Percentage

Foundation topic for profit/loss and SI/CI.

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Important Formulas

If A increases by x%, new value = A ? (1 + x/100)
[Core definition] Percentage means per hundred. If part = P and whole = W, percentage = (P/W) x 100, with W not equal to 0.
If A decreases by x%, new value = A ? (1 - x/100)
[Find the part] If x% of a quantity N is required, part = (x/100) x N.
% change = ((New - Old) / Old) ? 100
[Find the percentage] If a quantity A is compared with base B, A as a percentage of B = (A/B) x 100.
If A is x% more than B, then B is less than A by: (x / (100 + x)) ? 100 %
[Find the whole] If x% of a number is A, the number = (A x 100)/x.
If A is x% less than B, then B is more than A by: (x / (100 - x)) ? 100 %
[Percent-decimal conversion] x% = x/100. To convert a decimal to percent, multiply by 100.
Successive % change of a% then b%: Net = a + b + (ab/100)
[Fraction-percent conversion] For fraction a/b, percentage = (a/b) x 100. For p%, equivalent fraction = p/100 and then simplify.
If price increases by x%, consumption must decrease by (x/(100+x))?100 % to keep expenditure constant
[Interchange property] x% of y = y% of x. This is useful when one side is easier to calculate mentally.
If price decreases by x%, consumption can increase by (x/(100-x))?100 % to keep expenditure constant
[Percentage of a percentage] x% of y% of N = (xy/10000) x N.
Population after n years = P ? (1 + r/100)^n
[Ratio to percentage share] If A:B = m:n, A is 100m/(m+n)% of the total and B is 100n/(m+n)% of the total.
Population n years ago = P / (1 + r/100)^n
[One quantity as percentage of another] If A:B = m:n, then A is 100m/n% of B and B is 100n/m% of A.
If income is spent on A (a%), B (b%), savings = [100 - (a+b)] % of income
[Basic percentage increase] New value after p% increase = Original x (1 + p/100).
Percentage point difference ? percentage difference. If A=40% and B=30%, difference is 10 percentage points but 33.33% relative
[Basic percentage decrease] New value after p% decrease = Original x (1 - p/100).
[Reverse an increase] If final value F is obtained after p% increase, original = F x 100/(100+p).
[Reverse a decrease] If final value F is obtained after p% decrease, original = F x 100/(100-p).
[Percentage change] Percentage change = (New - Original)/Original x 100. The original value is always the base unless stated otherwise.
[Absolute decrease percentage] Percentage decrease = (Original - New)/Original x 100.
[Difference relative to smaller base] If A>B, A is (A-B)/B x 100% more than B.
[Difference relative to larger base] If A>B, B is (A-B)/A x 100% less than A.
[More-less conversion] If A is p% more than B, then B is 100p/(100+p)% less than A.
[Less-more conversion] If A is p% less than B, then B is 100p/(100-p)% more than A.
[Two successive changes] For signed percentage changes a% and b%, net change = a + b + ab/100 percent. Treat decreases as negative.
[Equal increase and decrease] An increase of x% followed by a decrease of x%, or vice versa, gives net decrease x^2/100%.
[Three successive changes] Net multiplier = (1+a/100)(1+b/100)(1+c/100). Net percentage = (multiplier-1) x 100.
[Repeated growth] After n periods of p% growth, final = initial x (1+p/100)^n.
[Repeated decay] After n periods of p% decline, final = initial x (1-p/100)^n.
[Reverse repeated growth] Initial = final/(1+p/100)^n.
[Reverse repeated decay] Initial = final/(1-p/100)^n.
[Compound annual growth rate] CAGR = [(Final/Initial)^(1/n)-1] x 100%.
[Compensating increase] After a p% decrease, the increase needed to restore the original = 100p/(100-p)%.
[Compensating decrease] After a p% increase, the decrease needed to restore the original = 100p/(100+p)%.
[Price-consumption constant expenditure] If price rises p%, consumption must fall 100p/(100+p)% to keep expenditure unchanged.
[Price-consumption constant expenditure after price fall] If price falls p%, consumption may rise 100p/(100-p)% without changing expenditure.
[Expenditure multiplier] If price changes by p% and quantity changes by q%, expenditure multiplier = (1+p/100)(1+q/100), using negative signs for falls.
[Revenue multiplier] Revenue = price x quantity. Therefore net revenue change under price and sales-volume changes uses the successive-change formula.
[Income-expenditure-savings identity] Savings = Income - Expenditure.
[Savings rate] Savings as percentage of income = Savings/Income x 100.
[Expenditure rate] Expenditure as percentage of income = Expenditure/Income x 100 = 100 - savings percentage.
[Changed savings] New savings = Old income x income multiplier - Old expenditure x expenditure multiplier.
[Recover old income from new income] If income becomes F after p% change, old income = F/(1+p/100), with signed p.
[Marks percentage] Percentage marks = Marks obtained/Maximum marks x 100.
[Marks from percentage] Marks obtained = Maximum marks x percentage/100.
[Pass-mark model] If pass percentage is p and maximum marks are M, pass marks = pM/100.
[Two-candidate pass equation] If one candidate is x marks below pass and another is y marks above pass, use their percentages to form two linear equations in total marks and pass marks.
[Election valid-vote model] Valid votes = Total votes x (100-invalid%)/100, after accounting for abstention when applicable.
[Election candidate votes] Candidate votes = Valid votes x candidate share/100.
[Election winning margin] Margin = Valid votes x difference in vote shares/100 for a two-candidate election.
[Population net percentage] When births, deaths, immigration and emigration are given on the same starting base, net signed percentage is their algebraic sum.
[Population sequential change] When demographic changes occur one after another on changed bases, multiply the successive factors instead of simply adding percentages.
[Mixture concentration] Solute quantity = Mixture quantity x concentration/100.
[Dilution by adding pure solvent] Solute stays constant. New concentration = old solute/new total mixture x 100.
[Concentration by evaporation] Solute stays constant while solvent decreases. New concentration = old solute/new total x 100.
[Adding pure solute] New concentration = (old solute + added solute)/(old total + added solute) x 100.
[Mixing two concentrations] Final concentration = (Q1C1 + Q2C2)/(Q1+Q2), where concentrations are in compatible percentage units.
[Repeated replacement] If fraction r of a mixture is removed and replaced n times, original component remaining = Initial component x (1-r)^n.
[Purity and impurity] Purity% + impurity% = 100%. Pure quantity = total quantity x purity/100.
[Profit percentage] Profit% = Profit/Cost price x 100.
[Loss percentage] Loss% = Loss/Cost price x 100.
[Selling price from profit/loss] SP = CP(1+profit%/100) or CP(1-loss%/100).
[Discount percentage] Discount% = (Marked price - Selling price)/Marked price x 100.
[Successive discounts] Net discount for discounts a% and b% = a+b-ab/100.
[Marked price-discount-profit relation] SP = MP x discount multiplier = CP x profit multiplier.
[Tax after discount] Tax is normally applied to the discounted selling price unless the question states another base.
[Percentage points] A change from 40% to 50% is an increase of 10 percentage points but a relative increase of 25%.
[Weighted percentage] Combined percentage = sum(group size x group percentage)/sum(group sizes).
[Aggregate result] Total successful cases = sum of successes in each group; divide by total cases, not by number of groups.
[Base effect] Equal absolute changes can produce different percentages because percentage change depends on the starting base.
[Percentage error] Percentage error = |Measured - True|/|True| x 100.
[Correction after wrong base] If a value was calculated as p% of wrong base W but should use true base T, corrected value = old value x T/W.
[Product change] If A changes by a% and B by b%, product AB changes by a+b+ab/100 percent, with signed changes.
[Quotient change exact] If numerator changes by a% and denominator by b%, new/old quotient = (1+a/100)/(1+b/100).
[Rectangle area change] If length changes by a% and breadth by b%, area changes by a+b+ab/100 percent.
[Square area change] If side changes by p%, area changes by 2p+p^2/100 percent.
[Circle area change] If radius changes by p%, area changes by 2p+p^2/100 percent.
[Cube volume change] If edge changes by p%, volume multiplier = (1+p/100)^3.
[Cuboid volume change] For signed changes a%, b%, c% in dimensions, volume multiplier = product of the three change factors.
[Cylinder volume change] Volume is proportional to r^2h, so multiplier = (radius factor)^2 x height factor.
[Fixed numerator fraction] If denominator increases p%, the fraction decreases by 100p/(100+p)%.
[Fixed denominator fraction] If numerator increases p%, the fraction increases by p%.
[Both numerator and denominator change] New fraction/old fraction = (1+a/100)/(1+b/100), where a and b are signed changes.
[Ratio after changes] If A:B=m:n and A, B change by a%, b%, new ratio = m(100+a):n(100+b).
[Percentage composition after subgroup changes] Compute new subgroup counts separately, then divide each by the new total.
[Set overlap] For two groups, A union B percentage = A% + B% - both%.
[Neither of two groups] Neither% = 100 - (A% + B% - both%).
[Three-set inclusion-exclusion] Union = A+B+C - AB-BC-CA + ABC, with all terms measured on the same base.
[Conditional percentage] Percentage of A among B = count(A and B)/count(B) x 100, not count(A)/total.
[Percentage contribution] A component's contribution to total change = component change/total original or total change, depending on the wording; identify the requested denominator.
[Commission] Commission = eligible sales x commission rate; for slabs, calculate each slab separately.
[Production rejection] Good output = total production x (100-rejection%)/100.
[Two-stage rejection] If rejection occurs at two independent sequential stages, good fraction = product of the two acceptance fractions.
[Attendance] Attendance% = attended sessions/total sessions x 100; required future attendance is solved using a new total denominator.
[Market share] Market share = company sales/total market sales x 100. If both company and market grow, compare their growth multipliers.
[Survey response] Response rate = completed responses/eligible invitations x 100 after excluding ineligible records only when stated.
[Approximation for small changes] For small p, q, product change is approximately p+q%, but hard problems require retaining pq/100.

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Shortcut Tricks

For x% increase followed by x% decrease: Net = -x?/100 % (always a loss)
[Base-100 method] For percentage comparisons, assume the original quantity is 100 unless an actual total is required. Convert every change into an easy multiplier.
Two successive increases of 10%: Net = 10+10+(10?10/100) = 21% ? memorize common pairs: 20%+20%=44%, 25%+25%=56.25%
[Multiplier method] Replace +p% by (100+p)/100 and -p% by (100-p)/100. Multiply factors instead of repeatedly finding percentages.
To find what % is X of Y: (X/Y)?100 ? practice converting: 1/8=12.5%, 1/6=16.67%, 1/7?14.28%, 3/8=37.5%
[Reverse multiplier] To move backward through a change, divide by the multiplier; never simply reverse the sign.
Multiplier method: 20% increase ? multiply by 1.2. 15% decrease ? multiply by 0.85. Faster than formula in exams
[Common fractions] Memorize 1/2=50%, 1/3=33.33%, 2/3=66.67%, 1/4=25%, 3/4=75%, 1/5=20%, 1/6=16.67%, 1/8=12.5%, 1/9=11.11%, 1/10=10%, 1/11=9.09%, 1/12=8.33%.
If A is 25% of B, then B is 400% of A. Reciprocal trick: if ratio is 1/n, reverse is n/1 ? 100%
[Useful eighths and sixteenths] 3/8=37.5%, 5/8=62.5%, 7/8=87.5%, 1/16=6.25%, 3/16=18.75%, 5/16=31.25%, 7/16=43.75%.
To find original value before x% increase: Original = New / (1 + x/100). Common trap in TCS/Infosys questions
[Useful twentieths and fortieths] 1/20=5%, 3/20=15%, 7/20=35%, 1/40=2.5%, 3/40=7.5%, 7/40=17.5%.
Fraction to % shortcuts: 1/3=33.33%, 2/3=66.67%, 3/8=37.5%, 5/8=62.5%, 7/8=87.5% ? memorize these for speed
[Swap trick] Use x% of y = y% of x, for example 16% of 25 = 25% of 16 = 4.
If A's salary is r% more than B, and you need B's salary as % of A: use 100?100/(100+r). E.g. r=25% ? 80%
[1% anchor] Find 1% first when the percentage is awkward, then scale. This is reliable for non-standard percentages.
For compound successive changes: multiply multipliers. 20% up then 30% down = 1.2 ? 0.7 = 0.84 ? net 16% decrease
[10%-5%-1% split] Build a percentage using easy pieces, such as 17%=10%+5%+2%.
Expenditure = Price ? Consumption. If one changes by x%, other must change by -x/(100?x)?100 to keep expenditure same
[Fraction first] Convert percentages like 12.5%, 16.67%, 20%, 25%, 33.33%, 37.5%, 62.5% into simple fractions before solving.
[Successive-change shortcut] For two changes, use a+b+ab/100 with signs rather than applying them separately.
[Equal up-down shortcut] Same x% up and down always gives x^2/100% loss.
[Restoration after loss] After p% loss, required gain = p/(100-p) x 100. The answer must be larger than p.
[Restoration after gain] After p% gain, required fall = p/(100+p) x 100. The answer must be smaller than p.
[More-less asymmetry] If A is p% more than B, immediately write ratio A:B=(100+p):100 before reversing the comparison.
[Less-more asymmetry] If A is p% less than B, write A:B=(100-p):100.
[Comparison denominator check] Words 'more than B' use B as base; words 'less than A' use A as base.
[Constant expenditure shortcut] Price factor x consumption factor = 1.
[Price rise mental shortcut] For a rise of p%, consumption fall = p/(100+p) x 100; use ratio 100:(100+p).
[Price fall mental shortcut] For a fall of p%, consumption rise = p/(100-p) x 100; use ratio 100:(100-p).
[Savings table] Assume income 100. If savings are s%, expenditure is 100-s. Apply separate multipliers to income and expenditure.
[Marks two-equation shortcut] Represent total marks by T and pass marks by P. Translate each candidate statement directly into percentage of T = P plus or minus marks.
[Election margin shortcut] In a two-candidate election, if winner gets x% of valid votes, margin percentage = (2x-100)% of valid votes.
[Election reverse shortcut] Valid votes = margin x 100/(difference in candidate percentages).
[Weighted-average warning] Never average percentages directly unless group sizes are equal.
[Weighted-average shortcut] Convert each group's percentage into actual successful count; add counts and divide by combined size.
[Mixture solute anchor] Track the actual amount of solute, not just concentration percentages.
[Dilution shortcut] When only water is added, old concentration x old volume = new concentration x new volume.
[Evaporation shortcut] When only water evaporates, solute amount remains fixed and concentration rises inversely with total volume.
[Repeated replacement shortcut] Original fraction left after n equal replacements = (1-r)^n.
[Successive discount shortcut] Net discount = sum of discounts - product/100 for two discounts.
[Markup-discount shortcut] Assume CP=100, compute MP from markup, then apply discount to get SP.
[Percentage-point check] Subtract rates for percentage points; divide the difference by the old rate for relative percent change.
[Product/area shortcut] For changes a% and b%, use a+b+ab/100. Do not omit the cross term in difficult questions.
[Square shortcut] Side change p% gives area change 2p+p^2/100%.
[Cube shortcut] Edge multiplier k gives volume multiplier k^3; percentages are faster through multipliers than expansion.
[Ratio update shortcut] Multiply each old ratio term by its own percentage-change factor; cancel common factors at the end.
[Fraction denominator shortcut] If denominator rises p% with numerator fixed, use p/(100+p) x 100% decrease.
[Fraction ratio shortcut] Treat a fraction as numerator divided by denominator and use the quotient multiplier.
[Overlap shortcut] For two sets, at least one = first + second - both; exactly one = first + second - 2(both).
[Three-set shortcut] Draw a compact inclusion-exclusion table to avoid counting the triple intersection incorrectly.
[Attendance target shortcut] Let x be future attended sessions. Required equation: (current attended+x)/(current total+x)=target rate.
[Rejected-output shortcut] Multiply acceptance rates, not rejection rates, across sequential quality stages.
[CAGR shortcut] Use growth factor Final/Initial, take nth root, then subtract 1. Do not divide total growth percentage by n.
[Reverse CAGR shortcut] Initial = Final/(1+r)^n.
[Base-effect shortcut] Compare every change against the correct original base before judging which percentage is larger.
[Hard DI shortcut] Convert all percentages to actual values first; delay rounding until the final step.
[Algebra shortcut] When x is p% of y and y is q% of z, x is pq/100% of z.
[Sanity check] After a decrease and restoration, the restoring percentage must be higher; after an increase and restoration, the restoring decrease must be lower.

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Solved Examples

Q1 A's income is 20% more than B. B's income is what % less than A?

Solution: B is less than A by (20/120)?100 = 16.67%

Q2 [Moderate | Successive changes] A salary is increased by 20%, reduced by 15%, and then increased by 10%. Find the overall percentage change.

Solution: Use multipliers: 1.20 x 0.85 x 1.10 = 1.122. Final salary is 112.2% of the original, so the net change is a 12.2% increase.

Q3 Price of rice increased by 25%. By what % must a family reduce consumption to keep expenditure unchanged?

Solution: Reduction = (25/125)?100 = 20%

Q4 [Moderate | Reverse percentage] After a 25% increase, a machine costs Rs 750,000. Find its original cost.

Solution: Original = 750,000/1.25 = Rs 600,000.

Q5 A number is first increased by 20% then decreased by 20%. What is net % change?

Solution: Net = 20 + (-20) + (20?(-20)/100) = -4%. Net 4% decrease

Q6 [Moderate | More-less comparison] A's score is 40% more than B's score. By what percentage is B's score less than A's?

Solution: Take B=100, so A=140. Required decrease = 40/140 x 100 = 28 4/7% (approximately 28.57%).

Q7 In an election, winner gets 55% of votes and wins by 1200 votes. Total valid votes?

Solution: Winner = 55%, Loser = 45%. Difference = 10% = 1200. Total = 12000

Q8 [Moderate | Restoration] A value falls by 30%. What percentage increase is required to restore the original value?

Solution: After the fall, 100 becomes 70. Required increase = 30/70 x 100 = 42 6/7% (approximately 42.86%).

Q9 A's salary is 40% of B's. B's salary is what % more than A's?

Solution: If A=40, B=100. B is more than A by (60/40)?100 = 150%

Q10 [Moderate | Percentage chain] x is 30% of y and y is 40% of z. x is what percentage of z?

Solution: x = 30% x 40% of z = 0.30 x 0.40z = 0.12z. Therefore x is 12% of z.

Q11 A student scores 30% and fails by 15 marks. Another scores 40% and passes by 35 marks. Find max marks.

Solution: Pass mark: 0.30x+15 = 0.40x-35 ? 0.10x=50 ? x=500

Q12 [Moderate | Ratio after changes] A:B=5:7. A increases by 20% and B decreases by 10%. Find the new ratio.

Solution: New ratio = 5 x 1.20 : 7 x 0.90 = 6:6.3 = 60:63 = 20:21.

Q13 A shopkeeper marks price 40% above cost and gives 25% discount. Profit or loss %?

Solution: SP = CP ? 1.4 ? 0.75 = 1.05 CP. Profit = 5%

Q14 [Difficult | Unknown percentage] A number is increased by x% and then reduced by 20%. The final value is 96% of the original. Find x.

Solution: (1+x/100) x 0.80 = 0.96. Hence 1+x/100 = 1.20, so x=20%.

Q15 Population of city is 8 lakh. It increases 10% in year 1 and decreases 10% in year 2. Final population?

Solution: 8,00,000 ? 1.1 ? 0.9 = 7,92,000. Net loss of 8000.

Q16 [Moderate | Three changes] A quantity rises by 10%, rises again by 20%, and then falls by 25%. Find the net percentage change.

Solution: Multiplier = 1.10 x 1.20 x 0.75 = 0.99. Therefore the net result is a 1% decrease.

Q17 In a mixture, milk is 80%. If 20L of water is added to 80L mixture, what is new % of milk?

Solution: Original milk = 64L in 80L. New total = 100L. Milk% = 64%

Q18 [Moderate | Population sequence] A town has 50,000 people. Population rises by 10%, then 20%, and then falls by 25%. Find the final population.

Solution: Final population = 50,000 x 1.10 x 1.20 x 0.75 = 49,500. The overall fall is 1%.

Q19 A's expenditure is 90% of income. Income rises 20%, expenditure 10%. Savings % change?

Solution: Old savings=10. New income=120, expenditure=99. New savings=21. Change=110%

Q20 [Difficult | CAGR] An investment grows from Rs 800,000 to Rs 1,064,800 in 3 years. Find the compound annual growth rate.

Solution: Growth factor = 1,064,800/800,000 = 1.331 = 1.1^3. Therefore CAGR = 10% per year.

Q21 If 15% of x = 20% of y, then x:y = ?

Solution: 15x = 20y ? x/y = 20/15 = 4:3

Q22 [Difficult | Price-consumption] The price of a commodity rises by 25%, but a family's expenditure on it rises by only 10%. Find the percentage change in consumption.

Solution: Consumption factor = expenditure factor/price factor = 1.10/1.25 = 0.88. Consumption decreases by 12%.

Q23 A bag has 600 coins of 25p and 50p. 60% are 25p coins. Total value?

Solution: 25p coins = 360, 50p = 240. Value = 90+120 = 210 rupees

Q24 [Moderate | Constant expenditure] Price falls by 20% and consumption rises by 25%. Find the change in expenditure.

Solution: Expenditure factor = 0.80 x 1.25 = 1. Therefore expenditure remains unchanged.

Q25 Two numbers are respectively 20% and 50% more than a third. What % is first of second?

Solution: Let third = 100. First=120, Second=150. % = (120/150)?100 = 80%

Q26 [Moderate | Revenue] A product's price rises by 20% while units sold fall by 10%. Find the percentage change in revenue.

Solution: Revenue factor = 1.20 x 0.90 = 1.08. Revenue increases by 8%.

Q27 A person spends 40% on food, 30% on rent, 20% on education. Saves ?1500. Find income.

Solution: Savings = 10% of income = 1500. Income = ?15,000

Q28 [Difficult | Income-expenditure-savings] A person earns Rs 80,000 and saves 25% of income. Income rises by 20% and expenditure rises by 10%. Find the new savings and the percentage increase in savings.

Solution: Old savings = 20,000 and old expenditure = 60,000. New income = 96,000; new expenditure = 66,000; new savings = Rs 30,000. Increase in savings = 10,000/20,000 x 100 = 50%.

Q29 TCS pattern: If 75% of workers are males and 40% of males are engineers, what % of total workers are male engineers?

Solution: 40% of 75% = 0.40 ? 0.75 = 0.30 = 30%

Q30 [Difficult | Savings change] A person saves 20% of income. Income rises by 15% and expenditure rises by 10%. Find the percentage increase in savings.

Solution: Assume income=100, expenditure=80, savings=20. New income=115 and new expenditure=88, so new savings=27. Increase = 7/20 x 100 = 35%.

Q31 [Very difficult | Comparative savings] A's income is 25% more than B's, A's expenditure is 20% more than B's, and A's savings are 40% more than B's. Find each person's savings as a percentage of income.

Solution: Let B's income=100 and expenditure=e. B saves 100-e. A's income=125, expenditure=1.2e, savings=125-1.2e. Equation: 125-1.2e=1.4(100-e). This gives e=75. B saves 25%, while A saves 35/125 x 100 = 28%.

Q32 [Difficult | Pass marks] One candidate scores 35% and fails by 24 marks. Another scores 47% and exceeds the pass mark by 36 marks. Find total marks and pass percentage.

Solution: The 12 percentage-point difference equals 24+36=60 marks. Total marks = 60/0.12 = 500. Pass marks = 35% of 500 + 24 = 199. Pass percentage = 199/500 x 100 = 39.8%.

Q33 [Moderate | Pass calculation] A candidate gets 220 marks and fails by 20 marks when the pass percentage is 40%. Find maximum marks. How many marks above pass would a 55% scorer obtain?

Solution: Pass marks=240. Since 240 is 40%, maximum marks=240/0.40=600. A 55% scorer gets 330, which is 90 marks above pass.

Q34 [Moderate | Weighted marks] A student scores 72% of 100, 64% of 150, 81% of 200, and 90% of 50. Find the overall percentage.

Solution: Marks obtained = 72+96+162+45=375. Maximum = 500. Overall percentage = 375/500 x 100 = 75%.

Q35 [Difficult | Election with abstention] In an election with 80,000 registered voters, 10% do not vote and 5% of votes cast are invalid. The winner gets 56% of valid votes. Find the winning margin.

Solution: Votes cast=72,000. Valid votes=72,000 x 0.95=68,400. In a two-candidate election, share difference=56%-44%=12%. Margin=68,400 x 0.12=8,208 votes.

Q36 [Moderate | Three-candidate election] In a three-candidate election, candidates receive 48%, 32%, and 20% of valid votes. The winner beats the runner-up by 6,400 votes. Find valid votes and the winner's votes.

Solution: Difference=16% of valid votes=6,400, so valid votes=40,000. Winner's votes=48% of 40,000=19,200.

Q37 [Difficult | Reverse election] In an election, 20% abstain and 10% of votes cast are invalid. The winner receives 60% of valid votes and wins by 8,640 votes. Find the number of registered voters.

Solution: Winner-loser difference=20% of valid votes, so valid votes=8,640/0.20=43,200. Valid votes are 0.80 x 0.90=72% of registered voters. Registered voters=43,200/0.72=60,000.

Q38 [Difficult | Sequential population] A city has 200,000 people. Births and deaths during the year are 4% and 1.5% of the starting population. Then immigration equals 2% of the population after natural change, and emigration equals 1% of the population after immigration. Find the final population.

Solution: Natural net addition=(4%-1.5%) x 200,000=5,000, giving 205,000. After immigration: 205,000 x 1.02=209,100. After emigration: 209,100 x 0.99=207,009.

Q39 [Difficult | Population composition] A town of 50,000 is 60% male. Male population grows 10% and female population grows 25%. Find the new male percentage.

Solution: Initial males=30,000 and females=20,000. New males=33,000; new females=25,000; total=58,000. Male percentage=33,000/58,000 x 100 = 56.90% approximately.

Q40 [Moderate | Dilution] A 60-litre solution contains 35% acid. How much water must be added to make it 21% acid?

Solution: Acid=60 x 0.35=21 litres. For 21% concentration, total volume must be 21/0.21=100 litres. Water added=40 litres.

Q41 [Difficult | Removal and replacement] An 80-litre solution contains 25% salt. Twenty litres are removed and replaced with water. Find the new concentration.

Solution: Salt initially=20 litres. Removed salt=20 x 25%=5 litres, so 15 litres remain. Final volume=80 litres. New concentration=15/80 x 100=18.75%.

Q42 [Moderate | Evaporation] A 100-litre solution contains 30% acid. Twenty litres of water evaporate. Find the new acid concentration.

Solution: Acid remains 30 litres; total volume becomes 80 litres. Concentration=30/80 x 100=37.5%.

Q43 [Difficult | Repeated replacement] A tank contains 64 litres of milk. Sixteen litres are removed and replaced with water, and the operation is repeated once. Find the final milk percentage.

Solution: Fraction of milk retained each time=48/64=3/4. Milk left=64 x (3/4)^2=36 litres. Milk percentage=36/64 x 100=56.25%.

Q44 [Moderate | Mixing concentrations] Mix 30 kg of a 40% solution with 20 kg of a 70% solution. Find the final concentration.

Solution: Solute=30 x 0.40 + 20 x 0.70 = 12+14=26 kg. Total=50 kg. Concentration=26/50 x 100=52%.

Q45 [Difficult | Adding pure solute] A 40 kg mixture is 30% chemical. How much pure chemical must be added to make it 50% chemical?

Solution: Initial chemical=12 kg. Let x kg be added. (12+x)/(40+x)=0.50. Thus 12+x=20+0.5x, giving x=16 kg.

Q46 [Difficult | Markup and discounts] An article costs Rs 1,000 and is marked 40% above cost. Successive discounts of 20% and 10% are offered. Find the profit percentage.

Solution: Marked price=1,400. Selling price=1,400 x 0.80 x 0.90=1,008. Profit=8, so profit percentage=0.8%.

Q47 [Moderate | Discount and tax] A product marked Rs 2,500 receives a 12% discount and then 18% tax is charged on the discounted price. Find the final amount.

Solution: Discounted price=2,500 x 0.88=2,200. Tax=396. Final amount=Rs 2,596.

Q48 [Very difficult | False weight] A trader marks goods 50% above cost, gives a 20% discount, and supplies only 900 g as 1 kg. Find the actual profit percentage.

Solution: Let cost of 1 kg=100. Charged price=100 x 1.50 x 0.80=120, while cost of 900 g=90. Profit=30 on cost 90, so actual profit=33 1/3%.

Q49 [Moderate | Weighted average] Forty students average 72%, and sixty students average 68%. Find the combined average percentage.

Solution: Combined percentage=(40 x 72 + 60 x 68)/100 = 69.6%.

Q50 [Difficult | Combined pass rate] Class A has 120 students with 75% passing, B has 80 with 65% passing, and C has 100 with 82% passing. Find the overall pass percentage.

Solution: Passes=90+52+82=224. Total=300. Overall pass percentage=224/300 x 100=74.67% approximately.

Q51 [Difficult | Weighted attrition] Departments of 200, 300, and 500 employees have attrition rates of 5%, 8%, and 12%. Find company-wide attrition.

Solution: Employees leaving=10+24+60=94. Total employees=1,000. Company-wide attrition=9.4%.

Q52 [Moderate | Percentage points] A conversion rate rises from 40% to 52%. State the percentage-point increase and the relative percentage increase.

Solution: Percentage-point increase=52-40=12 points. Relative increase=12/40 x 100=30%.

Q53 [Moderate | Measurement error] A length is measured as 48 cm instead of the true 50 cm. Find percentage error.

Solution: Percentage error=|48-50|/50 x 100=4%.

Q54 [Difficult | Wrong denominator] A quotient was calculated using a denominator 25% larger than the correct denominator. By what percentage is the computed quotient below the correct quotient, and by what percentage must it be raised to correct it?

Solution: Computed quotient factor=1/1.25=0.80, so it is 20% below correct. To move from 80 to 100, increase required=20/80 x 100=25%.

Q55 [Moderate | Product change] One factor rises by 15% and another falls by 20%. Find the percentage change in their product.

Solution: Product factor=1.15 x 0.80=0.92. Product decreases by 8%.

Q56 [Difficult | Quotient change] A numerator rises by 20% while its denominator falls by 25%. Find the percentage change in the fraction.

Solution: New/old fraction=1.20/0.75=1.60. The fraction increases by 60%.

Q57 [Moderate | Rectangle area] Length rises by 30% while breadth falls by 20%. Find the area change.

Solution: Area factor=1.30 x 0.80=1.04. Area increases by 4%.

Q58 [Moderate | Square] The perimeter of a square rises by 15%. Find the percentage rise in area.

Solution: Side also rises by 15%. Area factor=1.15^2=1.3225. Area rises by 32.25%.

Q59 [Moderate | Cube] The edge of a cube falls by 10%. Find the percentage fall in volume.

Solution: Volume factor=0.90^3=0.729. Volume falls by 27.1%.

Q60 [Difficult | Cylinder] The radius of a cylinder rises by 20% and height falls by 25%. Find the volume change.

Solution: Volume factor=1.20^2 x 0.75=1.44 x 0.75=1.08. Volume increases by 8%.

Q61 [Difficult | Fraction change] The numerator of a fraction rises by 25% and its denominator rises by 60%. Find the percentage change in the fraction.

Solution: New/old fraction=1.25/1.60=0.78125. The fraction decreases by 21.875%.

Q62 [Moderate | Ratio transformation] A:B=4:5. A rises by 50% and B falls by 20%. Find the new ratio.

Solution: New ratio=4 x 1.50 : 5 x 0.80 = 6:4 = 3:2.

Q63 [Difficult | Price-volume-quality] A factory raises price by 10%, increases production by 15%, but 4% of output is rejected. Assuming all good output is sold, find the revenue change.

Solution: Revenue factor=1.10 x 1.15 x 0.96=1.2144. Revenue increases by 21.44%.

Q64 [Difficult | Slab commission] Commission is 2% on the first Rs 200,000 of sales, 3% on the next Rs 300,000, and 5% above Rs 500,000. Find commission on Rs 650,000.

Solution: Commission=2% of 200,000 + 3% of 300,000 + 5% of 150,000 = 4,000+9,000+7,500 = Rs 20,500.

Q65 [Moderate | Price elasticity] Price falls by 12.5% and quantity sold rises by 20%. Find revenue change.

Solution: Revenue factor=0.875 x 1.20=1.05. Revenue increases by 5%.

Q66 [Difficult | Sequential rejection] A factory makes 50,000 units. Stage 1 rejects 6%, and stage 2 rejects 4% of the units surviving stage 1. Find good units and overall rejection percentage.

Solution: Good units=50,000 x 0.94 x 0.96=45,120. Rejected=4,880. Overall rejection=4,880/50,000 x 100=9.76%.

Q67 [Difficult | Defectives under growth] Production rises by 20%, while the defective rate falls from 8% to 5%. Find the percentage change in the number of defective units.

Solution: Let old production=N. Old defectives=0.08N. New defectives=1.20N x 0.05=0.06N. Defectives fall by (0.08-0.06)/0.08 x 100=25%.

Q68 [Very difficult | Effective capacity] A machine's rated capacity rises by 25%. Its downtime falls from 10% to 4%. Find the percentage increase in effective output.

Solution: Old effective factor=0.90. New effective factor=1.25 x 0.96=1.20. Relative increase=1.20/0.90-1=1/3=33 1/3%.

Q69 [Difficult | Market mix] Product A is 40% of company sales and the remaining products are 60%. A grows by 20% and the rest by 10%. Find total sales growth and A's new share.

Solution: Assume total=100. New A=48 and new rest=66, total=114. Total growth=14%. New A share=48/114 x 100=42.11% approximately.

Q70 [Moderate | Two-set overlap] In a placement batch, 68% clear coding, 54% clear aptitude, and 36% clear both. Find the percentage clearing at least one, neither, and exactly one.

Solution: At least one=68+54-36=86%. Neither=14%. Exactly one=68+54-2x36=50%.

Q71 [Very difficult | Three-set overlap] In a batch, 60% clear A, 55% clear B, and 50% clear C. Pairwise overlaps are 30%, 25%, and 20%, and 10% clear all three. Find percentages clearing at least one, exactly one, and exactly two.

Solution: At least one=60+55+50-30-25-20+10=100%. Exactly one=(60-30-20+10)+(55-30-25+10)+(50-20-25+10)=20+10+15=45%. Exactly two=(30+25+20)-3x10=45%.

Q72 [Difficult | Workforce composition] A company is 70% technical staff. Technical headcount rises by 20% and non-technical headcount by 50%. Find the new technical percentage.

Solution: Assume 100 staff: technical=70, non-technical=30. New counts=84 and 45, total=129. Technical percentage=84/129 x 100=65.12% approximately.

Q73 [Moderate | Combined subgroup rate] Boys are 60% of a batch and 75% of boys pass. Girls are 40% and 85% pass. Find overall pass percentage.

Solution: Overall pass percentage=0.60 x 75 + 0.40 x 85 = 45+34=79%.

Q74 [Moderate | Base effect] Sales rise from 200 to 250 and then fall back to 200. Find the rise percentage and fall percentage.

Solution: Rise=50/200 x 100=25%. Fall=50/250 x 100=20%. Equal absolute changes produce different percentages because the bases differ.

Q75 [Difficult | Percentage equation] x increased by 20% equals y reduced by 20%. Find x:y and state how much more y is than x.

Solution: 1.20x=0.80y, so x:y=2:3. Therefore y is (3-2)/2 x 100=50% more than x.

Q76 [Difficult | Equal repeated increase] A number becomes 144% of itself after two equal successive percentage increases. Find each increase.

Solution: (1+p/100)^2=1.44. Therefore 1+p/100=1.20 and p=20%.

Q77 [Difficult | Revenue target] Price is cut by 20%. By what percentage must sales volume rise so that revenue increases by 20%?

Solution: Required quantity factor=1.20/0.80=1.50. Sales volume must rise by 50%.

Q78 [Moderate | Chained comparison] A is 25% less than B, and C is 20% more than A. Compare C with B.

Solution: Let B=100. Then A=75 and C=90. Therefore C is 10% less than B.

Q79 [Very difficult | False weight plus discount] A trader marks goods 25% above cost, gives a 10% discount, and supplies 800 g as 1 kg. Find actual profit percentage.

Solution: Let cost of 1 kg=100. Charged price=100 x 1.25 x 0.90=112.5. Cost of 800 g=80. Profit=32.5 on 80, so profit percentage=40.625%.

Q80 [Moderate | Appreciation and depreciation] An asset appreciates 20%, depreciates 25%, and then appreciates 10%. Find net change.

Solution: Multiplier=1.20 x 0.75 x 1.10=0.99. Net change is a 1% decrease.

Q81 [Moderate | Restoration through value addition] A machine loses 20% of value. Repairs then increase its reduced value by 25%. Compare final value with original.

Solution: Multiplier=0.80 x 1.25=1. The machine returns exactly to its original value.

Q82 [Difficult | Stock-price sequence] A stock rises 40%, falls 30%, and rises 25%. Find net percentage change.

Solution: Multiplier=1.40 x 0.70 x 1.25=1.225. Net rise=22.5%.

Q83 [Difficult | Market share growth] A company has 25% market share. Its sales rise by 30% while the whole market rises by 20%. Find its new share and relative increase in share.

Solution: New share=25% x 1.30/1.20=27.083%. Increase=2.083 percentage points, or 2.083/25 x 100=8.333% relative.

Q84 [Difficult | Survey funnel] Of 5,000 invitations, 8% are ineligible. Of eligible people, 70% respond, and 5% of responses are incomplete. Find completed responses and completion rate on total invitations.

Solution: Eligible=4,600. Responses=3,220. Completed=3,220 x 0.95=3,059. Completion rate=3,059/5,000 x 100=61.18%.

Q85 [Moderate | Attendance target] A student has attended 72 of 90 classes. How many consecutive future classes must be attended to reach 85% attendance?

Solution: Let x be future classes attended. (72+x)/(90+x)=0.85. Thus 72+x=76.5+0.85x, so 0.15x=4.5 and x=30.

Q86 [Moderate | Transfer of mixture] Ten litres are transferred from a 50-litre 40% acid solution into 30 litres of a 20% acid solution. Find the concentration in the receiving vessel.

Solution: Transferred acid=10 x 0.40=4 litres. Receiving vessel initially has 6 litres acid. New acid=10 litres in 40 litres, so concentration=25%.

Q87 [Difficult | Changed pass-fail counts] A class has a 60% pass rate. If 20 additional students were counted as passing and 10 fewer as failing, the pass rate would be 70%. Find the original class size.

Solution: Let total=N. Original passes=0.60N. Modified passes=0.60N+20 and modified total=N+10. Equation: 0.60N+20=0.70(N+10). Hence 13=0.10N, so N=130.

Q88 [Difficult | Election exact margin] A constituency has 120,000 registered voters. Fifteen percent abstain, 4% of votes cast are invalid, and the winner gets 52.5% of valid votes. Find the winning margin.

Solution: Valid votes=120,000 x 0.85 x 0.96=97,920. Share difference=52.5%-47.5%=5%. Margin=97,920 x 0.05=4,896.

Q89 [Very difficult | Income and savings] A's income is 20% more than B's, A's expenditure is 10% more, and A's savings are 50% more. Find their savings rates.

Solution: Let B income=100 and expenditure=e. Then B saves 100-e. A income=120, expenditure=1.1e, and savings=120-1.1e. Equation: 120-1.1e=1.5(100-e), giving e=75. B saves 25%; A saves 37.5/120 x 100=31.25%.

Q90 [Difficult | Missing group average] Three groups form 40%, 35%, and 25% of a workforce. Their performance scores are 80%, 72%, and x%. Overall performance is 74%. Find x.

Solution: 0.40x80 + 0.35x72 + 0.25x = 74. Thus 32+25.2+0.25x=74, so x=67.2%.

Q91 [Very difficult | Repeated replacement] An 81-litre tank contains pure milk. Each time 27 litres are removed and replaced with water. The operation is performed three times. Find milk left and milk percentage.

Solution: Fraction retained each time=54/81=2/3. Milk left=81 x (2/3)^3=24 litres. Milk percentage=24/81 x 100=29.63% approximately.

Q92 [Difficult | Weighted examination] A test has papers of 150, 250, and 100 marks. A student scores 20% in the first and 60% in the second. What percentage must be scored in the third to obtain 48% overall?

Solution: Overall marks required=48% of 500=240. Scores in first two=30+150=180. Required in third=60 out of 100, so 60%.

Q93 [Difficult | Area reverse] The area of a rectangle increases by 32% while length increases by 20%. Find the percentage change in breadth.

Solution: Breadth factor=1.32/1.20=1.10. Breadth increases by 10%.

Q94 [Difficult | Markup-discount reverse] An article is marked 60% above cost. What discount gives a 12% profit?

Solution: Let CP=100, MP=160, required SP=112. Discount factor=112/160=0.70, so discount=30%.

Q95 [Difficult | Equal percentage values] Forty percent of A equals 60% of B equals 75% of C. Find A:B:C.

Solution: Let the common value be k. Then A=k/0.40=2.5k, B=k/0.60=5k/3, C=k/0.75=4k/3. Multiplying by 6 gives A:B:C=15:10:8.

Q96 [Very difficult | Revenue composition] Product revenues A, B, and C are 40%, 35%, and 25% of total. Next year A grows 25%, B falls 10%, and C grows 20%. Find total revenue growth and each new share.

Solution: Assume old total=100. New revenues: A=50, B=31.5, C=30; total=111.5. Growth=11.5%. New shares: A=44.84%, B=28.25%, C=26.91% approximately.

Q97 [Very difficult | Adult-child population] A city is 60% adult. Adults grow 10%, children fall 5%, and then 5% of the enlarged adult population migrates out. Find the final adult percentage.

Solution: Assume total=100. Adults after growth=66, then after migration=62.7. Children=40 x 0.95=38. Final total=100.7. Adult percentage=62.7/100.7 x 100=62.26% approximately.

Q98 [Difficult | Mixed revenue streams] Subscription revenue is 60% of total and service revenue is 40%. Subscription price rises 20% but subscribers fall 10%; service revenue rises 15%. Find total revenue change.

Solution: Subscription factor=1.20 x 0.90=1.08. Total factor=0.60x1.08 + 0.40x1.15 = 0.648+0.460=1.108. Total revenue rises 10.8%.

Q99 [Moderate | Fraction multiplier] A fraction's numerator rises 50% and denominator falls 20%. Find the percentage change.

Solution: Fraction multiplier=1.50/0.80=1.875. The fraction increases by 87.5%.

Q100 [Very difficult | Three-skill overlap] In a batch, 75% know Java, 65% Python, and 55% SQL. Pairwise overlaps are 45%, 35%, and 40%, and 25% know all three. Find at least one, exactly one, and exactly two.

Solution: At least one=75+65+55-45-35-40+25=100%. Exactly one=(75-45-40+25)+(65-45-35+25)+(55-40-35+25)=15+10+5=30%. Exactly two=(45+35+40)-3x25=45%.

Q101 [Difficult | Hiring funnel] A company receives 20,000 applications. Sixty percent clear aptitude, 75% of those clear coding, and 80% of those clear interviews. Find offers and overall selection rate. If 10% decline, find joiners.

Solution: Offers=20,000 x 0.60 x 0.75 x 0.80=7,200. Selection rate=36%. Joiners=7,200 x 0.90=6,480, equal to 32.4% of applicants.

Q102 [Very difficult | Reverse hiring funnel] A company finally gets 4,860 joiners. Ten percent of offers were declined; 75% of interviewees received offers; 80% of coding qualifiers reached interview; and 60% of applicants cleared aptitude to enter coding. Find initial applicants.

Solution: Applicants=4,860/(0.90 x 0.75 x 0.80 x 0.60)=15,000.

Q103 [Moderate | Score adjustment] A raw score of 72 is increased by 25% through normalization and then reduced by 10% as a penalty. Find final score and net percentage change.

Solution: Final score=72 x 1.25 x 0.90=81. Net multiplier=1.125, so net increase=12.5%.

Q104 [Difficult | Budget mix] A project budget allocates 30% to salaries, 25% to infrastructure, and 45% to other costs. Salaries exceed budget by 10%, infrastructure is 20% below budget, and other costs are exact. Find overall budget variance.

Solution: Assume total budget=100. Actual spend=30x1.10 + 25x0.80 + 45 = 33+20+45=98. The project is 2% under budget.

Q105 [Very difficult | Price-volume-tax] A product's pre-tax price falls 10%, quantity rises 20%, and tax rate rises from 5% to 8% of pre-tax revenue. Find the change in total customer payments including tax.

Solution: Old payment factor=1.05. New factor=0.90 x 1.20 x 1.08=1.1664. Relative factor=1.1664/1.05=1.110857. Total customer payments rise by about 11.09%.

Q106 [Difficult | Chained base comparisons] A is 20% more than B, B is 25% less than C, and C is 10% more than D. Compare A with D.

Solution: A/D=1.20 x 0.75 x 1.10=0.99. Therefore A is 1% less than D.

Q107 [Moderate | Future pass requirement] Two hundred candidates have completed a test and 70% passed. Another 100 will take it. What percentage of the next group must pass to make the final overall pass rate 75%?

Solution: Current passes=140. Final passes required=75% of 300=225. Required from next group=85, so 85%.

Q108 [Difficult | Weighted inflation] A household basket assigns weights 50%, 30%, and 20% to food, housing, and transport. Their prices change by +10%, -5%, and +25%. Find basket inflation.

Solution: Weighted change=0.50x10 + 0.30x(-5) + 0.20x25 = 5-1.5+5=8.5%.

Q109 [Difficult | Portfolio return] A portfolio invests 40%, 35%, and 25% in assets returning 15%, -10%, and 8%. Find portfolio return.

Solution: Portfolio return=0.40x15 + 0.35x(-10) + 0.25x8 = 6-3.5+2=4.5%.

Q110 [Moderate | Unknown second discount] Successive discounts of 20% and x% equal a single discount of 32%. Find x.

Solution: 20+x-(20x/100)=32. Hence 0.80x=12 and x=15%.

Q111 [Difficult | Unknown markup] An article is discounted 20% from marked price and still earns 12% profit. Find markup percentage over cost.

Solution: Let CP=100, required SP=112. Since SP=0.80MP, MP=140. Markup=40%.

Q112 [Difficult | Wrong operation] A value should have been divided by 1.2, but it was multiplied by 1.2. By what percentage is the wrong answer greater than the correct answer?

Solution: Wrong/correct=(1.2N)/(N/1.2)=1.44. Therefore the wrong answer is 44% greater.

Q113 [Moderate | Long percentage chain] A is 125% of B, B is 80% of C, and C is 150% of D. Compare A with D.

Solution: A/D=1.25 x 0.80 x 1.50=1.50. Therefore A is 50% more than D.

Q114 [Very difficult | Required attendance after absence] A student has attended 180 of 240 sessions. Over the next 40 sessions, the student will miss 5. How many additional consecutive sessions after that must be attended to reach 80% overall attendance?

Solution: After 40 sessions, attended=180+35=215 and total=280. Let x additional sessions all be attended. (215+x)/(280+x)=0.80. Thus 215+x=224+0.8x, so x=45.

Q115 [Difficult | Sales target with returns] Sales units rise 30%, but 8% of the new sales are returned. Price per retained unit rises 5%. Find net revenue change.

Solution: Net revenue factor=1.30 x 0.92 x 1.05=1.2558. Net revenue rises by 25.58%.

Q116 [Very difficult | Two-stage concentration] A 100-litre solution is 40% acid. Twenty litres are removed and replaced with water. Then 25 litres of the resulting mixture are removed and replaced with pure acid. Find final concentration.

Solution: After first replacement, acid=40 x 0.80=32 litres. The second removal takes 25/100 of 32=8 litres acid, leaving 24. Adding 25 litres pure acid gives 49 litres acid in 100 litres. Final concentration=49%.

Q117 [Difficult | Required defect rate] Production rises 25%. By what percentage must the defect rate fall from 8% so that the number of defectives falls by 20%?

Solution: Target defective count factor=0.80. Let new defect rate be r relative to old production: 1.25r = 0.80 x 0.08=0.064, so r=0.0512=5.12%. Rate falls by (8-5.12)/8 x 100=36%.

Q118 [Very difficult | Department growth and composition] A company has 1,000 employees: 40% in engineering, 35% in sales, and 25% in operations. Engineering grows 25%, sales falls 10%, and operations grows 20%. Find total headcount growth and new department shares.

Solution: New counts: engineering=500, sales=315, operations=300; total=1,115. Growth=11.5%. Shares are 44.84%, 28.25%, and 26.91% approximately.

Q119 [Difficult | Scholarship cutoff] A test has 800 marks. The pass cutoff is 42%, and scholarship cutoff is 25% above the pass marks. Find scholarship marks and scholarship percentage.

Solution: Pass marks=0.42 x 800=336. Scholarship marks=336 x 1.25=420. Scholarship percentage=420/800 x 100=52.5%.

Q120 [Very difficult | Two candidates and revised pass mark] A candidate scores 38% and fails by 28 marks. If the pass percentage were reduced by 4 percentage points, the candidate would pass by 12 marks. Find total marks and original pass percentage.

Solution: A 4 percentage-point reduction changes pass marks by 28+12=40. Therefore total marks=40/0.04=1,000. Candidate score=380. Original pass marks=408, so original pass percentage=40.8%.

Q121 [Difficult | Net discount and tax reversal] After a 20% discount and 18% tax on the discounted price, a customer pays Rs 4,720. Find marked price.

Solution: Payment factor=0.80 x 1.18=0.944. Marked price=4,720/0.944=Rs 5,000.

Q122 [Very difficult | Profit after return rate] A seller marks goods 30% above cost and gives 10% discount. Ten percent of sold units are returned for a full refund, but returned units can be resold later and are not damaged. For the completed sales in the period, find profit percentage on the cost of units that remain sold.

Solution: For each unit remaining sold, SP factor=1.30 x 0.90=1.17 of CP. Returns reduce both revenue and the cost of units treated as completed sales in the same proportion. Profit percentage on completed-sale cost remains 17%.

Q123 [Difficult | Response-rate correction] A survey reports 64% response based on 2,500 invitations. Later, 100 invitations are found ineligible and 80 completed responses are found duplicated and removed. Find corrected response rate among eligible invitations.

Solution: Reported responses=0.64 x 2,500=1,600. Correct responses=1,520. Eligible invitations=2,400. Corrected response rate=1,520/2,400 x 100=63.33% approximately.

Q124 [Very difficult | Overlap with neither] In a batch, 72% clear aptitude, 68% clear coding, and 18% clear neither. Find the percentage clearing both and exactly one.

Solution: At least one=82%. Both=72+68-82=58%. Exactly one=72+68-2x58=24%.

Q125 [Difficult | Required growth after decline] Revenue falls 20% in year 1. It must be 32% above the original by the end of year 2. What year-2 growth rate is required on the reduced revenue?

Solution: After year 1, factor=0.80. Required final factor=1.32. Year-2 factor=1.32/0.80=1.65. Required growth=65%.

Q126 [Very difficult | Average rate from counts] In a company, 30% of employees are women. Twenty percent of women and 12% of men work remotely. If 126 employees work remotely, find total employees.

Solution: Remote percentage=0.30x20% + 0.70x12% = 6%+8.4%=14.4%. Total employees=126/0.144=875.

Q127 [Difficult | Percentage allocation equation] A budget spends 35% on salaries and 24% on technology. Of the remainder, 40% goes to marketing. If marketing receives Rs 984,000, find total budget.

Solution: Remainder=41% of total. Marketing=40% of 41%=16.4% of total. Total budget=984,000/0.164=Rs 6,000,000.

Q128 [Very difficult | Successive removal with different fractions] A tank contains 120 litres of pure milk. First 20% is removed and replaced with water; then 25% of the mixture is removed and replaced with water. Find milk percentage.

Solution: Milk fraction remaining=0.80 x 0.75=0.60. Milk left=72 litres, so final milk percentage=60%.

Q129 [Difficult | Price, demand, and expenditure target] A commodity price rises 15%. A consumer wants expenditure to fall 8%. By what percentage must consumption change?

Solution: Consumption factor=0.92/1.15=0.80. Consumption must fall by 20%.

Q130 [Very difficult | Hiring percentage with reservation] Of all shortlisted candidates, 40% are from group A and 60% from group B. Selection rates are 30% and 20%. Group A forms what percentage of selected candidates?

Solution: Assume 100 shortlisted. Selected A=12 and selected B=12, total=24. Group A share among selected=50%.

Q131 [Difficult | Net productivity] Workforce falls 10%, average productivity per worker rises 25%, and defective output rises from 4% to 6%. Find the change in good output.

Solution: Old good-output factor=0.96. New factor=0.90 x 1.25 x 0.94=1.0575. Relative change=1.0575/0.96-1=10.15625%. Good output rises about 10.16%.

Q132 [Very difficult | Percentage equation with total] In a class, boys are 25% more numerous than girls. After 20% of boys and 10% of girls leave, 684 students remain. Find the original number of boys and girls.

Solution: Let girls=4k and boys=5k. Remaining=0.80 x 5k + 0.90 x 4k = 4k+3.6k=7.6k=684, so k=90. Original boys=450 and girls=360.

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