The ratio-unit method
A ratio a : b means the quantities are ak and bk for some k. Find k from one fact, then everything else follows.
Example. Split ₹1,200 in the ratio 2 : 3 : 5. There are 10 parts, each worth ₹120, so the shares are ₹240, ₹360 and ₹600.
Example. A class of 40 has boys and girls in the ratio 5 : 3. Eight parts make 40, so each part is 5 and there are 15 girls.
Combining ratios
To combine a : b and b : c, make the common term equal.
Example. a : b = 2 : 3 and b : c = 4 : 5. Make b equal to 12: a : b = 8 : 12 and b : c = 12 : 15, so a : b : c = 8 : 12 : 15.
Proportion
Four numbers are in proportion, a : b = c : d, when ad = bc.
Fourth proportional to a, b, c: d = bc/a. For 4, 6, 10: d = 60/4 = 15.
Mean proportional between a and b: √(ab). Between 9 and 16: √144 = 12.
Third proportional to a and b: b²/a.
Changing ratios
When the same amount is added to both terms, write an equation.
Example. Two numbers are 3x and 5x. Adding 9 to each gives the ratio 3 : 4, so 4(3x + 9) = 3(5x + 9). That gives 12x + 36 = 15x + 27 and x = 3. The numbers are 9 and 15.
Incomes, expenses and savings
Savings = income − expenditure. Write incomes as 4x, 3x and expenditures as 3y, 2y, then use the savings to form two equations.
Coins
Value per set × number of sets = total value. With 1-rupee, 50-paise and 25-paise coins in the ratio 2 : 3 : 4, one set is worth 2 + 1.50 + 1 = ₹4.50.
Partnership
Profit is shared in the ratio of capital × time.
Example. A invests ₹20,000 for 12 months and B invests ₹30,000 for 8 months. 240,000 : 240,000 = 1 : 1, so a profit of ₹9,000 is shared ₹4,500 each.
Shortcuts and tricks
The standard way is to write the quantities as 3x, 5x and so on and solve an equation. Many ratio questions can be answered from the parts alone.
Shares straight from the fraction
Example: ₹2,400 is divided in the ratio 3 : 5. Find the larger share.
Standard way: 3x + 5x = 2,400, so x = 300 and the larger share is 5 × 300 = ₹1,500.
Shortcut: the larger share is 5/8 of 2,400 = ₹1,500.
Chaining ratios by cross-multiplying
Example: a : b = 3 : 4 and b : c = 6 : 7. Find a : b : c.
Standard way: make b equal: a : b = 9 : 12 and b : c = 12 : 14, so a : b : c = 9 : 12 : 14.
Shortcut: a : b : c = (3 × 6) : (4 × 6) : (4 × 7) = 18 : 24 : 28 = 9 : 12 : 14.
Adding the same number keeps the difference
Example: two numbers are in the ratio 2 : 3. Adding 4 to each makes it 3 : 4. Find the numbers.
Standard way: (2x + 4)/(3x + 4) = 3/4 gives 8x + 16 = 9x + 12, so x = 4. The numbers are 8 and 12.
Shortcut: the difference is 1 part in both ratios, so the parts match directly: 2 → 3 is one part, and it equals 4. The numbers are 2 × 4 = 8 and 3 × 4 = 12.
Test the options
When a question gives several conditions, check each option against them instead of solving equations; usually only one option fits.
Exam tips
Never add ratios of different quantities without converting them to amounts.
Substituting the ratio values directly (A = 3, B = 4) is the fastest way to evaluate expressions like (2A + B) : (A + B).
In partnerships, a partner joining later contributes for fewer months.