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Ratio and Proportion

About 7 min · Mathematics · English

Time worked out from the lesson's 451 words at 150 words a minute, plus a minute for each question.

This topic tests your ability to compare quantities, share resources, and solve partnership/mixture problems. Mastery here directly improves your speed in profit-loss, time-work, and mixture-alligation questions since those topics build on ratio concepts.

Key ideas

• Ratio expresses how many times one quantity contains another. If A:B = 3:5, then A = 3x and B = 5x for some common multiplier x. The ratio does not tell absolute values, only relative proportions.
• Proportion is an equation of two ratios: a:b = c:d means a/b = c/d, which gives ad = bc (cross-multiplication property). Use this to find any one unknown when three terms are known.
• Compound ratio of two ratios a:b and c:d is ac:bd. For three ratios a:b, c:d, e:f, compound ratio is ace:bdf. This appears when combining multiple conditions (e.g., age ratios at different times).
• Duplicate ratio of a:b is a²:b². Triplicate ratio is a³:b³. Sub-duplicate ratio is √a:√b. These may feature in geometry mensuration problems.
• Mean proportional between a and c is b where a:b = b:c, giving b² = ac, so b = √(ac). Third proportional to a and b is c where a:b = b:c. Fourth proportional to a, b, c is d where a:b = c:d.

Watch out for

• Mistake: Adding ratios directly — thinking 2:3 plus 4:5 equals 6:8. Fix: You cannot add ratios like fractions. Convert each to actual quantities with a common total, or find a common term if linking two ratios.
• Mistake: Forgetting to include time in partnership problems — using only capital invested. Fix: Partnership profit always depends on (Capital × Time). If time isn't mentioned, assume equal duration for all partners.
• Mistake: Confusing mean proportional with average. If asked for mean proportional of 4 and 16, writing (4+16)/2 = 10. Fix: Mean proportional is √(4×16) = √64 = 8, not the arithmetic mean. The term "mean" here refers to geometric mean.

Quick reference

• a:b = ma:mb — Ratios remain unchanged when both terms are multiplied/divided by the same non-zero number.
• To divide amount M in ratio p:q → First part = p/(p+q) × M, Second part = q/(p+q) × M.
• Partnership: Profit ∝ (Capital × Time) — Both factors multiply, not add.
• Mean proportional between a and c is √(ac) — Not (a+c)/2.
• Cross-multiply to solve proportions: a:b = c:d ⇒ ad = bc — Then isolate the unknown.

Check yourself: 3 questions

This lesson is cut from the topic's note, not rewritten. Fact-checked on 18 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.

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