How to Use This Page
This page provides step-by-step solutions for every exercise in Chapter 3: Ratio and Proportion of the TN Samacheer Kalvi Class 6 Maths textbook (Term 1). Work through each solution on your own first, then check here. Every calculation is shown in full so you can follow the reasoning, not just the final answer.
Key Concepts (Quick Recap)
Ratio
A ratio compares two quantities of the same kind using division. If there are 20 girls and 25 boys in a class, the ratio of girls to boys is written as 20 : 25, which simplifies to 4 : 5.
- The first term is the antecedent, the second is the consequent.
- A ratio has no units — it is a pure number comparison.
- Order matters: 3 : 5 is not the same as 5 : 3.
Proportion
When two ratios are equal, they are said to be in proportion. If a : b = c : d, we write a : b :: c : d.
- Cross-product rule: a × d = b × c
- The terms a and d are called extremes; b and c are called means.
- In a proportion, product of extremes = product of means.
Unitary Method
1. Find the value of one unit first. 2. Multiply to get the value of the required number of units.
Direct Proportion
When one quantity increases, the other increases in the same ratio (or both decrease together).
- If 5 books cost ₹100, then 8 books cost ₹160 (more books → more cost).
Inverse Proportion
When one quantity increases, the other decreases in the same ratio.
- If 10 workers finish a job in 6 days, then 20 workers finish it in 3 days (more workers → fewer days).
Exercise 3.1 Solutions — Ratio
Question 1: Find the ratio of the following
Answer
(a) 15 to 20
Step 1: Write as a ratio → 15 : 20
Step 2: Find HCF of 15 and 20 → HCF = 5
Step 3: Divide both terms by 5
Answer3 : 4
(b) 24 to 36
Step 1: Write as a ratio → 24 : 36
Step 2: HCF of 24 and 36 = 12
Step 3: Divide both terms by 12
Answer2 : 3
(c) 75 holidays in 365 days (Additional Q3)
Step 1: Ratio of holidays to total days = 75 : 365
Step 2: HCF of 75 and 365 = 5
Step 3: Divide both terms by 5
Answer15 : 73
Question 2: 20 girls and 25 boys in a class. Find (a) Girls:Boys (b) Girls:Total (Additional Q4)
Answer
(a) Girls : Boys
Step 1: Number of girls = 20, Number of boys = 25
Step 2: Ratio = 20 : 25
Step 3: HCF of 20 and 25 = 5. Divide by 5.
Answer4 : 5
(b) Girls : Total students
Step 1: Total students = 20 + 25 = 45
Step 2: Ratio = 20 : 45
Step 3: HCF of 20 and 45 = 5. Divide by 5.
Answer4 : 9
Question 3: Simplify 20 : 5 (Additional Q5)
Answer
Step 1: Ratio = 20 : 5
Step 2: HCF of 20 and 5 = 5
Step 3: Divide both by 5
Answer4 : 1
Question 4: Moni walks 6 km/hr, Vimala walks 4 km/hr. Ratio of speeds (Additional Q2)
Answer
Step 1: Moni's speed = 6 km/hr, Vimala's speed = 4 km/hr
Step 2: Ratio = 6 : 4
Step 3: HCF of 6 and 4 = 2. Divide by 2.
Answer3 : 2
Question 5: Find equivalent ratios of 3 : 5
Answer
To find equivalent ratios, multiply or divide both terms by the same number.
Step 1: Multiply by 2 → 3 × 2 : 5 × 2 = 6 : 10
Step 2: Multiply by 3 → 3 × 3 : 5 × 3 = 9 : 15
Step 3: Multiply by 4 → 3 × 4 : 5 × 4 = 12 : 20
Answer6 : 10, 9 : 15, 12 : 20 are all equivalent to 3 : 5
Exercise 3.2 Solutions — Proportion
Question 1: Complete 3 : 5 :: ___ : 10 (Additional Q1)
Answer
Step 1: Let the missing term be x. So, 3 : 5 :: x : 10
Step 2: By cross-product rule: 3 × 10 = 5 × x
Step 3: 30 = 5x
Step 4: x = 30 ÷ 5 = 6
Answer3 : 5 :: 6 : 10
Question 2: Check whether 3 : 2 and 30 : 20 are in proportion (Additional Q18)
Answer
Step 1: Write as 3 : 2 :: 30 : 20
Step 2: Apply cross-product rule: Product of extremes = Product of means
Step 3: a × d = 3 × 20 = 60
Step 4: b × c = 2 × 30 = 60
Step 5: Since 60 = 60, the cross-products are equal.
AnswerYes, 3 : 2 and 30 : 20 are in proportion because ad = bc = 60.
Question 3: 16 : 24 :: x : 30, find x (Additional Q9)
Answer
Step 1: Set up the proportion: 16 : 24 :: x : 30
Step 2: Cross-multiply: 16 × 30 = 24 × x
Step 3: 480 = 24x
Step 4: x = 480 ÷ 24 = 20
Answerx = 20
Question 4: Fill in the blanks (Additional Q12)
Answer
(i) 30 : 24 :: 80 : ___
Step 1: Let the missing term be x. So, 30 : 24 :: 80 : x
Step 2: Cross-multiply: 30 × x = 24 × 80
Step 3: 30x = 1920
Step 4: x = 1920 ÷ 30 = 64
Answer30 : 24 :: 80 : 64
(ii) 5 : 6 :: 125 : ___
Step 1: Let the missing term be x. So, 5 : 6 :: 125 : x
Step 2: Cross-multiply: 5 × x = 6 × 125
Step 3: 5x = 750
Step 4: x = 750 ÷ 5 = 150
Answer5 : 6 :: 125 : 150
Question 5: 72 : y = 64 : x, find x : y (Additional Q11)
Answer
Step 1: Write as proportion: 72 : y :: 64 : x
Step 2: Cross-multiply: 72 × x = 64 × y
Step 3: x/y = 64/72
Step 4: Simplify 64/72. HCF of 64 and 72 = 8.
Step 5: 64 ÷ 8 = 8, 72 ÷ 8 = 9
Step 6: x/y = 8/9, so x : y = 8 : 9
Answerx : y = 8 : 9
Question 6: Breadth : Length = 2 : 5, find missing values (Additional Q7)
Answer
Step 1: The ratio is breadth : length = 2 : 5
Step 2: So breadth = 2k and length = 5k for some value k.
(a) When breadth = 8:
Step 3: 2k = 8, so k = 4
Step 4: Length = 5k = 5 × 4 = 20
(b) When breadth = 10:
Step 5: 2k = 10, so k = 5
Step 6: Length = 5k = 5 × 5 = 25
(c) When length = 50:
Step 7: 5k = 50, so k = 10
Step 8: Breadth = 2k = 2 × 10 = 20
(d) When length = 250:
Step 9: 5k = 250, so k = 50
Step 10: Breadth = 2k = 2 × 50 = 100
AnswerThe missing values are 20, 25, and 100 (for length) as the table requires.
Exercise 3.3 Solutions — Unitary Method
Question 1: 12 pens cost ₹96. How much do 8 pens cost? (Additional Q10)
Answer
Step 1: Cost of 12 pens = ₹96
Step 2: Cost of 1 pen = 96 ÷ 12 = ₹8
Step 3: Cost of 8 pens = 8 × 8 = ₹64
Answer8 pens cost ₹64
Question 2: 7 m of cloth costs ₹294. How much does 5 m cost? (Additional Q14)
Answer
Step 1: Cost of 7 m cloth = ₹294
Step 2: Cost of 1 m cloth = 294 ÷ 7 = ₹42
Step 3: Cost of 5 m cloth = 42 × 5 = ₹210
Answer5 m of cloth costs ₹210
Question 3: Divide ₹36 in the ratio 15 : 12 (Additional Q6)
Answer
Step 1: Simplify the ratio 15 : 12. HCF = 3. So, 15 ÷ 3 : 12 ÷ 3 = 5 : 4
Step 2: Sum of ratio terms = 5 + 4 = 9
Step 3: Value of one part = 36 ÷ 9 = ₹4
Step 4: First share = 5 × 4 = ₹20
Step 5: Second share = 4 × 4 = ₹16
AnswerSumi gets ₹20, Divya gets ₹16
Question 4: Divide ₹600 in the ratio 2 : 3 between Vimala and Yazhini (Additional Q8)
Answer
Step 1: Ratio = 2 : 3
Step 2: Sum of ratio terms = 2 + 3 = 5
Step 3: Value of one part = 600 ÷ 5 = ₹120
Step 4: Vimala's share = 2 × 120 = ₹240
Step 5: Yazhini's share = 3 × 120 = ₹360
Step 6: Yazhini gets more by 360 − 240 = ₹120
AnswerVimala gets ₹240, Yazhini gets ₹360. Yazhini gets ₹120 more than Vimala.
Question 5: A car travels 150 km using 10 litres of petrol. How far will it travel using 25 litres?
Answer
Step 1: Distance for 10 litres = 150 km
Step 2: Distance for 1 litre = 150 ÷ 10 = 15 km
Step 3: Distance for 25 litres = 15 × 25 = 375 km
Answer375 km
Question 6: If 8 workers can paint a wall in 6 hours, how many workers are needed to paint it in 4 hours? (Inverse proportion)
Answer
Step 1: This is inverse proportion — fewer hours means more workers.
Step 2: Total work = 8 workers × 6 hours = 48 worker-hours
Step 3: Workers needed = 48 ÷ 4 = 12 workers
Answer12 workers are needed
Exercise 3.4 Solutions — Direct and Inverse Proportion
Question 1: ₹1500 for 10 days. How much for 30 days? (Additional Q15 — Direct Proportion)
Answer
Step 1: More days → more money. This is direct proportion.
Step 2: Amount for 10 days = ₹1500
Step 3: Amount for 1 day = 1500 ÷ 10 = ₹150
Step 4: Amount for 30 days = 150 × 30 = ₹4500
Answer₹4500 for 30 days
Question 2: Temperature 15°C in 30 days. What temperature in 10 days? (Additional Q16 — Direct Proportion)
Answer
Step 1: This is direct proportion — fewer days, proportionally lower total.
Step 2: Temperature for 30 days = 15°C
Step 3: Temperature for 1 day = 15 ÷ 30 = 0.5°C
Step 4: Temperature for 10 days = 0.5 × 10 = 5°C
Answer5°C in 10 days
Question 3: ₹7500 for 3 months. Yearly expense? (Additional Q17 — Direct Proportion)
Answer
Step 1: More months → more expense. Direct proportion.
Step 2: Expense for 3 months = ₹7500
Step 3: Expense for 1 month = 7500 ÷ 3 = ₹2500
Step 4: Expense for 12 months (1 year) = 2500 × 12 = ₹30000
Answer₹30,000 per year
Question 4: Run rates — Aadisaran 5 runs/over, Mohan 6 runs/over. Who is better? (Additional Q13)
Answer
Step 1: Aadisaran's run rate = 5 runs per over
Step 2: Mohan's run rate = 6 runs per over
Step 3: Compare: 6 > 5
Step 4: Ratio of their rates = 5 : 6
AnswerMohan is the better scorer because his run rate (6 runs/over) is higher than Aadisaran's (5 runs/over).
Question 5: A train covers 360 km in 4 hours. How far will it cover in 7 hours at the same speed?
Answer
Step 1: Direct proportion — more time, more distance at the same speed.
Step 2: Distance in 4 hours = 360 km
Step 3: Distance in 1 hour = 360 ÷ 4 = 90 km
Step 4: Distance in 7 hours = 90 × 7 = 630 km
Answer630 km
Question 6: 6 taps fill a tank in 12 hours. How long will 9 taps take? (Inverse Proportion)
Answer
Step 1: More taps → less time. This is inverse proportion.
Step 2: Total work = 6 taps × 12 hours = 72 tap-hours
Step 3: Time for 9 taps = 72 ÷ 9 = 8 hours
Answer8 hours
Additional Questions — Detailed Solutions
AQ 1: Complete 3 : 5 :: ___ : 10
Covered in Exercise 3.2, Question 1 above.
Answer6 : 10 (since 3 × 10 = 5 × 6 = 30)
AQ 2: Moni walks 6 km/hr, Vimala walks 4 km/hr. Ratio?
Covered in Exercise 3.1, Question 4 above.
Answer3 : 2
AQ 3: 75 holidays in 365 days. Ratio?
Covered in Exercise 3.1, Question 1(c) above.
Answer15 : 73
AQ 4: 20 girls, 25 boys. Find ratios.
Covered in Exercise 3.1, Question 2 above.
Answer(a) Girls:Boys = 4:5, (b) Girls:Total = 4:9
AQ 5: Simplify 20 : 5
Covered in Exercise 3.1, Question 3 above.
Answer4 : 1
AQ 6: Divide ₹36 in ratio 15:12
Covered in Exercise 3.3, Question 3 above.
AnswerSumi ₹20, Divya ₹16
AQ 7: Breadth:Length = 2:5, missing values
Covered in Exercise 3.2, Question 6 above.
Answer20 and 100
AQ 8: ₹600 in ratio 2:3
Covered in Exercise 3.3, Question 4 above.
AnswerVimala ₹240, Yazhini ₹360. Yazhini gets ₹120 more.
AQ 9: 16:24 :: x:30, find x
Covered in Exercise 3.2, Question 3 above.
Answerx = 20
AQ 10: 12 pens cost ₹96, 8 pens cost?
Covered in Exercise 3.3, Question 1 above.
Answer₹64
AQ 11: 72:y = 64:x, find x:y
Covered in Exercise 3.2, Question 5 above.
Answerx:y = 8:9
AQ 12: Fill in blanks — proportion
Covered in Exercise 3.2, Question 4 above.
Answer(i) 30:24::80:64, (ii) 5:6::125:150
AQ 13: Run rates comparison
Covered in Exercise 3.4, Question 4 above.
AnswerMohan is better (6 runs/over vs 5 runs/over)
AQ 14: 7 m cloth costs ₹294, 5 m costs?
Covered in Exercise 3.3, Question 2 above.
Answer₹210
AQ 15: ₹1500 for 10 days, 30 days?
Covered in Exercise 3.4, Question 1 above.
Answer₹4500
AQ 16: 15°C in 30 days, 10 days?
Covered in Exercise 3.4, Question 2 above.
Answer5°C
AQ 17: ₹7500 for 3 months, yearly?
Covered in Exercise 3.4, Question 3 above.
Answer₹30,000
AQ 18: 3:2 and 30:20 in proportion?
Covered in Exercise 3.2, Question 2 above.
AnswerYes, ad = 3 × 20 = 60, bc = 2 × 30 = 60, so they are in proportion.
Common Mistakes
1. Forgetting to simplify ratios: Always divide both terms by their HCF. Writing 20 : 25 instead of 4 : 5 loses marks. Check: can both numbers be divided by a common factor greater than 1? If yes, simplify.
2. Mixing up the order of terms in a ratio: "Ratio of boys to girls" means boys : girls, not girls : boys. Read the question carefully and write terms in the exact order asked.
3. Wrong cross-multiplication in proportions: Remember the rule is a × d = b × c (product of extremes equals product of means). Some students write a × b = c × d — this is wrong. Always pair the outer terms and the inner terms.
4. Confusing direct and inverse proportion: If more of one quantity gives more of the other (more books → more cost), it is direct proportion. If more of one quantity gives less of the other (more workers → fewer days), it is inverse proportion. Always ask: "Does the second quantity go up or down when the first goes up?"
5. Skipping the unitary step: In the unitary method, always find the value of one unit first, then multiply. Skipping directly to the answer without showing this step can lead to calculation errors and loses method marks in exams.
6. Not adding ratio terms for division problems: When dividing an amount in a ratio, remember to find the total number of parts (sum of all ratio terms) first. If the ratio is 2 : 3, the total parts are 2 + 3 = 5, and each part equals the total amount divided by 5.
7. Adding units to a ratio: A ratio has no units. Writing "3 km : 5 km" as a final answer is wrong — it should be just "3 : 5". The units cancel out because a ratio is a comparison by division.
Exam Checklist
Before you write your exam, make sure you can do each of these:
- [ ] Simplify a ratio to its lowest terms using HCF
- [ ] Compare two ratios by converting them to equivalent ratios with a common second term (or comparing as fractions)
- [ ] Check if four terms are in proportion using the cross-product rule (ad = bc)
- [ ] Find a missing term in a proportion using cross-multiplication
- [ ] Divide a quantity in a given ratio — find total parts, then multiply each share
- [ ] Apply the unitary method — always find the value of one unit first
- [ ] Identify direct vs. inverse proportion — ask "does the other quantity increase or decrease?"
- [ ] Solve direct proportion problems — if x increases, y increases proportionally
- [ ] Solve inverse proportion problems — if x increases, y decreases proportionally (product stays constant)
- [ ] Write ratios without units — always simplify and remove units from the final answer
- [ ] Show all steps — examiners award marks for the method, not just the final answer
