A ratio a : b compares two quantities of the same kind by division (a/b). A proportion says two ratios are equal. These ideas run through sharing money, partnership profits, mixtures, ages, work and speed problems.
Ratio Basics
- A ratio does not change when both terms are multiplied or divided by the same number: 0.5 : 1.5 = 5 : 15 = 1 : 3.
- Increasing or decreasing both quantities by the same percentage keeps the ratio the same; adding or subtracting the same number changes it.
- Dividing T in the ratio a : b gives Ta/(a + b) and Tb/(a + b): 270 in 4 : 5 gives 120 and 150.
- To compare a : b with c : d, compare a × d with b × c.
- Ratios given as fractions: convert with the LCM of the denominators. 1/2 : 1/3 : 1/4 = 6 : 4 : 3.
- Combining ratios: if a : b = 2 : 3 and b : c = 6 : 7, make b equal: a : b : c = 4 : 6 : 7. In a chain p : q, q : r, r : s, p : s = product of the first terms : product of the second terms.
- If 2A = 3B = 4C, then A : B : C = 1/2 : 1/3 : 1/4 = 6 : 4 : 3.
| Term | Meaning | Example |
|---|---|---|
| Duplicate ratio | a² : b² | 2 : 3 → 4 : 9 |
| Sub-duplicate ratio | √a : √b | 49 : 81 → 7 : 9 |
| Triplicate ratio | a³ : b³ | 2 : 3 → 8 : 27 |
| Inverse ratio | b : a | 2 : 3 → 3 : 2 |
Proportion
- If a : b :: c : d, then product of extremes = product of means: a × d = b × c.
- Fourth proportional to a, b, c is bc/a. Third proportional to a and b is b²/a (6 and 12 → 24).
- Mean proportional between a and b is √(ab) (4 and 16 → 8). If a, b, c are in continued proportion, b² = ac.
- Componendo and dividendo: if a/b = c/d then (a + b)/(a − b) = (c + d)/(c − d). Useful for questions like (x² + y²)/(x² − y²) = 17/8, which gives x²/y² = 25/9 and x : y = 5 : 3.
| Variation | Meaning | Relation |
|---|---|---|
| Direct | One rises, the other rises in the same ratio | x₁/y₁ = x₂/y₂ |
| Inverse | One rises, the other falls so the product stays fixed (men and days) | x₁y₁ = x₂y₂ |
| Joint | Depends on two or more quantities, e.g. A = kB² | Find k first |
Partnership
Sharing profit
Profit ratio = (capital × time) of A : (capital × time) of B : …
If capitals are invested for the same time, profit is shared in the ratio of capitals. If a partner adds or withdraws money during the year, add up capital × months for each period. A working partner first takes his salary or commission, and the rest is shared in the capital ratio.
Worked Example
P starts with Rs 6000; Q joins after 4 months with Rs 8000. Year's profit Rs 17,000. P : Q = 6000 × 12 : 8000 × 8 = 72 : 64 = 9 : 8, so P gets 9/17 × 17,000 = Rs 9000.
Mixtures and Alligation
Rule of alligation
Cheaper : dearer = (dearer price − mean price) : (mean price − cheaper price)
Water (price 0) mixed with milk at Rs 12/L to give Rs 8/L: water : milk = (12 − 8) : (8 − 0) = 1 : 2. The same rule works for concentrations: 30% and 70% acid mixed to 50% gives 1 : 1.
Repeated replacement
Pure liquid left = V × (1 − x/V)ⁿ
V = total volume, x = amount taken out and replaced each time, n = number of times. 20 L of milk, 4 L replaced with water twice: 20 × (16/20)² = 12.8 L milk, so milk : water = 12.8 : 7.2 = 16 : 9.
- Adding water to change a ratio: the milk stays the same. 40 L in 3 : 1 has 30 L milk; for 3 : 2 water must be 20 L, so add 10 L.
- Mixing equal volumes of two mixtures: add the fractions of each part. Unequal volumes: scale each mixture to its actual volume first.
Income and Expenditure Pattern
Incomes 5 : 4, expenditures 3 : 2, each saves Rs 1200: write 5x − 3y = 1200 and 4x − 2y = 1200. Solving gives x = 600, y = 600, so A's income is Rs 3000. This two-equation pattern appears often.
Common Mistakes
Ignoring time in partnership; adding the same number to both terms and expecting the ratio to stay the same; forgetting to make the common term equal before combining ratios; giving the working partner's salary out of his capital share instead of first.