NCERT Solutions for Class 6 Maths Chapter 12 — Ratio and Proportion
Complete step-by-step NCERT exercise solutions for CBSE Class 6 Maths Chapter 12 Ratio and Proportion. Covers ratios, simplification, equivalent ratios, proportion, unitary method, and word problems with detailed worked examples, common mistakes, and exam tips.
Key Concepts
| Concept | Definition / Formula |
|---|---|
| Ratio | A comparison of two quantities by division; written as a:b or a/b; a and b must be in the same units |
| Simplifying Ratios | Divide both terms by their HCF to get the simplest form; e.g., 12:18 → HCF=6 → 2:3 |
| Equivalent Ratios | Ratios that are equal; obtained by multiplying or dividing both terms by the same number; e.g., 2:3 = 4:6 = 6:9 |
| Proportion | Two ratios are in proportion if they are equal; if a:b = c:d, then a, b, c, d are in proportion; written as a:b :: c:d |
| Unitary Method | Find the value of 1 unit first, then multiply to find the value of the required units |
| Comparison using Ratio | To compare two quantities, convert them to the same unit, then express as a ratio |
| Ratios in Different Situations | Ratios can compare prices, speeds, ages, weights, etc. — always ensure same units |
Exercise 12.1 — Ratio: Concept and Simplification
Question 1
There are 20 girls and 15 boys in a class.
(a) What is the ratio of number of girls to the number of boys?
Number of girls = 20
Number of boys = 15
Ratio of girls to boys = 20 : 15
HCF of 20 and 15 = 5
Simplified ratio = (20 ÷ 5) : (15 ÷ 5) = 4 : 3
(b) What is the ratio of number of boys to the total number of students in the class?
Total students = 20 + 15 = 35
Ratio of boys to total students = 15 : 35
HCF of 15 and 35 = 5
Simplified ratio = (15 ÷ 5) : (35 ÷ 5) = 3 : 7
Question 2
Express the following ratios in simplest form:
(a) 24 : 36
HCF of 24 and 36 = 12
Simplified ratio = (24 ÷ 12) : (36 ÷ 12) = 2 : 3
(b) 45 : 30
HCF of 45 and 30 = 15
Simplified ratio = (45 ÷ 15) : (30 ÷ 15) = 3 : 2
(c) 72 : 60
HCF of 72 and 60 = 12
Simplified ratio = (72 ÷ 12) : (60 ÷ 12) = 6 : 5
(d) 150 : 250
HCF of 150 and 250 = 50
Simplified ratio = (150 ÷ 50) : (250 ÷ 50) = 3 : 5
Question 3
In a school, there are 1200 students, out of which 480 are girls. Find the ratio of:
(a) Girls to total students
Girls = 480, Total = 1200
Ratio = 480 : 1200
HCF of 480 and 1200 = 240
Simplified ratio = (480 ÷ 240) : (1200 ÷ 240) = 2 : 5
(b) Boys to total students
Boys = 1200 − 480 = 720
Ratio = 720 : 1200
HCF of 720 and 1200 = 240
Simplified ratio = (720 ÷ 240) : (1200 ÷ 240) = 3 : 5
(c) Boys to girls
Ratio = 720 : 480
HCF of 720 and 480 = 240
Simplified ratio = (720 ÷ 240) : (480 ÷ 240) = 3 : 2
Question 4
The length of a keyboard is 45 cm and its breadth is 20 cm. Find the ratio of:
(a) Length to breadth
Ratio = 45 : 20
HCF of 45 and 20 = 5
Simplified ratio = (45 ÷ 5) : (20 ÷ 5) = 9 : 4
(b) Breadth to length
Ratio = 20 : 45
HCF of 20 and 45 = 5
Simplified ratio = (20 ÷ 5) : (45 ÷ 5) = 4 : 9
Question 5
Divide Rs 60 in the ratio 1 : 2 between Kriti and Kiran.
Total ratio parts = 1 + 2 = 3
Total amount = Rs 60
Value of 1 part = 60 ÷ 3 = Rs 20
Kriti's share = 1 × 20 = Rs 20
Kiran's share = 2 × 20 = Rs 40
Verification: 20 + 40 = 60 ✓ and 20:40 = 1:2 ✓
Question 6
Divide Rs 560 between Ravi and Rani in the ratio 3 : 5.
Total ratio parts = 3 + 5 = 8
Total amount = Rs 560
Value of 1 part = 560 ÷ 8 = Rs 70
Ravi's share = 3 × 70 = Rs 210
Rani's share = 5 × 70 = Rs 350
Verification: 210 + 350 = 560 ✓ and 210:350 = 3:5 ✓
Question 7
The ratio of the weight of Anu to the weight of her mother is 1 : 3. If her mother weighs 60 kg, what is Anu's weight?
Anu's weight : Mother's weight = 1 : 3
Mother's weight = 60 kg
3 parts = 60 kg
1 part = 60 ÷ 3 = 20 kg
Anu's weight = 20 kg
Question 8
Compare the ratios 3 : 4 and 2 : 3. Which is greater?
To compare, convert to equivalent ratios with a common denominator.
LCM of 4 and 3 = 12
3 : 4 = (3 × 3) : (4 × 3) = 9 : 12
2 : 3 = (2 × 4) : (3 × 4) = 8 : 12
Since 9 > 8, we have 9/12 > 8/12
Therefore, 3 : 4 is greater than 2 : 3.
Question 9
Convert the ratio 3 : 5 into a percentage.
Ratio = 3 : 5 = 3/5
Percentage = (3/5) × 100
= 300/5
= 60%
Question 10
A map is drawn to the scale 1 : 1,000,000. If the distance between two cities on the map is 5 cm, what is the actual distance?
Scale: 1 cm on map = 1,000,000 cm in reality
5 cm on map = 5 × 1,000,000 = 50,00,000 cm
Convert to km: 1 km = 1,00,000 cm
Actual distance = 50,00,000 ÷ 1,00,000 = 50 km
Exercise 12.2 — Proportion
Question 1
Determine if the following are in proportion:
(a) 15, 45, 40, 120
Ratio of first two terms: 15 : 45 = 1 : 3
Ratio of last two terms: 40 : 120 = 1 : 3
Since 1 : 3 = 1 : 3, the numbers are in proportion.
15 : 45 :: 40 : 120
(b) 12, 18, 20, 30
Ratio of first two terms: 12 : 18 = 2 : 3
Ratio of last two terms: 20 : 30 = 2 : 3
Since 2 : 3 = 2 : 3, the numbers are in proportion.
12 : 18 :: 20 : 30
(c) 22, 33, 16, 27
Ratio of first two terms: 22 : 33 = 2 : 3
Ratio of last two terms: 16 : 27
HCF of 16 and 27 = 1 (they are already in simplest form)
16 : 27 ≠ 2 : 3
The numbers are NOT in proportion.
(d) 18, 8, 27, 12
Ratio of first two terms: 18 : 8 = 9 : 4
Ratio of last two terms: 27 : 12 = 9 : 4
Since 9 : 4 = 9 : 4, the numbers are in proportion.
18 : 8 :: 27 : 12
Question 2
Write True (T) or False (F):
(a) 12 : 18 :: 20 : 30 is a proportion.
12 : 18 = 2 : 3
20 : 30 = 2 : 3
(b) 6 : 8 :: 15 : 20 is a proportion.
6 : 8 = 3 : 4
15 : 20 = 3 : 4
(c) 30 : 45 :: 50 : 75 is a proportion.
30 : 45 = 2 : 3
50 : 75 = 2 : 3
(d) 16 : 24 :: 20 : 32 is a proportion.
16 : 24 = 2 : 3
20 : 32 = 5 : 8
Question 3
Are the following statements true?
(a) 40 persons : 200 persons = Rs 15 : Rs 75
Left ratio: 40 : 200 = 1 : 5
Right ratio: 15 : 75 = 1 : 5
(b) 7.5 litres : 15 litres = 5 kg : 10 kg
Left ratio: 7.5 : 15 = 1 : 2
Right ratio: 5 : 10 = 1 : 2
(c) 2 kg : 80 kg = 25 g : 625 g
Left ratio: 2 : 80 = 1 : 40
Right ratio: 25 : 625 = 1 : 25
Question 4
In a proportion, the first, second and fourth terms are 6, 18 and 42 respectively. Find the third term.
Let the third term be x.
The proportion is: 6 : 18 :: x : 42
In a proportion, product of extremes = product of means
6 × 42 = 18 × x
252 = 18x
x = 252 ÷ 18
x = 14
The third term is 14.
Verification: 6 : 18 = 1 : 3 and 14 : 42 = 1 : 3 ✓
Question 5
Find the value of x in the following proportions:
(a) x : 6 :: 15 : 30
Product of extremes = Product of means
x × 30 = 6 × 15
30x = 90
x = 90 ÷ 30
x = 3
(b) 4 : 8 :: x : 20
Product of extremes = Product of means
4 × 20 = 8 × x
80 = 8x
x = 80 ÷ 8
x = 10
(c) 9 : 15 :: 12 : x
Product of extremes = Product of means
9 × x = 15 × 12
9x = 180
x = 180 ÷ 9
x = 20
Question 6
The ratio of the age of a father to the age of his son is 7 : 3. If the father is 42 years old, find the age of the son.
Father's age : Son's age = 7 : 3
Father's age = 42 years
7 parts = 42
1 part = 42 ÷ 7 = 6
Son's age = 3 × 6 = 18 years
Question 7
The ratio of iron and carbon in an alloy is 5 : 2. If 3.5 kg of carbon is used, how much iron is used?
Iron : Carbon = 5 : 2
2 parts = 3.5 kg
1 part = 3.5 ÷ 2 = 1.75 kg
Iron = 5 × 1.75 = 8.75 kg
Exercise 12.3 — Unitary Method and Word Problems
Question 1
If the cost of 5 pens is Rs 40, find the cost of 8 pens.
Question 2
A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?
Question 3
The cost of 6 kg of rice is Rs 270. What will be the cost of 10 kg of rice?
Question 4
8 workers can complete a task in 12 days. How many days will 6 workers take to complete the same task?
Question 5
A train covers 350 km in 5 hours. How long will it take to cover 490 km?
Question 6
The weight of 40 books is 8 kg. What is the weight of 75 such books?
Question 7
A shopkeeper sells 12 pencils for Rs 36. How much will he charge for 20 pencils?
Question 8
If 15 boxes can hold 180 chocolates, how many boxes are needed to hold 300 chocolates?
Question 9
The ratio of boys to girls in a school is 5 : 7. If there are 600 boys, how many girls are there?
Boys : Girls = 5 : 7
Number of boys = 600
5 parts = 600
1 part = 600 ÷ 5 = 120
Number of girls = 7 × 120 = 840 girls
Question 10
Rahul earns Rs 45,000 in 6 months. How much does he earn in a year?
Question 11
A recipe requires 3 cups of flour for 24 cookies. How many cups of flour are needed for 56 cookies?
Question 12
The ratio of the present ages of Ravi and Raju is 4 : 5. If Ravi is 24 years old, what will be Raju's age after 3 years?
Current ages: Ravi : Raju = 4 : 5
Ravi's current age = 24 years
4 parts = 24
1 part = 24 ÷ 4 = 6
Raju's current age = 5 × 6 = 30 years
Raju's age after 3 years = 30 + 3 = 33 years
Common Mistakes
1. Not simplifying ratios fully: Always divide by the HCF. 12:18 simplifies to 2:3, not 4:6. Keep simplifying until both terms have no common factor other than 1.
2. Comparing quantities with different units: To compare 50 paise and Rs 2, you must first convert both to the same unit. 50 paise and Rs 2 = 200 paise. Ratio = 50 : 200 = 1 : 4.
3. Confusing direct and inverse proportion: In the unitary method, if more quantity means more cost, it is direct proportion. If more workers means fewer days, it is inverse proportion. Always think about whether the relationship increases or decreases.
4. Writing ratios in wrong order: "Ratio of boys to girls" means boys:girls. "Ratio of girls to boys" means girls:boys. The order matters — they are different ratios.
5. Forgetting to verify proportion: Always check your answer. If a : b :: c : d, then verify that a × d = b × c (product of extremes = product of means).
6. Unit conversion errors in the unitary method: Always ensure you are calculating per unit correctly. "Per unit" means dividing by the given quantity, not multiplying. Cost of 5 items = Rs 50 means cost of 1 item = Rs 10, not Rs 250.
Exam Checklist
- [ ] I know that a ratio compares two quantities of the same kind by division (a:b).
- [ ] I can simplify a ratio by dividing both terms by their HCF.
- [ ] I can find equivalent ratios by multiplying or dividing both terms by the same number.
- [ ] I understand that two ratios are in proportion when they are equal (a:b :: c:d).
- [ ] I can verify proportion using the rule: product of extremes = product of means.
- [ ] I always convert to the same unit before forming a ratio (e.g., grams and kilograms, paise and rupees).
- [ ] I can divide a quantity in a given ratio by finding the value of one part first.
- [ ] I can apply the unitary method: find the value of 1 unit, then multiply for the required number.
- [ ] I distinguish between direct and inverse proportion before solving.
- [ ] I verify my answer by checking whether the ratio or proportion holds true with the calculated values.
Quick Reference — Key Formulas
| Concept | Formula / Rule |
|---|---|
| Ratio | a : b = a/b (both in same units) |
| Simplify ratio | Divide both terms by HCF |
| Equivalent ratio | Multiply/divide both terms by same number |
| Proportion | a : b :: c : d means a/b = c/d |
| Extremes and Means | a : b :: c : d → a × d = b × c |
| Divide in ratio m:n | One part = Total ÷ (m + n); shares = m × part and n × part |
| Unitary Method | Find value of 1 unit first, then multiply |
| Convert ratio to percentage | (a/b) × 100% |
These solutions follow the NCERT textbook pattern for Class 6 Maths Chapter 12 — Ratio and Proportion. Practice each question step-by-step and verify your answers by checking the proportion or recalculating.