Proportion

1. Meaning, notation and core identities

A ratio compares two quantities through division. A proportion states that two such ratios are equal. Thus, 6 : 9 = 10 : 15 is a proportion because both ratios simplify to 2 : 3. Write the quantities in the same order on both sides: if the first ratio compares rice with wheat, the second must also compare rice with wheat.

For a : b = c : d, cross-multiplication gives ad = bc. This equality is the basic test of proportion and the quickest way to find a missing term. If 8 : 12 = x : 30, then 12x = 240 and x = 20. Alternatively, 12 becomes 30 by multiplication by 2.5, so 8 must become 20 by the same factor.

The fourth proportional to a, b and c is d such that a : b = c : d; hence d = bc/a. The third proportional to a and b is c such that a : b = b : c; hence c = b²/a. The mean proportional between positive quantities a and c is b = √(ac). For example, the mean proportional between 4 and 9 is 6.

Useful transformations include inversion, b/a = d/c, and alternation, a/c = b/d, whenever the denominators are non-zero. Componendo and dividendo gives (a + b)/(a − b) = (c + d)/(c − d), when defined. Use these identities selectively; cross-multiplication or a common multiplier is usually simpler.

• Convert comparable quantities into common units before forming a dimensionless ratio: 1.5 metres : 75 centimetres = 150 : 75 = 2 : 1.
• A rate can compare unlike units, such as kilometres per hour; retain the units consistently throughout the calculation.
• If a : b = 3 : 5, write a = 3k and b = 5k rather than assuming that a and b equal 3 and 5.

2. Direct proportion and the unitary method

Two quantities are directly proportional when their ratio remains constant. If y = kx, doubling x doubles y and multiplying x by any factor multiplies y by the same factor. Examples include cost and quantity at a fixed unit price, distance and time at a fixed speed, and wages and hours at a fixed hourly rate.

The unitary method first finds the value corresponding to one unit and then scales it. If 8 notebooks cost ₹240, one notebook costs ₹30 and 15 notebooks cost ₹450. The equivalent proportional equation is 240/8 = C/15. Before using either approach, check that discounts, fixed charges or changing rates do not alter the relationship.

Direct proportion differs from a general increasing relationship. A taxi fare consisting of a ₹50 fixed charge plus ₹12 per kilometre increases with distance but is not directly proportional to distance. For 5 kilometres the fare is ₹110; for 10 kilometres it is ₹170, not ₹220. The fixed component prevents a constant fare-to-distance ratio.

• A direct-proportion graph is a straight line through the origin.
• Percentage scaling is proportional: a 20% increase multiplies a quantity by 1.20.
• When a total T is divided in the ratio m : n, the shares are Tm/(m + n) and Tn/(m + n).

3. Inverse proportion and fixed-product reasoning

Two quantities are inversely proportional when their product remains constant. If xy = k, doubling x halves y. The identifying question is: what total is being held fixed? For a fixed journey, speed × time equals distance. For a fixed job performed by equally efficient workers, workers × days is constant when daily working hours are unchanged.

Suppose 12 workers complete a job in 15 days. The job requires 180 worker-days under the stated assumptions. With 20 equally efficient workers, the time becomes 180/20 = 9 days. Writing 12/20 = 15/D would incorrectly treat the relationship as direct. The reliable equation is 12 × 15 = 20 × D.

Inverse change is not an equal and opposite percentage change. If speed increases by 25%, it becomes 1.25 times its original value. For the same distance, time becomes 1/1.25 = 0.8 times the original time, a reduction of 20%. Similarly, a 20% speed reduction increases time by 25%. Reciprocal multipliers prevent this frequent error.

• Equal worker efficiency and unchanged daily hours must be stated or reasonably implied.
• Adding workers may not proportionately reduce real-world project time if tasks cannot be divided or coordination causes delays.
• Food stock and duration are inversely related to the number of consumers only when consumption per person per day remains constant.

4. Compound proportion and related applications

Compound proportion involves more than two changing quantities. Express the underlying relationship before manipulating ratios. For uniform production, output is proportional to workers × days × hours per day × efficiency. Consequently, time is directly proportional to required output and inversely proportional to workers, daily hours and efficiency.

Suppose 12 workers working 6 hours daily complete 360 units in 5 days. How long will 15 equally efficient workers working 8 hours daily take to complete 720 units? Required days = 5 × (720/360) × (12/15) × (6/8) = 6 days. Each factor has a meaning: more output increases time, while more workers and longer working hours reduce it.

Map scales are another direct application. At a scale of 1 : 50,000, one centimetre on the map represents 50,000 centimetres, or 0.5 kilometre, on the ground. A map distance of 6 centimetres therefore represents 3 kilometres. The scale is a ratio of lengths, not areas.

For similar figures, areas vary as the square of corresponding lengths and volumes as the cube. If every length is multiplied by 3, area is multiplied by 9 and volume by 27. In mixtures, maintain the distinction between component ratios and fractions of the total: milk : water = 3 : 2 means milk forms 3/5, not 3/2, of the mixture.

• In ratio chains, equalise the common term: A : B = 2 : 3 and B : C = 4 : 5 give A : B : C = 8 : 12 : 15.
• When equal quantities are added to both terms of a ratio, the ratio generally changes; equal multiplication preserves it.
• Use squared or cubed scale factors only when geometric similarity or the relevant formula justifies them.

5. CSAT problem-solving strategy and common traps

Begin by naming the unknown and recording the fixed conditions. Decide whether a constant ratio, constant product or a multi-factor relationship applies. This classification matters more than memorising a rule of three. A short equation often exposes information that is missing or irrelevant.

Predict the direction of change before calculating. More workers should require fewer days for the same work; a larger order should cost more at a fixed unit price. Cancel common factors before multiplying, and retain fractions until the final step. This reduces arithmetic effort and rounding errors.

Finally, check units, magnitude and the original condition. In multiple-choice questions, estimation can eliminate impossible answers, but it cannot replace choosing the correct relationship. When a price contains a fixed fee, efficiency changes, or demand varies unpredictably, ordinary direct or inverse proportion may not apply.

• Do not confuse the ratio of two shares with either share's fraction of the total.
• Do not assume that equal percentage increases and decreases cancel.
• Do not mix minutes with hours, grams with kilograms or map centimetres with ground kilometres.
• For successive changes, multiply scale factors rather than adding percentage changes.

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