Class 6 Maths - Ratio and Proportion NCERT Solutions

Chapter 12: Ratio and Proportion (NCERT Solutions)

Exercise 12.1 (Ratio)

Q1. There are 20 girls and 15 boys in a class.
(a) What is the ratio of girls to boys? (b) What is the ratio of girls to total students?

(a) Ratio of girls to boys = 20 : 15 = 4 : 3. (Dividing both by HCF = 5.) Answer: 4 : 3.

(b) Total students = 20 + 15 = 35. Ratio of girls to total = 20 : 35 = 4 : 7. Answer: 4 : 7.

Q2. Out of 30 students in a class, 6 like football, 12 like cricket, and remaining like tennis. Find the ratio of:
(a) Number of students liking football to number liking tennis.
(b) Number of students liking cricket to total number of students.

Tennis = 30 − 6 − 12 = 12 students.

(a) Football : Tennis = 6 : 12 = 1 : 2.

(b) Cricket : Total = 12 : 30 = 2 : 5.

Q3. See the figure, and write the ratio of:
(a) Number of shaded triangles to total triangles (b) Number of shaded to unshaded triangles.

(Assuming: total 6 triangles, 4 shaded, 2 unshaded.)
(a) Shaded to total = 4 : 6 = 2 : 3.

(b) Shaded to unshaded = 4 : 2 = 2 : 1.

Q5. Find the ratio of the following:
(a) 81 to 108 (b) 98 to 63 (c) 33 km to 121 km (d) 30 minutes to 45 minutes

(a) HCF(81,108) = 27. 81:108 = 3 : 4.

(b) HCF(98,63) = 7. 98:63 = 14 : 9.

(d) HCF(30,45) = 15. 30:45 = 2 : 3.

Q6. Find the ratio of the following:
(a) 30 minutes to 1.5 hours (b) 40 cm to 1.5 m (c) 55 paise to Rs. 1 (d) 500 mL to 2 litres

(a) 1.5 hours = 90 minutes. 30 : 90 = 1 : 3.

(b) 1.5 m = 150 cm. 40 : 150 = 4 : 15. 4 : 15.

(d) 2 litres = 2000 mL. 500 : 2000 = 1 : 4.

Q9. Divide 20 pens between Sheela and Sangeeta in the ratio 3 : 2.

Total parts = 3 + 2 = 5.
Sheela's share = (3/5) × 20 = 12 pens.
Sangeeta's share = (2/5) × 20 = 8 pens.

Q10. Mother wants to divide Rs. 36 between her daughters Shreya and Bhoomika in the ratio of their ages. If age of Shreya is 15 years and age of Bhoomika is 12 years, find how much each will get.

Ratio of ages = 15 : 12 = 5 : 4. Total parts = 9.
Shreya gets = (5/9) × 36 = Rs. 20.
Bhoomika gets = (4/9) × 36 = Rs. 16.

Q11. Present age of father is 42 years and that of his son is 14 years. Find the ratio of: (a) present age of father to present age of son (b) age of the father to the age of son, when son was 12 years old.

(a) 42 : 14 = 3 : 1.

(b) When son was 12, son was 2 years younger (14 − 12 = 2). So father was 42 − 2 = 40.
Ratio = 40 : 12 = 10 : 3.

Exercise 12.2 (Proportion)

Q1. Determine if the following are in proportion:
(a) 15, 45, 40, 120 (b) 33, 121, 9, 96 (c) 24, 28, 36, 48 (d) 32, 48, 70, 210 (e) 4, 6, 8, 12 (f) 33, 44, 75, 100

Rule: a : b :: c : d if a × d = b × c (product of extremes = product of means).

(a) 15 × 120 = 1800; 45 × 40 = 1800. Yes, in proportion.

(b) 33 × 96 = 3168; 121 × 9 = 1089. No.

(d) 32 × 210 = 6720; 48 × 70 = 3360. No.

(e) 4 × 12 = 48; 6 × 8 = 48. Yes, in proportion.

(f) 33 × 100 = 3300; 44 × 75 = 3300. Yes, in proportion.

Q2. Write true or false for each of the following:
(a) 16 : 24 :: 20 : 30 (b) 21 : 6 :: 35 : 10 (c) 12 : 18 :: 28 : 12 (d) 8 : 9 :: 24 : 27 (e) 5.2 : 3.9 :: 3 : 4

(a) 16×30 = 480; 24×20 = 480. True.

(b) 21×10 = 210; 6×35 = 210. True.

(d) 8×27 = 216; 9×24 = 216. True.

(e) 5.2×4 = 20.8; 3.9×3 = 11.7. False.

Q4. Fill in the blanks:
(a) 8 : 12 = ___ : 6 (b) 15 : 4 = 45 : ___ (c) ___ : 18 = 3 : 9 (d) 12 : 16 = ___ : 20

(a) 8/12 = x/6 → x = (8×6)/12 = 4.

(b) 15/4 = 45/x → x = (45×4)/15 = 12.

(d) 12/16 = x/20 → x = (12×20)/16 = 15.

Exercise 12.3 (Unitary Method)

Q1. If the cost of 6 cans of juice is Rs. 210, then what will be the cost of 4 cans of juice?

Cost of 1 can = 210 / 6 = Rs. 35.
Cost of 4 cans = 35 × 4 = Rs. 140.

Q2. A motorbike travels 220 km in 5 litres of fuel. How much distance will it cover in 1.5 litres?

Distance in 1 litre = 220 / 5 = 44 km.
Distance in 1.5 litres = 44 × 1.5 = 66 km.

Q3. If the cost of a dozen soaps is Rs. 153.60, what will be the cost of 15 such soaps?

Cost of 1 soap = 153.60 / 12 = Rs. 12.80.
Cost of 15 soaps = 12.80 × 15 = Rs. 192.

Q4. A car requires 1.5 litres of petrol to travel 30 km. How many litres are needed for 210 km?

Petrol for 1 km = 1.5 / 30 = 0.05 litres.
Petrol for 210 km = 0.05 × 210 = 10.5 litres.

Q5. Annapurna reads 20 pages of a book in 30 minutes. How many pages will she read in 1 hour 15 minutes?

1 hour 15 minutes = 75 minutes.
Pages in 1 minute = 20/30 = 2/3.
Pages in 75 minutes = (2/3) × 75 = 50 pages.

Q8. In a year, Seema earns Rs. 1,50,000 and saves Rs. 50,000. Find the ratio of Money earned to money spent.

Money spent = 1,50,000 − 50,000 = Rs. 1,00,000.
Ratio of earned to spent = 1,50,000 : 1,00,000 = 3 : 2.

Class 6 Maths - Ratio and Proportion Practice Questions

Chapter 12: Ratio and Proportion (Practice Questions)

RD Sharma / HOT Practice

Q1. Express the following ratios in simplest form:
(a) 24 : 36 (b) 91 : 26 (c) 5 min : 45 sec (d) 1 km : 250 m

(a) HCF(24,36) = 12. 2 : 3

(b) HCF(91,26) = 13. 91/13=7, 26/13=2. 7 : 2

(d) 1 km = 1000 m. 1000 : 250, HCF = 250. 4 : 1

Q2. The ratio of boys to girls in a class is 5 : 3. If there are 40 students total, how many boys and girls are there?

Total parts = 5 + 3 = 8. One part = 40/8 = 5.
Boys = 5 × 5 = 25. Girls = 3 × 5 = 15.

Q3. If 3 : 5 = x : 35, find x.

3/5 = x/35 → x = (3 × 35)/5 = 21.

Q4. Are 4, 9, 16, and 36 in proportion?

Product of extremes: 4 × 36 = 144. Product of means: 9 × 16 = 144.
144 = 144. Yes, they are in proportion.

Q5. Divide Rs. 560 between Ravi and Kavya in the ratio 3 : 4.

Total parts = 7. One part = 560 / 7 = 80.
Ravi = 3 × 80 = Rs. 240. Kavya = 4 × 80 = Rs. 320.

Q6. If 15 men repair a road in 8 days, in how many days will 5 men do the same work? (Indirect proportion)

Less men → more days (indirect proportion).
15 × 8 = 5 × d → d = 120/5 = 24 days.

Q7. The ratio of sodium to chlorine in common salt is 23 : 35.5. If a sample has 46 g of sodium, find the amount of chlorine.

23 : 35.5 = 46 : x → x = (46 × 35.5) / 23 = 71 g of chlorine.

Q8. A recipe uses 3 cups of flour for every 2 cups of sugar. For 12 cups of flour, how many cups of sugar are needed?

3 : 2 = 12 : x → x = (12 × 2) / 3 = 8 cups of sugar.

Q9. A car travels 150 km in 3 hours. How far will it travel in 5 hours at the same speed?

Speed = 150/3 = 50 km/h. Distance in 5 hours = 50 × 5 = 250 km.

Q10. If 4 : 5 = 24 : x, find x. Verify by cross multiplication.

x = (24 × 5) / 4 = 30.
Verify: 4 × 30 = 120; 5 × 24 = 120. ✓

Q11. Three students scored 40, 60, and 80 marks. Find the ratio of:
(a) first to second (b) second to third (c) first to third

(a) 40 : 60 = 2 : 3

(b) 60 : 80 = 3 : 4

Q12. The lengths of two sticks are 45 cm and 75 cm. What is the ratio of the longer to the shorter stick in simplest form?

HCF(75, 45) = 15. 75 : 45 = 5 : 3.

Q13. 18 workers can build a wall in 30 days. How many workers are needed to build it in 10 days?

More workers → fewer days (indirect proportion).
18 × 30 = x × 10 → x = 540/10 = 54 workers.

Q14. Divide 90 into two parts in the ratio 7 : 2.

Total parts = 9. One part = 90/9 = 10.
Parts = 7×10 = 70 and 2×10 = 20.

Q15. If 8 pencils cost Rs. 48, find the cost of 15 pencils.

Cost of 1 pencil = 48/8 = Rs. 6.
Cost of 15 pencils = 6 × 15 = Rs. 90.

Q16. Is 7 : 21 the same as 1 : 3? Are they in proportion?

7 : 21 = 1 : 3 (dividing both by 7). Yes, they are equivalent ratios.
7 × 3 = 21; 21 × 1 = 21. Yes, they are in proportion.

Q17. The ratio of monthly savings to monthly expenses of a family is 2 : 9. If monthly savings are Rs. 4000, find the total monthly income.

Savings : Expenses = 2 : 9. 2 parts = Rs. 4000. One part = Rs. 2000.
Expenses = 9 × 2000 = Rs. 18,000.
Total income = Savings + Expenses = 4000 + 18000 = Rs. 22,000.

Q18. If 6 : x is in proportion with x : 54, find x.

6 : x :: x : 54 (continued proportion). x² = 6 × 54 = 324. x = 18.

Q19. A map is drawn to the scale 1 cm = 5 km. If two cities are 8 cm apart on the map, what is the actual distance?

Actual distance = 8 × 5 = 40 km.

Q20. In a class test, a student scored 36 out of 60. What is the ratio of marks scored to total marks? What percentage did the student get?

Ratio = 36 : 60 = 3 : 5.
Percentage = (36/60) × 100 = 60%.

Class 6 Maths - Ratio and Proportion Summary

Chapter 12: Ratio and Proportion (Concepts & Summary)

1. Ratio

Ratio is a comparison of two quantities of the same kind by division.
Written as a : b or a/b. Read as "a is to b".
a is the antecedent (first term); b is the consequent (second term).
Both quantities must be in the same unit before finding the ratio.
A ratio can be simplified by dividing both terms by their HCF.
Ratio has no units.

Example: 15 : 25 = 15/25 = 3/5 = 3 : 5 (HCF = 5).

2. Proportion

Proportion: When two ratios are equal, they are said to be in proportion.
a : b :: c : d (read as "a is to b as c is to d").
Test: a : b :: c : d ↔ a × d = b × c.
a and d are called extremes; b and c are called means.
Product of extremes = Product of means.

Example: 2 : 3 :: 8 : 12. Check: 2×12 = 24; 3×8 = 24. ✓

3. Finding Missing Terms

If a : b = c : x, then: x = (b × c) / a

If a : x = c : d, then: x = (a × d) / c

Example: 5 : 8 = 25 : x → x = (8 × 25) / 5 = 40.

Dividing a quantity in a given ratio:
To divide Q in ratio m : n → First part = m/(m+n) × Q; Second part = n/(m+n) × Q.

4. Unitary Method

Steps:

Find the value for 1 unit (divide).
Multiply by the required number of units.

Direct proportion: As one quantity increases, the other also increases. (More items → more cost.)

Inverse proportion: As one quantity increases, the other decreases. (More workers → fewer days.)

Example: 5 pens cost Rs. 30. Cost of 1 pen = Rs. 6. Cost of 8 pens = Rs. 48.

5. Quick Reference

Concept	Formula / Rule
Simplify ratio a : b	Divide both by HCF(a, b)
Test proportion	a × d = b × c (cross product)
Find missing term x	Product of extremes = Product of means
Divide Q in m : n	Parts = m×Q/(m+n) and n×Q/(m+n)
Unitary method (direct)	Value for 1 unit × required units
Continued proportion	a : b :: b : c ⇒ b² = a × c

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