach 710.22 → x = 800 − 2 = 798.Verification / Alternative check:Plug back: 0.89 * 798 = 710.22; add 365 gives 1075.22, matching exactly.Why Other Options Are Wrong:897, 898, 752: Substitution fails to reach 1075.22 when processed through 0.89x + 365.Common Pitfalls:Dividing by 89 instead of 0.89, or rounding too early and losing accuracy. Keep at least two decimals until the final check.Final Answer:798

Question 19

Percentage equivalence: If 15% of A equals 20% of B, then 25% of A is equal to what percent of B?

Accepted Answer

Introduction / Context:This problem tests equivalence transformations with percentages and proportional reasoning. When two different percentages of two quantities are equal, we can form a ratio between the quantities and then translate another percentage accordingly.Given Data / Assumptions:0.15 * A = 0.20 * B.We need 25% of A expressed as a percent of B.All quantities are positive real numbers; percentage arithmetic uses base 100.Concept / Approach:From 0.15A = 0.20B, divide both sides by 0.15B to get A/B = 0.20/0.15. Once A/B is known, compute (25% of A) as a fraction of B and convert to a percentage. Scaling by a constant preserves ratios.Step-by-Step Solution:0.15A = 0.20B ⇒ A/B = 0.20/0.15 = 20/15 = 4/325% of A = 0.25A = (0.25 * A/B) * B0.25 * (A/B) = 0.25 * (4/3) = 1/3Therefore, 25% of A = (1/3) * B = 33 1/3% of BVerification / Alternative check:Pick convenient numbers: let B = 3, then A = 4 (from A/B = 4/3). 25% of A = 1. That is 1/3 of B = 33 1/3%, confirming the result.Why Other Options Are Wrong:30% and 35% arise from arithmetic slips; 25% assumes A = B; 40% overstates the ratio beyond 4/3 scaling by 0.25.Common Pitfalls:Mixing “percent of A” with “percent of B”. Always convert to a common base (here, B) before stating the final percentage.Final Answer:33 1/3%

Question 20

A number increased by 137 1/2% has an increment of 33. What is the original number?

Accepted Answer

Introduction / Context:This question assesses understanding of percentage increase where the increment (absolute change) is specified. Converting a mixed percentage to a decimal allows direct computation of the base value.Given Data / Assumptions:Increase rate = 137 1/2% = 137.5% = 1.375 as a decimal.Increment amount (final − original) = 33.Concept / Approach:If a quantity N is increased by r%, the increment is (r/100) * N. Here, 33 corresponds to 137.5% of N. Solve N = 33 / 1.375. Mixed percentages should be converted carefully to decimals or fractions before solving.Step-by-Step Solution:r = 137.5% = 1.375Increment = r * N ⇒ 33 = 1.375 * NN = 33 / 1.375 = 24Verification / Alternative check:Check: 137.5% of 24 = 24 * 1.375 = 33. The increment matches the stated value, confirming N = 24.Why Other Options Are Wrong:25 and 27 are close but yield increments 34.375 and 37.125; 22 gives 30.25; 28 gives 38.5, none equal to 33.Common Pitfalls:Confusing the increment with the final value, or interpreting 137 1/2% as 1.37 instead of 1.375. Always convert the mixed percent correctly.Final Answer:24

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