This chapter explores the relationship between quantities through comparison, emphasizing how they scale relative to one another. A firm understanding of ratio and proportion is not only essential for mathematics but also forms a critical base for subjects like physics, chemistry, economics, and everyday problem-solving.
This comprehensive guide focuses on clearly explaining the concepts of ratio, proportion, continued proportion, and their applications using relevant mathematical principles. Through simplified explanations, solved examples, and application-oriented thinking, students can develop confidence and accuracy in this chapter.
What is a Ratio? (অনুপাত কী?)
A ratio is a quantitative relationship between two numbers that shows how many times one value contains or is contained within the other. It is represented in the form a : b or as a fraction ab, where a and b are quantities of the same kind and b ≠ 0.
For example:
- If the number of boys to girls in a class is 3 and 2, the ratio is written as 3:2.
- This means for every 3 boys, there are 2 girls.
Key Features of Ratio:
- Both quantities must be in the same unit.
- Ratios can be simplified just like fractions.
- It expresses relative size, not absolute value.
Example:
Simplify the ratio 60 cm to 2 m.
Convert 2 m = 200 cm
⇒ Ratio = 60 : 200 = 3 : 10
Types of Ratios (অনুপাতের প্রকারভেদ)
Ratios can be classified as:
- Simple Ratio (সরল অনুপাত): Comparison between two quantities.
Example: 4:5 - Compound Ratio (যৌগিক অনুপাত): The product of two or more ratios.
Example: If A:B = 2:3 and C:D = 4:5, then compound ratio = (2×4):(3×5) = 8:15 - Duplicate Ratio (দ্বিগুণ অনুপাত): Ratio of the squares of two quantities.
If a:b = 3:4, then duplicate ratio = 9:16 - Triplicate Ratio (ত্রিগুণ অনুপাত): Ratio of the cubes of two quantities.
If a:b = 2:5, triplicate ratio = 8:125
What is Proportion? (সমানুপাত কী?)
A proportion is an equation stating that two ratios are equal. If a : b = c : d, then a, b, c, and d are in proportion. It is written as a : b :: c : d or ab = cd.
Here, a and d are called extremes and b and c are means.
Basic Rule:
In a proportion, the product of the means equals the product of the extremes.
So, a × d = b × c
Example:
Is 4, 8, 15, and 30 in proportion?
Check: 4 × 30 = 120, 8 × 15 = 120 ⇒ Hence, they are in proportion.
Continued Proportion (ক্রমাগত সমানুপাত)
Three quantities a, b, c are in continued proportion if a : b = b : c.
This implies b² = ac
Example:
Check if 2, 4, 8 are in continued proportion.
4² = 16, 2 × 8 = 16 ⇒ Yes, they are in continued proportion.
Properties of Proportion (সমানুপাতের গুণাবলী)
Understanding the properties of proportion helps in solving problems efficiently:
- Invertendo Rule:
If ab = cd, thenba = dc - Alternendo Rule:
If ab = cd, then ac = bd - Componendo and Dividendo:
If ab = cd, then
(a+b)(a-b) = (c+d)(c-d) - Subtraction or Addition of Ratios:
If a : b = c : d, then
(ab)b = (cd)d
These identities are often applied in competitive exams and algebraic simplifications.
Application of Ratio and Proportion (অনুপাত ও সমানুপাতের প্রয়োগ)
This chapter has wide application in solving practical and theoretical problems:
- Mixing Problems:
Example: In what ratio should 30% acid be mixed with 50% acid to get a solution of 40% acid? - Scale Drawings:
Maps, blueprints, and model designs use ratios for scaling real-life objects. - Financial Distribution:
Dividing money or goods in a particular ratio.
Example: Divide ₹500 in the ratio 3:2 ⇒ ₹300 and ₹200. - Speed, Time, and Distance:
Speed and time are inversely proportional when the distance is constant. - Work and Time:
If A can complete a task in 5 days and B in 10, their work ratio is 2:1.
Important Formulas and Conversions
- Ratio of two quantities: a : b = ab
- Proportion: a : b = c : d ⇒ a × d = b × c
- Continued proportion: b² = a × c
- Compound ratio: (a : b) × (c : d) = ac : bd
- Direct proportion: a ∝ b ⇒ ab = constant
- Inverse proportion: a ∝ 1b ⇒ a × b = constant
Common Mistakes to Avoid
- Always ensure quantities in a ratio have the same unit before comparing.
- Don’t confuse between ratio and fraction.
- In a proportion, double-check the cross multiplication.
- Be cautious with unit conversions in application-based problems.
Sample Problems
Example 1:
Divide ₹720 between A and B in the ratio 5:7.
Total parts = 5 + 7 = 12
A’s share = (512) × 720 = ₹300
B’s share = (712) × 720 = ₹420
Example 2:
If a:b = 3:4 and b:c = 2:5, find a:b:c
⇒ a:b = 3:4, b:c = 2:5
Make b common: LCM of 4 and 2 = 4
So, a:b = 3:4, b:c = 4:10
⇒ a:b:c = 3:4:10
Conclusion
The chapter “Ratio and Proportion” in Class 10 Mathematics builds essential skills in comparing and evaluating quantities. Mastering this topic equips students to handle a variety of real-world and mathematical problems. With strong conceptual understanding, systematic practice, and familiarity with core formulas and properties, students can solve questions with confidence and precision.
This topic also strengthens the groundwork for higher-level math and practical applications in science, economics, and daily life—making it one of the most relevant chapters in the syllabus.