Ratio and Proportion

Ratio and Proportion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Ratio compares two same-kind quantities by division, written a:b, where a is the antecedent and b the consequent. The ratio is dimensionless, so both quantities must share the same unit. Proportion declares two ratios equal: a:b = c:d means a/b = c/d, equivalently ad = bc by cross-multiplication.

Must-know results for HAT-UG:

• Dividing a quantity Q in the ratio a:b gives parts aQ/(a+b) and bQ/(a+b).
• Mean proportional between a and b: x = √(ab).
• Third proportional to a and b: c = b²/a.
• Direct proportion y = kx; inverse proportion y = k/x.

High-yield traps: keep ratio order fixed when forming compound ratios, and never equate a = c, b = d individually — only the cross-product identity holds.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definitions and Notation

A ratio a:b is read as “a is to b” and equals the fraction a/b when b ≠ 0. Two ratios are in proportion when they are numerically equal: a:b = c:d ⟺ a/b = c/d ⟺ ad = bc. The four numbers a, b, c, d are called proportionals, with a and d as the extremes and b and c as the middle terms.

Dividing a Quantity in a Ratio

To split Q into parts with ratio a:b, the larger share is a/(a+b) × Q and the smaller share is b/(a+b) × Q. This single relation powers most HAT-UG division problems (income, age, profit-sharing).

Compound, Duplicate, and Continued Ratios

• Compound ratio of a:b and c:d = ac:bd.
• Duplicate ratio of a:b = a²:b²; triplicate ratio = a³:b³.
• Three ratios a:b, c:d, e:f are in continued proportion when a:b = b:c = c:d.

Mean and Third Proportionals

For numbers a and b, the mean proportional x satisfies a:x = x:b, giving x = √(ab). The third proportional c satisfies a:b = b:c, giving c = b²/a.

Chain of Ratios

If A:B = m:n and B:C = p:q, then A:B:C = mp : np : nq. Always align the shared term (B) so its coefficient matches across both ratios.

Direct and Inverse Variation

In direct proportion y = kx, both quantities grow together; the ratio y₁/y₂ = x₁/x₂ stays constant. In inverse proportion y = k/x, the product x₁y₁ = x₂y₂ stays constant. Time-and-work problems use the inverse form (more workers, less time), while partnership profit uses direct division.

Mixtures (Allegation)

For a mixture of two ingredients at cheaper price C and dearer price D giving mean price M, the mixing ratio is (M − C) : (D − M). This is the HAT-UG shortcut for “in what ratio must tea at Rs 280 and Rs 340 be mixed to sell at Rs 300/kg?”.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Working with Multi-Step Ratio Changes

When a ratio a:b changes to a′:b′ after a gain or loss, the original amounts satisfy the difference rule: original A / original B is recoverable from the identity

a′ · B − b′ · A = (a′b − ab′) · (common base).

Concretely, if A:B = 3:4 and after A gains Rs 100 the new ratio becomes 5:4, then let A = 3k and B = 4k, giving (3k+100)/4k = 5/4, so k = 50 and A = Rs 150, B = Rs 200. This structure recurs in HAT-UG age and money problems.

Partnership and Time Problems

In a partnership, profits split by capital × time, so when capitals are c₁, c₂ invested for times t₁, t₂, the profit ratio is c₁t₁ : c₂t₂. If only capitals differ, it reduces to direct proportion; if only times differ, inverse-proportion logic applies (longer investment, larger share).

Common Mistakes and Traps

1. Unit mismatch — comparing 1 m to 50 cm without converting gives a wrong “ratio” of 1:50 instead of 2:1. Convert first.
2. Order swap in compound ratios — swapping c and d changes the compound ratio from ac:bd to ad:bc.
3. a = c fallacy — from a:b = c:d one can deduce only ad = bc; the individual equalities a = c and b = d are false in general (e.g., 2:4 = 3:6 but 2 ≠ 3).
4. Inverse vs direct confusion — “more men, fewer days” is inverse (product constant); “more speed, more distance in same time” is direct (ratio constant).
5. (a−c)/(b−d) misuse — the property (a−c)/(b−d) = a/b = c/d holds only when the subtracted constants are equal multipliers; otherwise cross-multiply.

Adjacent Connections

Ratio feeds directly into percentages, profit and loss, simple & compound interest (ratios of P, R, T determine interest ratios), and probability (odds = favourable:unfavourable). Mastery here is prerequisite for Averages (allegation form) and Time-Speed-Distance (inverse proportion of speed and time at fixed distance).

HAT-UG Practice Prompts

1. If x:y = 5:7 and y:z = 14:9, what is x:y:z? (Hint: chain rule.)
2. Two varieties of rice costing Rs 42/kg and Rs 63/kg are mixed so that the mixture sells at Rs 54/kg. In what ratio must they be combined?

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