Ratio and Proportion
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Important Formulas & Concepts
Study MaterialRatio and Proportion
Ratio and Proportion is one of the most fundamental and important arithmetic topics in Quantitative Aptitude. Questions from this chapter are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement examinations.
This chapter helps candidates understand comparison between quantities and relationships among numbers. It is widely used in:
- Partnership problems
- Mixture and Allegation
- Time and Work
- Percentage calculations
- Age problems
- Speed and distance concepts
Why Ratio and Proportion is Important?
- Highly scoring arithmetic topic.
- Foundation for advanced aptitude chapters.
- Frequently asked in competitive exams.
- Improves analytical and logical thinking.
- Used extensively in real-life calculations.
What is Ratio?
When two quantities of the same kind are compared, the comparison is called a ratio.
Ratio shows how many times one quantity contains another quantity.
Ratio of x and y = x : y or x/y
In a ratio:
- x is called the Antecedent
- y is called the Consequent
Example:
Ratio of 10 and 15:
= 10 : 15
= 2 : 3
✔ Quantities compared in a ratio must have the same units.
Important Properties of Ratio
- Ratio has no unit.
- Ratio can be simplified like fractions.
- If both terms are multiplied or divided by the same number, the ratio remains unchanged.
- Ratios are compared using fractions.
Types of Ratio
1. Duplicate Ratio
If the ratio of two numbers is x : y, then the ratio of their squares is called duplicate ratio.
Duplicate Ratio of x : y = x² : y²
Example:
Duplicate ratio of 3 : 5
= 9 : 25
2. Sub-Duplicate Ratio
The ratio of square roots of two quantities is called sub-duplicate ratio.
Sub-duplicate Ratio of x : y = √x : √y
Example:
Sub-duplicate ratio of 16 : 81
= 4 : 9
3. Triplicate Ratio
The ratio of cubes of two quantities is called triplicate ratio.
Triplicate Ratio of x : y = x³ : y³
Example:
Triplicate ratio of 2 : 3
= 8 : 27
4. Sub-Triplicate Ratio
The ratio of cube roots of two quantities is called sub-triplicate ratio.
Sub-triplicate Ratio of x : y = ∛x : ∛y
5. Inverse Ratio
When antecedent and consequent are interchanged, the resulting ratio is called inverse ratio.
Inverse Ratio of x : y = y : x
Example:
Inverse ratio of 4 : 7
= 7 : 4
6. Compound Ratio
When two or more ratios are multiplied together, the resulting ratio is called compound ratio.
If a : b and c : d, then compound ratio = ac : bd
Example:
Compound ratio of 2 : 3 and 5 : 7
= 10 : 21
Comparison of Ratios
Ratios are compared by converting them into fractions.
If a : b and c : d are two ratios:
a : b > c : d if ad > bc
a : b < c : d if ad < bc
a : b = c : d if ad = bc
Example:
Compare 5 : 8 and 9 : 14
5 × 14 = 70
8 × 9 = 72
Since 70 < 72
Therefore:
5 : 8 < 9 : 14
What is Proportion?
Equality of two ratios is called proportion.
a : b :: c : d
or
a/b = c/d
Here:
- a and d are called Extremes
- b and c are called Means
Fundamental Property of Proportion
Product of Means = Product of Extremes
bc = ad
Example:
2 : 5 :: 8 : 20
5 × 8 = 2 × 20
40 = 40
Componendo and Dividendo
If:
a/b = c/d
Then:
(a + b)/(a − b) = (c + d)/(c − d)
This concept is widely used in advanced ratio and proportion problems.
Direct Proportion
Two quantities are directly proportional if increase in one causes increase in another.
x ∝ y
x = ky
Example:
More workers → More work completed
Inverse Proportion
Two quantities are inversely proportional if increase in one causes decrease in another.
x ∝ 1/y
xy = k
Example:
More workers → Less time required
Important Formulae Summary
| Concept | Formula |
|---|---|
| Ratio | x : y |
| Duplicate Ratio | x² : y² |
| Sub-duplicate Ratio | √x : √y |
| Triplicate Ratio | x³ : y³ |
| Sub-triplicate Ratio | ∛x : ∛y |
| Inverse Ratio | y : x |
| Compound Ratio | ac : bd |
| Proportion | a : b :: c : d |
| Means = Extremes | bc = ad |
Common Mistakes in Ratio & Proportion
- Comparing quantities with different units.
- Incorrect simplification of ratios.
- Errors in cross multiplication.
- Confusing direct and inverse proportion.
- Mistakes in compound ratio calculations.
Important Exam Tips
- Always convert quantities into same units before comparison.
- Simplify ratios to smallest form.
- Use cross multiplication in proportion problems.
- Memorize duplicate and triplicate ratio formulas.
- Practice direct and inverse variation problems regularly.
- Use fraction method for faster calculations.
- Verify calculations carefully in ratio comparisons.
Ratio and Proportion is one of the most important arithmetic chapters in Quantitative Aptitude. Strong understanding of ratios, proportions, variations, and comparison techniques helps candidates solve competitive exam questions quickly and accurately.