Ratio and proportion is a short chapter with a big reach. Partnership, mixtures, ages, time and work, and data interpretation all depend on it. The formulas are simple, and most mistakes come from rushing, so this page gives you the rules, worked examples and practice questions.
What a ratio is
A ratio compares two quantities of the same kind. The ratio of a to b is written a : b, which is the fraction a/b. Ratios have no units, and they stay the same if you multiply or divide both terms by the same number, so 6 : 9 is the same as 2 : 3.
Core rules
- Compound ratio of a : b and c : d is ac : bd.
- Duplicate ratio of a : b is a² : b², and the sub-duplicate ratio is √a : √b.
- Proportion: if a : b = c : d, then ad = bc.
- Mean proportional between a and b is √(ab).
- Third proportional to a and b is b²/a.
- Fourth proportional to a, b and c is (b × c)/a.
Example 1. The mean proportional between 4 and 25 is √100 = 10. The third proportional to 4 and 6 is 36/4 = 9. The fourth proportional to 3, 4 and 9 is (4 × 9)/3 = 12.
Dividing a quantity in a ratio
To divide a quantity in the ratio a : b : c, add the parts (a + b + c), find the value of one part by dividing the total by that sum, and multiply.
Example 2. Divide ₹7,200 in the ratio 3 : 4 : 5. Total parts = 12, so one part = 600. The shares are ₹1,800, ₹2,400 and ₹3,000.
Combining two ratios
If A : B = 2 : 3 and B : C = 4 : 5, make the B value the same in both. Multiply the first ratio by 4 and the second by 3. Then A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.
Ratio and age problems
Write ages as multiples of one unknown. If the ages of A and B are in the ratio 3 : 5 and the sum is 40, then 8 parts = 40, one part = 5, and the ages are 15 and 25. Remember that when years pass, the same number is added to both ages, but the ratio changes.
Partnership
Profit is shared in the ratio of investment × time.
Example 3. A invests ₹20,000 for 12 months and B invests ₹30,000 for 8 months. A's share is 20,000 × 12 = 2,40,000 and B's share is 30,000 × 8 = 2,40,000, so the ratio is 1 : 1. On a profit of ₹15,000, each gets ₹7,500.
Alligation (mixtures)
When two ingredients with prices c (cheaper) and d (dearer) are mixed to get a mean price m, the quantities are in the ratio cheaper : dearer = (d − m) : (m − c).
Example 4. Rice at ₹40 per kg and ₹60 per kg is mixed to get a mixture worth ₹52 per kg. Ratio = (60 − 52) : (52 − 40) = 8 : 12 = 2 : 3.
Direct and inverse proportion
Two quantities are in direct proportion if one rises when the other rises, such as cost and number of items. They are in inverse proportion if one falls when the other rises, such as number of workers and days needed.
When quantities are added or removed
When you add or remove an amount, the ratio changes, so recalculate from the actual quantities. Example 5. A 50-litre mixture has milk and water in the ratio 3 : 2. Milk = 30 litres and water = 20 litres. If 10 litres of water is added, the new ratio is 30 : 30 = 1 : 1.
Income, expenditure and savings
Example 6. The incomes of A and B are in the ratio 5 : 4 and their expenditures are in the ratio 3 : 2. Each saves ₹1,600. Let the incomes be 5x and 4x and the expenditures 3y and 2y. Then 5x − 3y = 1,600 and 4x − 2y = 1,600. From the second equation y = 2x − 800. Substituting in the first gives 5x − 6x + 2,400 = 1,600, so x = 800. A's income is ₹4,000 and B's is ₹3,200. Check: expenditures are 2,400 and 1,600, so both save 1,600.
A quick tip on setting up
Always write the ratio as parts first (for example 3x and 5x), and form the equation from the information in the question. This habit works for ages, incomes, mixtures and partnerships alike.
Common mistakes to avoid
- Comparing quantities in different units. Convert them first.
- Reversing the order of the terms. a : b is not the same as b : a.
- Forgetting to match the common term when combining ratios.
- In alligation, subtracting in the wrong order.
Practice questions
- Divide 900 in the ratio 2 : 3 : 4.
- Find the fourth proportional to 5, 8 and 15.
- Find the mean proportional between 9 and 16.
- If A : B = 3 : 4 and B : C = 6 : 7, find A : B : C.
- In what ratio should rice at ₹30 per kg and ₹50 per kg be mixed to get ₹42 per kg?
Answers: 1) 200, 300, 400, 2) 24, 3) 12, 4) 9 : 12 : 14, 5) cheaper : dearer = 2 : 3.
How to practise
Do ten mixed questions a day, then see how ratios appear in profit and loss and time and work. Test yourself in our mock tests.
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