Quantitative Aptitude
Ratio & Proportion Guide & Practice
Learn ratio simplification, proportion, mean/third proportional, mixture and alligation with solved examples and free topic-wise mock tests. Explore dynamic solver blueprints, master fundamental equations, examine step-by-step solved examples, and practice with real exam-grade mock test sets.
1. Fundamentals & Definitions
- Ratio: A ratio is a comparison of two quantities of the same kind and in the same units. It is denoted by the symbol ':'. The ratio between two quantities 'a' and 'b' is written as a : b or a/b. In the ratio a : b, 'a' is called the antecedent and 'b' is called the consequent.
- Proportion: A proportion is an equality of two ratios. If a : b = c : d, then a, b, c, and d are said to be in proportion. This can also be written as a : b :: c : d. Here, 'a' and 'd' are called extremes, while 'b' and 'c' are called means. In a proportion, the product of extremes is equal to the product of means (a × d = b × c).
- Compounded Ratio: When two or more ratios are multiplied term by term, the resulting ratio is called a compounded ratio. For example, the compounded ratio of (a : b) and (c : d) is (ac : bd).
- Duplicate Ratio: The compounded ratio of a ratio with itself. The duplicate ratio of a : b is a² : b².
- Triplicate Ratio: The triplicate ratio of a : b is a³ : b³.
- Sub-duplicate Ratio: The sub-duplicate ratio of a : b is √a : √b.
- Sub-triplicate Ratio: The sub-triplicate ratio of a : b is ³√a : ³√b.
- Inverse Ratio (or Reciprocal Ratio): The inverse ratio of a : b is b : a.
- Continued Proportion: Three quantities a, b, c are said to be in continued proportion if a : b = b : c. In this case, b is called the mean proportional between a and c, and c is the third proportional to a and b.
- Mean Proportional: If a : b = b : c, then b is the mean proportional between a and c. The formula is b² = ac.
- Third Proportional: If a : b = b : c, then c is the third proportional to a and c. The formula is c = b²/a.
- Fourth Proportional: If a : b = c : d, then d is the fourth proportional to a, b, and c. The formula is d = (b × c)/a.
2. Core Concepts & Formulas
| Concept | Description | Formula / Rule |
|---|---|---|
| Product of Means and Extremes | In a proportion a : b :: c : d, the product of the mean terms equals the product of the extreme terms. | b × c = a × d |
| Continued Proportion | Three quantities a, b, c are in continued proportion. | a/b = b/c or b² = ac |
| Mean Proportional | The mean proportional x between two numbers a and c. | x = √(ac) |
| Third Proportional | The third proportional x to a and b. | x = b²/a |
| Fourth Proportional | The fourth proportional x to a, b, and c. | x = (b × c) / a |
| Componendo | If a/b = c/d, then (a+b)/b = (c+d)/d. | |
| Dividendo | If a/b = c/d, then (a-b)/b = (c-d)/d. | |
| Componendo and Dividendo | If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). | |
| Invertendo | If a/b = c/d, then b/a = d/c. | |
| Alternendo | If a/b = c/d, then a/c = b/d. | |
| Combining Ratios | If A : B = x : y and B : C = p : q. | To find A : B : C, make the 'B' term equal in both ratios. Multiply the first ratio by p and the second by y. Then A : B : C = xp : yp : yq. |
| Income, Expenditure, Savings | If incomes are in ratio a:b and expenditures are c:d. | Let incomes be ax and bx, expenditures be cy and dy. Then Savings = Income - Expenditure. Savings₁ = ax - cy, Savings₂ = bx - dy. |
| Mixtures (Alligation) | Adding or removing a quantity changes the ratio. | If a container has x liters of a mixture with A:B = m:n, the quantity of A is (m/(m+n)) * x and B is (n/(m+n)) * x. If y liters of B are added, the new ratio is (m/(m+n)) * x : (n/(m+n)) * x + y. |
Solved Examples
1Example 1 (Easy)
Question: The salaries of A, B, and C are in the ratio 2 : 3 : 5. If their salaries were increased by 15%, 10%, and 20% respectively, what will be the new ratio of their salaries?
2Example 2 (Moderate)
Question: A bag contains Rs. 410 in the form of Rs. 5, Rs. 2, and Re. 1 coins. The number of coins are in the ratio 4 : 6 : 9. Find the number of Rs. 2 coins.
3Example 3 (Hard)
Question: In a mixture of 60 litres, the ratio of milk and water is 2 : 1. If this ratio is to be 1 : 2, then what is the quantity of water to be further added?