Ratio and proportion
A ratio compares quantities of the same kind, and sharing in a ratio means finding the value of one part. Mixtures, alligation and partnership shares are all ratio questions where you must decide what each part stands for.
Formulas and when they apply
- Share of a part = (its ratio number ÷ sum of ratio numbers) × total
- If a : b = m : n and b : c = p : q, then a : b : c = mp : np : nq
- Alligation: cheaper : dearer = (dearer − mean) : (mean − cheaper)Applies when: the mean price lies between the two prices; quantities in the same unit
- Partners share profit in the ratio of (money × time) for each partner
- Direct proportion: if a rises, b rises in the same ratio. Inverse proportion: a × b stays constantApplies when: decide first whether more of one means more or less of the other
- After n replacements of x from a vessel of capacity V, original liquid left = V × (1 − x/V)ⁿApplies when: the same volume removed and replaced each time, with the mixture uniform
- Equal ratios: a/b = c/d means ad = bc
Worked examples
Example 1. ₹ 7,200 is divided in the ratio 2 : 3 : 7. How much does the person with 7 parts get?
Solution. Total parts = 2 + 3 + 7 = 12, so one part is 7,200 ÷ 12 = ₹ 600. The person with 7 parts gets 7 × 600 = ₹ 4,200.
Example 2. Wheat at ₹ 30 per kg is mixed with wheat at ₹ 36 per kg to make a mixture worth ₹ 34 per kg. In what ratio (cheaper : dearer) are they mixed?
Solution. Cheaper : dearer = (36 − 34) : (34 − 30) = 2 : 4 = 1 : 2. Check: 1 kg at 30 and 2 kg at 36 cost 30 + 72 = ₹ 102 for 3 kg, which is ₹ 34 per kg.
Example 3. Xavier invests ₹ 20,000 for 12 months and Yash invests ₹ 30,000 for 4 months. They share a profit of ₹ 9,000. What does Yash get?
Solution. Xavier: 20,000 × 12 = 2,40,000. Yash: 30,000 × 4 = 1,20,000. The ratio is 2 : 1, so Yash gets 1/3 of ₹ 9,000 = ₹ 3,000.
Shortcuts and their limits
- Use alligation instead of forming equations for any mixture of two items at different prices or strengths.Does not apply when: It works for exactly two items and only when the mean lies between them. For three items, combine two first.
- In a ratio question, set the unknown multiplier as k and write each quantity as a multiple of k.Does not apply when: It applies only when the ratio is between quantities of the same kind and you know one total or one difference.
- In inverse proportion, multiply the two quantities and keep the product fixed.Does not apply when: It is wrong for direct proportion, where it is the ratio, not the product, that stays fixed.
Common mistakes
- Taking a ratio number as the actual amount instead of finding the value of one part.
- Reversing the alligation ratio, so the larger share goes to the wrong item.
- Ignoring time when partners join at different months.
- Using direct proportion for a situation that is inverse, such as more people finishing food faster.
- Treating the volume removed from a mixture as pure original liquid in the second removal.
Independent practice made by EduVistaHQ with original questions. It is not an official examination paper, it is not affiliated with or endorsed by any examination body or company, and a practice score does not predict a real result.