60-second revision
- Combine A:B and B:C by making B equal: A:B = 3:5 and B:C = 2:7 give A:B:C = 6:10:35.
- Share of A when X is divided in a:b:c = X × a/(a + b + c).
- Fourth proportional to a, b, c = bc/a; third proportional to a, b = b²/a.
- Mean proportional between a and b = square root of ab.
- Duplicate ratio of a:b = a²:b²; sub-duplicate = sqrt a : sqrt b; triplicate = a³:b³.
- Compound ratio of a:b and c:d = ac:bd.
- If a/b = c/d then (a + b)/(a - b) = (c + d)/(c - d) (componendo-dividendo).
What a ratio means
A ratio a:b compares two quantities of the same kind: a:b = a/b. Multiplying or dividing both terms by the same number does not change it (12:18 = 2:3). Always reduce to lowest terms and use the same units on both sides (40 paise : Rs 2 = 40:200 = 1:5).
Combining ratios
To join A:B and B:C, make the common term B equal. Example: A:B = 3:5 and B:C = 2:7. Multiply the first by 2 and the second by 5: A:B = 6:10 and B:C = 10:35, so A:B:C = 6:10:35.
Dividing an amount in a ratio
Add the parts, then take each share as a fraction of the total. Rs 4,500 in the ratio 2:3:4 has 9 parts of Rs 500, so the shares are Rs 1,000, Rs 1,500 and Rs 2,000.
If you know one share or the difference between two shares, find the value of one part first. Example: in 4:7, the difference of shares is Rs 600, so 3 parts = 600 and one part = Rs 200; shares are Rs 800 and Rs 1,400.
Proportion
Four numbers a, b, c, d are in proportion (a:b = c:d) when ad = bc (product of extremes = product of means).
| Term | Formula | Example |
|---|---|---|
| Fourth proportional to a, b, c | d = bc/a | to 4, 10, 6: 60/4 = 15 |
| Third proportional to a, b | c = b²/a | to 9, 12: 144/9 = 16 |
| Mean proportional between a, b | square root of ab | between 4 and 25: sqrt 100 = 10 |
Special ratios
| Name | Of a:b | Example |
|---|---|---|
| Duplicate | a²:b² | 3:4 gives 9:16 |
| Sub-duplicate | sqrt a : sqrt b | 49:64 gives 7:8 |
| Triplicate | a³:b³ | 3:4 gives 27:64 |
| Sub-triplicate | cube root a : cube root b | 8:27 gives 2:3 |
| Inverse | b:a (or 1/a : 1/b) | 3:4 gives 4:3 |
| Compound of a:b and c:d | ac:bd | 3:5 and 10:9 gives 30:45 = 2:3 |
Changing a ratio by adding or subtracting
Write the numbers as ax and bx, apply the change, and set up the new ratio. Example: two numbers are in the ratio 3:5. If 4 is subtracted from each, the ratio becomes 1:3. Then (3x - 4)/(5x - 4) = 1/3, so 9x - 12 = 5x - 4 and x = 2. The numbers are 6 and 10.
Income, expenditure and savings
Income = expenditure + savings. Use one variable for incomes and another for expenditures. Example: incomes are 5:3, expenditures 9:5, and each person saves Rs 2,000. Then 5x - 9y = 2,000 and 3x - 5y = 2,000. Solving gives y = 2,000 and x = 4,000, so the incomes are Rs 20,000 and Rs 12,000.
Coins and notes
When coins are in a ratio, find the value of one "set". Example: Rs 1, 50 paise and 25 paise coins in the ratio 2:3:4 total Rs 90. One set is worth 2 × 1 + 3 × 0.5 + 4 × 0.25 = Rs 4.50, so there are 20 sets: 40, 60 and 80 coins.
Variation
- Direct: x is proportional to y when x/y stays constant (cost and quantity at a fixed rate).
- Inverse: x is inversely proportional to y when xy stays constant (workers and days for the same job, speed and time for the same distance).
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