Percentage, Ratio and Proportion

Percentage, ratio and proportion is the quiet workhorse of every judiciary and CLAT-PG aptitude paper. It rarely appears as a single labelled question; instead it hides inside profit-and-loss sums, data interpretation tables, reservation-roster puzzles and even the very judgments you will one day write. A judge who cannot tell whether a 27% quota added to an existing 22% breaches the 50% ceiling is no better placed than a candidate who cannot convert 3 : 2 into 60% and 40%. This chapter builds the topic from first principles, drills the high-yield shortcuts examiners reward, and then does something most aptitude notes never attempt: it grounds the arithmetic in the actual percentage reasoning of the Supreme Court, from M.R. Balaji to the 2021 Maratha verdict.

What a percentage really is

A percentage is nothing more exotic than a fraction whose denominator is fixed at 100. To say a quantity is x per cent is to say it equals x/100 of the whole. This single idea unlocks every conversion you will ever need: a fraction becomes a percentage by multiplying by 100, and a percentage becomes a fraction by dividing by 100. Thus 1/4 is 25%, 1/8 is 12.5%, and 5/6 is 83.33%. Committing the common fraction-percentage equivalents to memory is the highest-return investment in the whole arithmetic syllabus, because it turns multi-step division into instant recall.

The reason percentages dominate competitive papers is that they are dimensionless. A marks-out-of-50 score and a marks-out-of-200 score cannot be compared directly, but converted to percentages they sit on the same scale. This comparability is exactly why courts and commissions express reservation, attendance, and majority thresholds as percentages rather than raw counts. Before attempting speed problems, see the companion chapter on number system and arithmetic foundations, which establishes the fraction fluency this topic assumes.

Two framings recur. The first is "x is what per cent of y", answered by (x/y) x 100. The second is "what is x per cent of y", answered by (x/100) x y. Candidates lose marks by confusing the base. The base is always the quantity after the word "of", and it is the denominator. When a question says "20% of the candidates failed", the base is the total candidates, not the pass figure.

Percentage increase, decrease and the base trap

Percentage change is measured against the original value, never the final one. A rise from 80 to 100 is a 25% increase (20/80), but the fall back from 100 to 80 is only a 20% decrease (20/100). This asymmetry, the base trap, is the single most exploited idea in aptitude papers. A price that rises 25% and then falls 25% does not return to its starting point; it ends at 93.75% of the original, a net loss of 6.25%.

The general rule for two successive changes of a% and b% is a net change of (a + b + ab/100) per cent, where reductions carry a negative sign. So a 25% rise followed by a 25% fall gives 25 - 25 + (25 x -25)/100 = -6.25%. This formula collapses an entire family of "by what per cent must salary be reduced after a rise to restore the original" questions into a single line. If salary rises by r%, the restoring reduction is 100r/(100 + r) per cent.

The base trap reappears constantly in legal arithmetic. When a roster already carries 22% reservation and a State adds 27% more, the addition is computed on the total cadre, not on the unreserved remainder, so the two simply sum to 49%. Misreading the base is precisely the error that inflated the Mysore quota to 68% in M.R. Balaji v. State of Mysore, discussed below. For the commercial cousin of these sums, see profit, loss and discount, where successive discounts behave identically.

Successive percentage changes and the multiplier method

The cleanest way to handle chains of percentage change is the multiplier. A rise of p% multiplies a quantity by (1 + p/100); a fall by (1 - p/100). To apply several changes, multiply the multipliers. A value subjected to +10%, then -20%, then +25% becomes original x 1.10 x 0.80 x 1.25 = original x 1.10, a net 10% gain. The multiplier method eliminates sign errors and scales to any number of stages.

It also answers population and depreciation problems instantly. A town growing 5% a year for three years multiplies by 1.05 cubed; a machine depreciating 10% a year for two years multiplies by 0.9 squared. Because compound interest is the same multiplier idea applied to money, the technique carries straight over to simple and compound interest. A useful approximation for small successive changes is that the net is close to the sum of the individual percentages, with the product term (ab/100) supplying the correction; examiners design distractors around candidates who forget that correction.

A final caution: multipliers must act on the same base in sequence, each new multiplier applying to the running result. Mixing a percentage of the original with a percentage of the running total inside one chain is a classic trap, and it is the arithmetic analogue of the post-based versus vacancy-based counting dispute the Supreme Court resolved in R.K. Sabharwal v. State of Punjab.

The language of ratio

A ratio compares two quantities of the same kind by division, written a : b and read "a to b". It is a pure number; the units cancel, which is why a ratio of lengths and a ratio of incomes look identical on paper. A ratio is unchanged when both terms are multiplied or divided by the same non-zero number, so 4 : 6, 2 : 3 and 10 : 15 are the same ratio. Reducing to lowest terms is the first move in almost every ratio problem.

Three derived ratios appear often. The duplicate ratio of a : b is a-squared : b-squared, used in area comparisons. The sub-duplicate ratio is the square-root form, used to recover sides from areas. The inverse ratio simply swaps the terms to b : a, and it is indispensable in inverse-proportion problems such as men-versus-days in time and work. When several ratios must be chained, say A : B and B : C with different B-values, make the common term equal by scaling, then read off A : B : C.

Ratios are the natural language of legal apportionment. Maintenance shared between dependants, partnership profit divided by capital, and seats distributed among categories are all ratio operations. The skill the exam tests, dividing a whole in a given ratio, is the same skill a court exercises when it splits compensation among legal heirs.

Dividing a quantity in a given ratio

To divide an amount T in the ratio a : b : c, add the parts to get a + b + c, then give each share its fraction of the total: the first gets T x a/(a+b+c), and so on. To split Rs 12,000 in 3 : 4 : 5, the parts sum to 12, the unit value is Rs 1,000, and the shares are Rs 3,000, Rs 4,000 and Rs 5,000. The unit-value method, find one part then scale, is faster and less error-prone than computing each fraction separately.

The reverse operation is just as common: given one share and the ratio, recover the total. If the smallest share in a 2 : 3 : 5 split is Rs 4,000, then one part is Rs 2,000 and the total is Rs 20,000. Examiners like to give a difference between two shares rather than a share itself; the difference corresponds to the difference in parts, so a gap of Rs 6,000 across a 3 : 5 split means two parts equal Rs 6,000, one part is Rs 3,000.

This is exactly the arithmetic of a reservation roster. A 100-point roster filled in the proportion 15 : 7.5 : 27 for SC, ST and OBC allots posts at fixed points, and the question of whether a category's count is read against the cadre total or the vacancies of a single year is what divided the Court in R.K. Sabharwal. Data-heavy versions of these splits appear in data interpretation pie charts, where the whole is 360 degrees and each sector is a ratio share.

Proportion and the rule of three

A proportion is a statement that two ratios are equal, a : b :: c : d, meaning a/b = c/d. Its defining property is that the product of the means equals the product of the extremes: b x c = a x d. This cross-multiplication, the rule of three, is the engine behind unitary-method problems. If 5 pens cost Rs 60, then 8 pens cost (60/5) x 8 = Rs 96, because the price-to-quantity ratio is held constant.

Quantities may vary in direct proportion, where one rises as the other rises (cost and quantity), or in inverse proportion, where one rises as the other falls (workers and time for a fixed job). The skill is diagnosing which applies before setting up the equation; treating an inverse relationship as direct is the commonest conceptual error in time, distance and speed problems, where speed and time are inversely proportional for a fixed distance.

Continued proportion, a : b :: b : c, gives b-squared = a x c, so b is the mean proportional (geometric mean) of a and c, and c is the third proportional to a and b. These appear in geometry-flavoured aptitude items and in valuation problems where a court must interpolate a fair value between two benchmarks.

The 50% ceiling: percentages in constitutional law

No body of Indian law turns on a percentage more famously than reservation jurisprudence. In M.R. Balaji v. State of Mysore, AIR 1963 SC 649, a five-judge Bench struck down a Mysore order reserving 68% of seats for backward classes, holding that special provisions under Articles 15(4) and 16(4) must stay within reasonable limits and that, speaking generally, reservation ought not to exceed 50%. The Court's reasoning is pure proportion: reservation is an exception to the equality rule, and an exception that swallows more than half the whole ceases to be an exception.

The principle hardened into doctrine in Indra Sawhney v. Union of India, AIR 1993 SC 477 (the Mandal Commission case), where a nine-judge Bench fixed 50% as the normal ceiling on total reservation, breachable only in extraordinary situations and never under the carry-forward rule. Indra Sawhney also introduced the creamy-layer exclusion, itself a percentage and income-threshold exercise: those above a defined income are filtered out of the backward class before the quota is computed.

The arithmetic is unforgiving. If a State already operates 22% for SC/ST and adds 27% for OBC, the running total is 49%, just inside the ceiling. Any further category, however deserving, must either fit the residual 1% or claim the extraordinary-situations exception. This is the base-and-sum logic of the earlier sections applied to fundamental rights.

Roster arithmetic and post-based counting

How a percentage is counted matters as much as its size. In R.K. Sabharwal v. State of Punjab (decided 10 February 1995), the Supreme Court held that the prescribed reservation percentage operates on the total number of posts in a cadre, not on the vacancies arising in a particular year. Once the reserved category's representation in the cadre reaches the roster figure, the reserved points are treated as filled, and further vacancies at those points go back to the category only on the retirement or exit of an incumbent. Reserved candidates selected on merit against general posts are counted as general, not added to the quota.

This is the post-based roster, and it is a direct application of the dividing-in-a-ratio method to a fixed denominator. The cadre strength is the whole; the percentage fixes how many points belong to each category; the roster is simply the ordered allocation of those points. The dispute that reached the Court was, at bottom, a base-trap: counting the percentage against annual vacancies (a small, shifting base) instead of the cadre total (the correct, stable base) silently inflated reserved numbers beyond the sanctioned proportion.

The case is a model of why aptitude precision is a judicial skill. A judge who internalises that a percentage is meaningless without a clearly identified base will not be misled by a roster that quietly changes its denominator midway, exactly the discipline the multiplier and unit-value methods instil.

Quantifiable data and proportional justification

In M. Nagaraj v. Union of India, (2006) 8 SCC 212, a five-judge Constitution Bench upheld the constitutional amendments enabling reservation in promotion for SC/ST, but conditioned their exercise on the State collecting quantifiable data showing backwardness, inadequacy of representation, and that the measure would not impair administrative efficiency, all while respecting the 50% ceiling and the creamy-layer concept. Each of these is a proportional test: representation is a percentage of cadre strength, adequacy is that percentage measured against population share, and efficiency is a constraint on how far the proportion may be pushed.

The judgment effectively makes statistics a precondition of constitutional power. A State cannot assert backwardness in the abstract; it must show, in numbers, what proportion of posts a community holds against what proportion of the eligible population it represents. This is the comparability virtue of percentages from the opening section, deployed as a check on arbitrary policy.

For the aspirant, Nagaraj is a reminder that the same ratio can be expressed as a fraction, a percentage, or a per-lakh figure, and that choosing the right denominator (cadre, population, applicant pool) is what makes the comparison honest. The administrative-efficiency caveat also echoes the inverse-proportion intuition: beyond a point, raising one quantity (quota) may depress another (efficiency).

Quotas in education and the limits of the percentage

Percentages also frame the autonomy of educational institutions. In T.M.A. Pai Foundation v. State of Karnataka, decided in 2002 by an eleven-judge Bench, the dispute arose from a State directive reserving 40% of seats as a government quota in private professional colleges. The Court held that unaided institutions, minority or not, could not be compelled to surrender their admissions to a State-imposed percentage, though aided institutions could be required to set aside a reasonable share for weaker sections.

The case shows percentages operating as instruments of policy and as limits on it simultaneously. A 40% quota is an arithmetic claim on a finite whole; the constitutional question is whether the claimant (the State) may make that claim against a particular base (a privately funded institution). The ratio analysis, what fraction of seats, against what total, funded by whom, is inseparable from the rights analysis.

For exam purposes, the lesson mirrors the partnership-and-share sums: the same percentage produces very different absolute numbers depending on the size and nature of the base, and a well-set question will vary the base precisely to test whether the candidate is computing against the right whole.

The 50% ceiling reaffirmed: the Maratha verdict

The percentage logic reached its most recent landmark in Jaishri Laxmanrao Patil v. Chief Minister of Maharashtra, decided 5 May 2021. The Maharashtra SEBC Act, 2018 had granted the Maratha community a separate reservation that pushed the State's total to roughly 68%, the very figure struck down nearly six decades earlier in Balaji. A five-judge Constitution Bench struck down the Maratha reservation, holding unanimously that the 50% ceiling laid down in Indra Sawhney remained good law, did not need to be revisited or referred to a larger Bench, and that the Maratha case disclosed no extraordinary situation justifying a breach.

The symmetry between 1963 and 2021, two attempts to cross 50%, both at about 68%, both struck down, is the clearest possible demonstration that a constitutional principle can be, at heart, an arithmetic one. The exception to equality may not consume the majority it qualifies. Every successive-percentage and base-trap drill in this chapter is, in miniature, the same vigilance the Court exercised: does the total, properly summed against the correct whole, stay within the permitted half?

Returning to the broader syllabus, the aptitude and reasoning hub sequences this topic before profit-loss and interest precisely because percentage fluency underwrites them all. A candidate fluent in converting, comparing and summing percentages reads both a discount cascade and a reservation roster with the same eye.

High-yield shortcuts and exam strategy

Speed in this topic comes from pattern recognition, not from longer calculation. Memorise the fraction-percentage table up to 1/12; it converts most "x% of y" sums into single-digit multiplication. Use the multiplier method for every chain of changes, and the net-change formula a + b + ab/100 for any two-step problem. For "by what per cent is A more or less than B" comparisons, anchor firmly on B as the base, since reversing A and B is the examiner's favourite trap.

For ratio problems, always reduce to lowest terms first, then use the unit-value method to find one part before scaling to all. When a difference between shares is given, equate it to the difference in ratio parts. For proportion, set up cross-multiplication and pause to confirm whether the relationship is direct or inverse before solving, the diagnostic step that separates correct from confidently wrong answers.

Finally, sanity-check magnitudes. A percentage answer above 100 is possible only when the part exceeds the base (a growth or markup), and a reserved total above 50% should trigger the same alarm in an aptitude question as it does in a constitution Bench. Cultivating that instinct, that a number must be measured against the right whole and kept within sensible limits, is the through-line from this chapter to the rest of the data interpretation and commercial-arithmetic syllabus.