Question 1

What percentage of numbers from 1 to 70 have 1 or 9 in the unit's digit?

Accepted Answer

Given DataRange of integers: 1 to 70 (inclusive)We need the percentage of numbers whose unit digit is 1 or 9Key IdeaIn every block of 10 consecutive integers (e.g., 1–10, 11–20, …), the unit digits cycle through 0–9 exactly once. Therefore, in each block of 10 numbers, exactly two numbers end with 1 and 9.Step-by-StepTotal blocks of 10 between 1 and 70 = 7 blocks (1–10, 11–20, …, 61–70)Count with unit digit 1 or 9 per block = 2Total such numbers = 7 × 2 = 14Total numbers overall = 70Percentage = (14 ÷ 70) × 100 = 20%Common PitfallsForgetting that 70 is included; the last block is 61–70 and still contains 61 and 69.Counting per 10 but then not scaling the count across all 7 blocks.Final AnswerRequired percentage = 20%.

Question 2

In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was:

Accepted Answer

Given DataTotal votes = 7500Invalid votes = 20%One candidate gets 55% of valid votes; we need the other candidate's valid votesStep 1: Compute valid votesValid votes = 80% of 7500 = 0.80 × 7500 = 6000Step 2: Split between two candidatesCandidate A = 55% of 6000 = 0.55 × 6000 = 3300Candidate B = 6000 − 3300 = 2700Check & NotesPercentages apply only to valid votes; remove invalid votes first.Alternatively, Candidate B = 45% of 6000 = 2700 gives the same result.Final AnswerValid votes polled by the other candidate = 2700.

Question 3

Two numbers A and B are such that the sum of 5% of A and 4% of B is two-third of the sum of 6% of A and 8% of B. Find the ratio of A : B.

Accepted Answer

Given Relation0.05A + 0.04B equals two-thirds of (0.06A + 0.08B)Write the Equation0.05A + 0.04B = (2/3) × (0.06A + 0.08B)0.05A + 0.04B = 0.04A + 0.053333…BCollect Like Terms(0.05 − 0.04)A = (0.053333… − 0.04)B0.01A = 0.013333…BA/B = 0.013333… ÷ 0.01 = 1.3333… = 4/3Alternative Fractional MethodMultiply all terms by 300 to clear decimals:15A + 12B = (2/3) × (18A + 24B) ⇒ 15A + 12B = 12A + 16B3A = 4B ⇒ A:B = 4:3Final AnswerA:B = 4:3.

Question 4

Gauri went to the stationers and bought things worth Rs. 25, out of which 30 paise went on sales tax on taxable purchases. If the tax rate was 6%, then what was the cost of the tax free items?

Accepted Answer

Given DataTotal bill paid = Rs. 25Sales tax paid on taxable items = Rs. 0.30Tax rate = 6% on taxable purchasesFind: cost of tax-free itemsStep 1: Recover pre-tax amount for the taxable portionTax = Rate × (Taxable value)0.30 = 0.06 × (Taxable value)Taxable value (pre-tax) = 0.30 ÷ 0.06 = Rs. 5.00Step 2: Separate the components from the totalAmount paid towards taxable items (incl. tax) = 5.00 + 0.30 = 5.30Amount left for tax-free items = Total − 5.30 = 25.00 − 5.30 = Rs. 19.70Checks & NotesTax applies only to the taxable subset; tax-free items do not attract the 6% charge.Always back out the pre-tax value from the tax amount using Tax = Rate × Value.Final AnswerCost of tax-free items = Rs. 19.70.

Question 5

A student multiplied a number by 3⁄5 instead of 5⁄3. What is the percentage error in the calculation?

Accepted Answer

GivenIncorrect multiplier used = 3/5Correct multiplier should be = 5/3Step 1: Compare wrong result to correct resultRatio (Wrong ÷ Correct) = (3/5) ÷ (5/3) = (3/5) × (3/5) = 9/25 = 0.36Step 2: Compute percentage errorError fraction = 1 − 0.36 = 0.64Percentage error = 0.64 × 100 = 64% (less than the correct value)InterpretationThe computed answer is 36% of the correct one; hence it is short by 64%.Final AnswerPercentage error = 64% less.

Question 6

A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total score did he make by running between the wickets?

Accepted Answer

Given Score BreakdownTotal runs = 110Boundaries (fours) = 3 ⇒ runs from fours = 3 × 4 = 12Sixes = 8 ⇒ runs from sixes = 8 × 6 = 48Step 1: Runs made without boundaries/sixesRuns by running between wickets = Total − (Fours + Sixes)= 110 − (12 + 48) = 110 − 60 = 50Step 2: Percentage of totalPercentage = (50 ÷ 110) × 100 = (5 ÷ 11) × 100 ≈ 45.45%Notes(5 ÷ 11) × 100% is a recurring decimal 45.4545…; report to two decimals.Final AnswerHe scored 45.45% of his runs by running between the wickets.

Question 7

Two tailors X and Y are paid a total of Rs. 550 per week by their employer. If X is paid 120 percent of the sum paid to Y, how much is Y paid per week?

Accepted Answer

GivenTotal weekly pay to X and Y = Rs. 550X is paid 120% of what Y is paidSet UpLet Y's pay = yThen X's pay = 1.2yTotal = y + 1.2y = 2.2y = 550Solvey = 550 ÷ 2.2 = Rs. 250CheckX = 1.2 × 250 = 300; X + Y = 300 + 250 = 550 ✓Final AnswerY is paid Rs. 250 per week.

Question 8

Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are:

Accepted Answer

GivenTwo students' marks differ by 9The higher mark is 56% of the sum of their marksLet the marks beHigher = x, Lower = y, with x = y + 9Also given: x = 56% of (x + y) ⇒ x = 0.56(x + y)Solve the Systemx = 0.56x + 0.56y ⇒ 0.44x = 0.56yx = (0.56/0.44) y = (56/44) y = (14/11) yBut x = y + 9 ⇒ (14/11) y = y + 9(14/11 − 1) y = 9 ⇒ (3/11) y = 9 ⇒ y = 33x = y + 9 = 42CheckSum = 42 + 33 = 75; 56% of 75 = 42 ✓Final AnswerMarks obtained = 42 and 33.

Question 9

Rajeev buys good worth Rs. 6650. He gets a rebate of 6% on it. After getting the rebate, he pays sales tax @ 10%. Find the amount he will have to pay for the goods.

Accepted Answer

GivenMarked value of goods = Rs. 6650Rebate (discount) = 6%After rebate, sales tax = 10% on the discounted amountStep 1: Apply rebateDiscounted price = 6650 × (1 − 0.06) = 6650 × 0.94 = Rs. 6251.00Step 2: Add sales tax on discounted priceTax = 10% of 6251.00 = 0.10 × 6251.00 = Rs. 625.10Final payable = 6251.00 + 625.10 = Rs. 6876.10NotesTax is computed after rebate (on the reduced base).Order matters: apply discount first, then tax.Final AnswerAmount to be paid = Rs. 6876.10.

Question 10

If 20% of a = b, then b% of 20 is the same as:

Accepted Answer

Given Relation20% of a equals b ⇒ b = 0.20aWe need the value of: b% of 20Compute b% of 20b% of 20 = (b/100) × 20 = 20b/100 = b/5Since b = 0.20a ⇒ b/5 = (0.20a)/5 = 0.04a = a/25Equivalent Phrasinga/25 is the same as 4% of a.Final Answerb% of 20 = a/25.

Question 11

A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, he had:

Accepted Answer

Given dataHe sells 40% of the apples.Apples left after selling = 420 (this is the remaining 60%).Step 1: Relate remainder to the wholeIf 40% were sold, then 60% remained.60% of original = 420Step 2: Solve for the original count0.60 × Original = 420Original = 420 ÷ 0.60 = 700Common pitfallsDo not subtract 40 from 420; 40% is a percentage, not a count.Recognize that 420 represents the remaining 60%.Final AnswerOriginally, he had 700 apples.

Question 12

In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is ⅔ of the number of students of 8 years of age which is 48. What is the total number of students in the school?

Accepted Answer

Given data20% are below 8 years.Exactly 8 years old = 48 students.Above 8 years = ⅔ of the number of 8-year-olds = ⅔ × 48 = 32.Step 1: Express the totalLet total students = T.Breakdown: Below 8 = 20% of T = 0.2T; Exactly 8 = 48; Above 8 = 32.Step 2: Form the equationT = 0.2T + 48 + 32T − 0.2T = 800.8T = 80 ⇒ T = 80 ÷ 0.8 = 100Checks & notesPercent applies to the total T, not just to the non-8-year-olds.Above-8 count is directly computed from the 8-year-old group (2/3 of 48).Final AnswerTotal number of students = 100.

Question 13

If A = x% of y and B = y% of x, then which of the following is true?

Accepted Answer

DefinitionsA = x% of y = 𝑥 100 × 𝑦 100 x ​ ×yB = y% of x = 𝑦 100 × 𝑥 100 y ​ ×xStep 1: Compare A and BA = 𝑥 100 × 𝑦 100 x ​ ×y = 𝑥 𝑦 100 100 xy ​ B = 𝑦 100 × 𝑥 100 y ​ ×x = 𝑥 𝑦 100 100 xy ​ ConclusionBoth expressions are identical for all real x and y.Final AnswerA = B.

Question 14

Three candidates contested an election and received 1136, 7636 and 11628 votes respectively. What percentage of the total votes did the winning candidate get?

Accepted Answer

Given votesCandidate 1 = 1136Candidate 2 = 7636Candidate 3 = 11628 (winning candidate)Step 1: Total votesTotal = 1136 + 7636 + 11628 = 20400Step 2: Percentage for winnerPercentage = 11628 20400 × 100 20400 11628 ​ ×100= 969 1700 × 100 1700 969 ​ ×100 ≈ 57.0588…%RoundingRounded to the nearest percent ≈ 57%.Final AnswerWinning candidate got 57% of the total votes.

Question 15

The population of a town increased from 1,75,000 to 2,62,500 in a decade. The average percent increase of population per year is:

Accepted Answer

GivenInitial population P0 = 1,75,000Population after 10 years P10 = 2,62,500ConceptThe average annual percent increase is the CAGR: ( 𝑃 10 / 𝑃 0 ) 1 / 10 − 1 (P 10 ​ /P 0 ​ ) 1/10 −1.Step 1: Compute the growth factorGrowth factor = 2 , 62 , 500 1 , 75 , 000 = 1.5 1,75,000 2,62,500 ​ =1.5Step 2: Compute CAGRCAGR = 1.5 1 / 10 − 1 1.5 1/10 −1Using ln/exponent: ln(1.5) ≈ 0.405465; divide by 10 ⇒ 0.0405465; exp ⇒ 1.04141CAGR ≈ 1.04141 − 1 = 0.04141 = 4.14% per yearFinal AnswerAverage percent increase ≈ 4.14% per year.

Question 16

Cost–profit switch: loss to profit swing method: If an item would yield ₹12 more when sold at 8% profit instead of 8% loss, what is the cost price of the furniture?

Accepted Answer

Introduction / Context: Changing from an 8% loss to an 8% profit is a total swing of 16 percentage points on the cost price. The rupee change in outcome (₹12) therefore equals 16% of the cost. This “swing method” is a fast technique for many profit–loss questions where two scenarios differ only by percentage on the same base.Given Data / Assumptions:Scenario A: loss = 8% of cost.Scenario B: profit = 8% of cost.Difference in outcomes between B and A = ₹12.Same item, same cost base.Concept / Approach: The change from −8% to +8% is a net change of 16% of cost. Therefore 0.16 * Cost = ₹12. Solve for the cost to get a neat integer. No selling price needs to be computed explicitly.Step-by-Step Solution:Let C be cost price.Net swing = 8% + 8% = 16% = 0.16 of C.Given swing equals ₹12 ⇒ 0.16C = 12.C = 12 / 0.16 = 12 * (100/16) = 1200/16 = ₹75.Verification / Alternative check: At C = 75: 8% loss ⇒ loss = ₹6 ⇒ SP = 69. 8% profit ⇒ profit = ₹6 ⇒ SP = 81. Difference in outcomes = 81 − 69 = ₹12, matching the given.Why Other Options Are Wrong: ₹72, ₹70, ₹80, ₹60 would yield swings of 11.52, 11.2, 12.8, and 9.6 respectively, not ₹12.Common Pitfalls: Applying the 16% swing to selling price rather than cost. The swing is always on the same base as the stated percentages, here the cost price.Final Answer: ₹ 75

Question 17

Dilution with water: from 80% acid to 50% acid (assume 100 cc initial solution): How much water must be added to an 80% solution of sulphuric acid to obtain a 50% solution?

Accepted Answer

Introduction / Context: This is a classic dilution (percentage) problem. Because the original question omits the initial volume, we adopt the standard recovery assumption of a 100 cc starting solution for clarity and comparability. The key idea is that the amount of pure acid remains constant while water is added, so the new concentration depends only on the changed total volume.Given Data / Assumptions:Initial volume assumed = 100 cc (Recovery-First standardization).Initial concentration = 80% acid.Target concentration = 50% acid.Water added = w cc (unknown).Concept / Approach: Amount of pure acid is conserved during dilution. Set up the conservation equation: initial acid amount = final acid amount. Then solve for the volume of water to be added. Express the final result in cubic centimeters (cc).Step-by-Step Solution:Initial acid = 80% of 100 cc = 0.8 * 100 = 80 cc.Let water added be w cc. Final volume = 100 + w cc.Target: final acid fraction = 50% ⇒ 80 / (100 + w) = 0.5.Cross-multiply: 80 = 0.5(100 + w) = 50 + 0.5w ⇒ 30 = 0.5w ⇒ w = 60 cc.Verification / Alternative check: Final mixture volume = 160 cc with 80 cc acid ⇒ 80/160 = 50%, as required.Why Other Options Are Wrong: 80, 75, 70, and 50 cc would produce final concentrations of 44.4%, 51.4%, 51.6%, and 61.5% respectively (on a 100 cc starting base), not the desired 50%.Common Pitfalls: Treating “percent” as additive rather than conserving the mass of acid, or forgetting to specify/assume an initial volume when the problem statement omits it.Final Answer: 60 cc

Question 18

Composition percentages in a ternary alloy: A metallic alloy contains 20 g copper, 20 g tin, and 40 g silver. What is the percentage of tin in the alloy (by mass)?

Accepted Answer

Introduction / Context: Percentage composition by mass is a staple concept in mixtures and alloys. You find the total mass of the mixture and then compute the share of a component as a percentage of this total. The arithmetic is simple but the concept is widely applicable in chemistry and aptitude tests.Given Data / Assumptions:Mass of copper = 20 g.Mass of tin = 20 g.Mass of silver = 40 g.All quantities are additive; no mass loss.Concept / Approach: Percentage of a component = (component mass / total mass) * 100. First, compute the total mass. Next, divide the tin mass by the total mass, and then multiply by 100 to convert to percent.Step-by-Step Solution:Total mass = 20 + 20 + 40 = 80 g.Tin mass = 20 g.Tin percentage = (20 / 80) * 100 = 25%.Verification / Alternative check: The alloy is composed in the ratio Cu:Sn:Ag = 1:1:2. The tin fraction is 1 part out of 4 total parts ⇒ 1/4 = 25%, confirming the arithmetic.Why Other Options Are Wrong: 20% or 30% result from miscounting parts; 40% confuses tin with silver; 10% would imply tin is the smallest fraction by far, which it is not.Common Pitfalls: Forgetting to sum all components to find the total, or mixing up which component’s mass is being asked for.Final Answer: 25%

Question 19

Successive price increase on petrol: The price of petrol is first increased by 20% and then again by 40%. If the original price was ₹25 per litre, what will be the final price per litre after both increases?

Accepted Answer

Introduction / Context:This percentage question checks understanding of successive percentage changes. When two increases are applied one after another, we must multiply the factors rather than add the percentages. The context is a fuel price update where the base price is known and two sequential hikes are announced. Given Data / Assumptions: Original petrol price = ₹25 per litre. First increase = 20%. Second increase = 40% (applied after the first increase). Concept / Approach:For an increase of r%, multiply the current value by (1 + r/100). Two increases r1% and r2% in succession yield a net factor (1 + r1/100) * (1 + r2/100). This multiplicative compounding is essential; adding 20% and 40% as 60% would be incorrect here. Step-by-Step Solution: First increase: New price after +20% = 25 * 1.20 = ₹30 Second increase: Apply +40% on ₹30 ⇒ 30 * 1.40 = ₹42 Final price = ₹42 per litre Verification / Alternative check:Multiply once using combined factor: 25 * (1.20 * 1.40) = 25 * 1.68 = 42. This confirms the stepwise computation and highlights the compounding nature of successive percentage changes. Why Other Options Are Wrong:₹45 and ₹48 come from misapplying or over-compounding the rates; ₹40 often appears when someone adds 20% and 40% as 60% and applies 25 * 1.60 = 40, which is wrong because the second increase is on the increased price, not the original price. ₹35 is unrelated to any correct approach. Common Pitfalls:Adding percentages directly (20 + 40) without compounding. Forgetting that the second percentage acts on the updated price leads to systematic errors in retail and finance problems. Final Answer:₹42

Question 20

Single percentage increase on price: The price of a product is increased by 20%. If the original price is ₹300, what is the final price after the increase?

Accepted Answer

Introduction / Context:A straightforward computation: increasing a known base price by a given percentage. This type of question is common in shopping, taxation (e.g., GST), and inflation computations, where the new amount must be calculated from the original value and a single rate of change. Given Data / Assumptions: Original price = ₹300. Increase = 20% of the original price. Concept / Approach:For a single increase of r%, multiply the original value by (1 + r/100). Here r = 20, so factor = 1.20. This compact method avoids separate steps of calculating the increment first and then adding it back, though both methods yield the same result. Step-by-Step Solution: Increase amount = 20% of 300 = 0.20 * 300 = ₹60 Final price = 300 + 60 = ₹360 Equivalently, 300 * 1.20 = ₹360 Verification / Alternative check:Reverse check: If final is 360, the increase over 300 is 60, and 60/300 * 100% = 20%. This confirms accuracy. Why Other Options Are Wrong:₹350, ₹365, and ₹370 correspond to incorrect increase amounts (16.7%, 21.7%, 23.3% respectively). ₹340 implies only a 13.3% increase, not 20%. Common Pitfalls:Mixing up the base for percentage increase or mistakenly applying further changes. Ensure the percentage is always applied to the stated original price unless clearly specified otherwise. Final Answer:₹360

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