TNPSC Group IV: Ratio, Proportion, Direct and Inverse Variation — Complete Short Notes, Formulas, Shortcuts and 8 Worked Examples
Today's topic is Ratio and Proportion, which is explicitly listed in the official TNPSC Group IV Aptitude syllabus. Yesterday's lesson covered Simplification. This lesson does not complete the entire Aptitude syllabus; interest, mensuration, time and work, logical reasoning, puzzles and number series remain to be studied.
1. What Is a Ratio? A ratio compares two quantities of the same kind by division. The ratio of a to b is written a:b or a/b, where b ≠ 0. • a is the antecedent (first term); b is the consequent (second term). • Order matters: 2:3 is different from 3:2. • Convert both quantities to the same unit before comparing them. • A ratio of quantities measured in the same unit has no unit. • Simplify 12:18 by dividing both terms by their HCF, 6: the answer is 2:3.
2. Simplifying Ratios Whole numbers: Divide both terms by their common factor. Decimals: Multiply both terms by 10, 100, etc., to remove decimal places. Fractions: Multiply both terms by a suitable common denominator (often the LCM). A ratio remains unchanged if both terms are multiplied or divided by the same non-zero number.
Worked Example 1 — Convert Units Find the ratio 2.4 metres : 60 centimetres. 2.4 m = 240 cm. Therefore, 240:60 = 4:1. Answer: 4:1. Never compare 2.4 directly with 60 without converting units.
Worked Example 2 — Decimal Ratio 0.75 : 1.25 = 75:125 = 3:5 (divide by 25). Answer: 3:5.
3. Dividing an Amount in a Given Ratio To divide a total T in the ratio a:b: First share = T × a/(a+b). Second share = T × b/(a+b). For three shares a:b:c, the total number of parts is a+b+c.
Worked Example 3 — Sharing Money Divide ₹1,680 in the ratio 3:4. Total parts = 3+4 = 7. First share = 1680 × 3/7 = ₹720. Second share = 1680 × 4/7 = ₹960. Check: 720+960 = 1680. Answer: ₹720 and ₹960.
4. What Is a Proportion? Two ratios form a proportion when they are equal. If a:b :: c:d, then a/b = c/d, so a×d = b×c (cross-multiplication). • a and d are called the extremes. • b and c are called the means. • Product of extremes = product of means. • Fourth proportional: if a:b = c:x, then x = (b×c)/a (a ≠ 0). • In continued proportion, a:b = b:c, so b² = ac; b is the mean proportional.
Worked Example 4 — Missing Term If x:18 = 5:6, then x/18 = 5/6. 6x = 18×5 = 90, so x = 15. Answer: x = 15.
5. Combining Two Ratios If A:B = 4:7 and B:C = 14:9, make the common term B equal in both ratios. Multiply the first ratio by 2: A:B = 8:14. Combine them to get A:B:C = 8:14:9.
Worked Example 5 — Three People Given A:B = 4:7 and B:C = 14:9, find A:B:C. A:B = 8:14; B:C = 14:9. Answer: 8:14:9.
6. Direct Proportion Two quantities are in direct proportion if one increases or decreases by the same factor as the other. Example: Number of notebooks and total cost, when the price per notebook is fixed. x/y is constant; x₁/y₁ = x₂/y₂. Shortcut: Find the value of one unit, then multiply by the required number of units.
Worked Example 6 — Direct Proportion Eight notebooks cost ₹120. What is the cost of 14 notebooks at the same rate? One notebook costs 120/8 = ₹15. Fourteen notebooks cost 14×15 = ₹210. Answer: ₹210.
7. Inverse Proportion Two quantities are in inverse proportion if increasing one by a factor decreases the other by the same factor. Example: Number of equally efficient taps and the time required to fill the same tank. x×y is constant; x₁y₁ = x₂y₂. Important: The total job and individual efficiency must remain unchanged.
Worked Example 7 — Inverse Proportion Six equally efficient taps fill a tank in 10 hours. How long will 15 taps take? 6×10 = 15×t. t = 60/15 = 4 hours. Answer: 4 hours.
8. Changing Ratios and Mixtures If two ingredients are in the ratio a:b, there are a+b total parts. When one ingredient is added or removed, calculate the new actual quantities before finding the new ratio. Do not simply change the old ratio terms without checking the quantities.
Worked Example 8 — Milk and Water A 28-litre mixture has milk:water = 5:2. Four litres of water are added. Find the new ratio. Total parts = 7; one part = 28/7 = 4 litres. Milk = 5×4 = 20 litres; water = 2×4 = 8 litres. New water quantity = 8+4 = 12 litres. New ratio = 20:12 = 5:3. Answer: 5:3.
9. Important Formulas and Shortcuts • If a:b = m:n and a+b = S, then a = Sm/(m+n), b = Sn/(m+n). • If a:b = m:n and a−b = D, where m>n, then a = Dm/(m−n), b = Dn/(m−n). • If a:b = c:d, then ad = bc. • If a:b = m:n, write a = km and b = kn for a common multiplier k. • If a:b = m:n, then a/b = m/n, but a/(a+b) = m/(m+n). • A ratio of 2:3 means the first share is 2/5 of the total, NOT 2/3. • Direct proportion: More → more, for a constant unit price. • Inverse proportion: More → less, for a fixed job and equal efficiency.
10. Common Exam Mistakes • Comparing metres and centimetres without converting them to the same unit. • Assuming that the first share of a 3:5 ratio is 3/5 of the total; it is 3/8. • Reversing the order of the ratio terms. • Using an inverse-proportion formula for a direct-proportion problem. • Combining A:B and B:C without making their B terms equal. • Incorrect cross-multiplication in a proportion.
11. One-Minute Revision Ratio = Comparison of two quantities of the same kind. Proportion = Equality of two ratios. a:b :: c:d → ad = bc. To split a total in the ratio a:b, use a/(a+b) and b/(a+b). Direct proportion → x/y remains constant. Inverse proportion → xy remains constant. Always standardise units first.
12. Textbook and Book-Back Progress Relevant textbook: Tamil Nadu Samacheer Kalvi Class 6 Mathematics, Term 1, Chapter 3 — Ratio and Proportion. Older textbook copies and secondary references identify Exercises 3.1, 3.2, 3.3 and 3.4. However, the complete original 2026–27 government textbook exercises, including every sub-question, have not yet been directly verified. Verified book-back questions completed: 0. Total book-back questions: Source not yet fully verified; no total is claimed. All eight worked examples in this lesson are original instructional examples. None is labelled a verified TNPSC PYQ or book-back question. Topics still pending in Aptitude: Simple and Compound Interest, Area, Volume, Time and Work, Logical Reasoning, Puzzles, Dice, Visual Reasoning, Alphanumeric Reasoning and Number Series.
13. Study Plan and Spaced Revision Solve all eight examples again without looking at the answers. First identify whether each word problem uses direct or inverse proportion. Planned 1/3/7/14-day revision dates: 17, 19, 23 and 30 October 2026. Tonight's 7 PM set of 15 headline MCQs will include two Ratio and Proportion questions.
