Core ideas
A ratio a : b compares two quantities of the same kind; it does not tell you the actual values until you know one value, the total or a difference. Write quantities as ak and bk and find k. Target: 30 to 45 seconds.
Key formulas
| Term | Formula |
|---|---|
| Share of A in total S (ratio a : b : c) | S x a/(a + b + c) |
| Mean proportional of a, b | sqrt(ab) |
| Third proportional to a, b | b²/a |
| Fourth proportional to a, b, c | bc/a |
| Duplicate ratio of a : b | a² : b² |
| Sub-duplicate ratio | sqrt(a) : sqrt(b) |
| Componendo-dividendo | if a/b = c/d then (a + b)/(a - b) = (c + d)/(c - d) |
Combining ratios
If A : B = 2 : 3 and B : C = 4 : 5, make B equal: multiply the first by 4 and the second by 3. A : B : C = 8 : 12 : 15.
Shortcut (chain): A : B : C = (a1 x b2) : (b1 x b2) : (b1 x c2) for A : B = a1 : b1 and B : C = b2 : c2.
Typical question types
- Adding/subtracting the same number: (5x - 9)/(7x - 9) = 7/11; cross-multiply and solve.
- Incomes and expenditures: incomes 4x, 3x; expenditures 3y, 2y; savings equations give x and y.
- Coins: the coin-count ratio is not the value ratio. Multiply count ratio by face values: 5 : 6 : 7 coins of 50p, Re 1, Rs 2 gives value 2.5 : 6 : 14.
- Salary increments: new ratio = (old x (1 + r1)) : (old x (1 + r2)).
- Students joining/leaving: set up (7x - 40)/(5x + 20) = 4/3.
Worked example
Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Find the numbers.
(3x + 10)/(5x + 10) = 5/7, so 21x + 70 = 25x + 50, x = 5. Numbers: 15 and 25.
Worked example 2
Rs. 5,200 is divided among A, B, C in the ratio 1/2 : 1/3 : 1/4. Find A's share.
Multiply each fraction by the LCM 12: 6 : 4 : 3 (sum 13). A = 5,200 x 6/13 = Rs. 2,400 (B = 1,600, C = 1,200).
Common traps
- Dividing in the ratio of fractions without converting to whole numbers.
- Applying a ratio to the wrong total (for example, using the value ratio for coins).
- Forgetting that adding the same number to both terms changes the ratio.
Option trick: the answer must be a multiple of the ratio term. If boys : girls = 7 : 5, the total must be a multiple of 12 - eliminate other options instantly.