1. What Is a Ratio?
A ratio compares two quantities of the same kind by division. For a and b, where b is not zero, a : b = a/b. The first term is the antecedent and the second is the consequent. Order matters: 2 : 3 is different from 3 : 2. Convert quantities to the same unit before comparing them. The common units cancel in the final ratio.
Example: 2 metres : 50 centimetres = 200 : 50 = 4 : 1, not 2 : 50.
2. Simplest and Equivalent Ratios
Divide both terms by their Highest Common Factor (HCF) to express a ratio in its simplest form. Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio.
For example, 18 : 30 = 3 : 5. Equivalent ratios include 6 : 10 and 9 : 15.
3. What Is a Proportion?
Two ratios form a proportion when they are equal.
a : b :: c : d ⇒ a/b = c/d ⇒ a × d = b × c, where b and d are non-zero.
The first and fourth terms, a and d, are the extremes. The middle terms, b and c, are the means. The product of the extremes equals the product of the means.
4. Direct and Inverse Proportion
Direct proportion: If one quantity is multiplied by k and the other also changes by the same factor k, their ratio remains constant. For a fixed unit price, quantity and total cost are directly proportional. Formula: y₁/x₁ = y₂/x₂.
Inverse proportion: If one quantity is multiplied by k and the other is divided by k, their product remains constant. For the same job and equally efficient workers, number of workers and number of days are inversely proportional. Formula: x₁y₁ = x₂y₂.
Important: Two quantities moving in opposite directions does not by itself prove inverse proportion; their product must stay constant.
5. Important Formulas and Shortcuts
Application Formula or Method ------------------------- --------------------------------------------- Ratio a : b = a/b Simplest ratio Divide both terms by their HCF Proportion a : b :: c : d ⇒ ad = bc Divide S in a : b First share = Sa/(a+b); second = Sb/(a+b) Divide S in a : b : c Shares = Sa/(a+b+c), Sb/(a+b+c), Sc/(a+b+c) Direct proportion y₁/x₁ = y₂/x₂ Inverse proportion x₁y₁ = x₂y₂ Combine a : b and b : c Make the common term b equal using LCM
6. Worked Example 1 – Simplifying a Ratio
Question: Simplify 24 : 36.
Step 1: HCF of 24 and 36 = 12. Step 2: 24 ÷ 12 : 36 ÷ 12 = 2 : 3. Answer: 2 : 3.
7. Worked Example 2 – Converting Units
Question: Find the ratio 2.5 metres : 75 centimetres.
Step 1: 2.5 metres = 250 centimetres. Step 2: Ratio = 250 : 75; HCF = 25. Step 3: 250/25 : 75/25 = 10 : 3. Answer: 10 : 3.
8. Worked Example 3 – Dividing an Amount
Question: Divide ₹840 in the ratio 3 : 4.
Step 1: Total parts = 3 + 4 = 7. Step 2: One part = 840 ÷ 7 = ₹120. Step 3: First share = 3 × 120 = ₹360; second share = 4 × 120 = ₹480. Check: 360 + 480 = 840. Answer: ₹360 and ₹480.
9. Worked Example 4 – Finding a Missing Term
Question: If 4 : 7 :: 12 : x, find x.
Step 1: 4/7 = 12/x. Step 2: Cross-multiply: 4x = 7 × 12 = 84. Step 3: x = 84/4 = 21. Answer: 21.
10. Worked Example 5 – Checking a Proportion
Question: Are 6 : 9 and 10 : 15 in proportion?
Step 1: Product of extremes = 6 × 15 = 90. Step 2: Product of means = 9 × 10 = 90. Answer: Yes. Both cross-products are equal.
11. Worked Example 6 – Direct Proportion
Question: Five kilograms of rice cost ₹240. What is the cost of eight kilograms at the same rate?
Step 1: Cost per kilogram = 240 ÷ 5 = ₹48. Step 2: Cost of 8 kg = 8 × 48 = ₹384. Answer: ₹384. Cost rises in the same ratio as quantity when unit price is fixed.
12. Worked Example 7 – Inverse Proportion
Question: Eight equally efficient workers finish a job in 15 days. How many days will 12 workers take?
Step 1: Workers × Days = Constant. Step 2: 8 × 15 = 12 × x. Step 3: x = 120/12 = 10. Answer: 10 days. Assume the job, daily working hours and individual efficiency remain unchanged.
13. Worked Example 8 – Combining Ratios
Question: A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.
Step 1: Make the common term B equal. LCM of 3 and 4 = 12. Step 2: Multiply A : B by 4 → 8 : 12. Step 3: Multiply B : C by 3 → 12 : 15. Answer: A : B : C = 8 : 12 : 15.
14. Worked Example 9 – Dividing into Three Shares
Question: Divide ₹1,200 in the ratio 2 : 3 : 5.
Step 1: Total parts = 2 + 3 + 5 = 10. Step 2: One part = ₹1,200/10 = ₹120. Step 3: Shares = ₹240, ₹360 and ₹600. Check: 240 + 360 + 600 = 1,200. Answer: ₹240, ₹360 and ₹600.
15. Worked Example 10 – A Changed Ratio
Question: Two numbers are in the ratio 3 : 5, and their sum is 64. If 8 is added to each number, what is the new ratio?
Step 1: Total parts = 3 + 5 = 8; one part = 64/8 = 8. Step 2: Original numbers = 24 and 40. Step 3: New numbers = 32 and 48. Step 4: 32 : 48 = 2 : 3. Answer: 2 : 3.
16. Examination Shortcuts and Common Mistakes
1. Units: Convert metres to centimetres or rupees to paise before comparing. 2. Order: Boys : girls is not the same as girls : boys. 3. Sharing: Divide the total by the sum of the ratio terms to find one part. 4. Proportion: Use ad = bc for a : b :: c : d. 5. Direct: At a fixed unit price, quantity and total cost are directly proportional. 6. Inverse: For equal workers and the same job, workers and days are inversely proportional. 7. Combining ratios: Equalise the common term using the LCM. 8. Frequent error: If a : b = 3 : 5, then a/(a+b) = 3/8, not 3/5.
17. One-Minute Revision
Ratio: a : b. Equivalent ratios: 2 : 3 = 4 : 6. Proportion: a : b :: c : d. Cross-product rule: ad = bc. Direct proportion: Ratio remains constant. Inverse proportion: Product remains constant. Sharing: Total × relevant parts / total parts.
18. Textbook References and Book-Back Progress
Primary textbook: Tamil Nadu Samacheer Kalvi Class 6 Mathematics, Term 1, Chapter 3 – Ratio and Proportion. Exercises identified: 3.1, 3.2, 3.3, 3.4 and 3.5. Supplementary textbook: Class 7 Mathematics, Term 1, Chapter 4 – Direct and Inverse Proportion; Exercises 4.1, 4.2 and 4.3.
------------------------------------------------------------------------ Exercise Items Identified in Original-Verified Completed Independent Guides --------------------- --------------------- ---------------------------- Class 6 – 3.1 11 main numbered 0 questions; several subquestions
Class 6 – 3.2 12 main numbered 0 questions; several subquestions
Class 6 – 3.3 9 main numbered 0 questions; several subquestions
Class 6 – 3.4 Full count not yet 0 verified
Class 6 – 3.5 Full count not yet 0 verified
Class 7 – 4.1, 4.2, Full counts not yet 0 4.3 verified ------------------------------------------------------------------------
Source status: The exercise identifiers were located in independent textbook guides, but the complete original 2026–27 government textbook exercises, diagrams and subquestions are source not yet verified. All ten worked examples above are original instructional examples, not verified TNPSC previous-year questions or original textbook questions. No book-back items are marked complete today.
Plan: After original-source verification, every relevant numbered question and subquestion in Exercises 3.1–3.5 and 4.1–4.3 will be tracked and divided into separately numbered 7:00 PM batches (1/N, 2/N, etc.). Unverified items will not be presented as verified book-back questions.
19. Spaced Revision Schedule
Interval Date ---------- ----------------- +1 day 22 October 2026 +3 days 24 October 2026 +7 days 28 October 2026 +14 days 4 November 2026
Today’s cumulative review: Day 08 Time and Work (+1); Day 06 Aptitude topic (+3, as recorded in the earlier lesson); Day 02 Aptitude topic (+7, as recorded in the earlier lesson).
20. ETN Push Notification
Title: 🔢 TNPSC Group IV | Ratio and Proportion!
Message: 📚 Master ratios, proportions, direct and inverse variation, and dividing amounts with 10 fully worked examples. Read Day 09 Aptitude on ETN News!
