Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
NCERT Solutions for Class 6 Maths Chapter 7 — Fractions
Key Concepts
| Concept | Definition | Example |
|---|---|---|
| Fraction | A part of a whole expressed as numerator/denominator | 3/5 means 3 parts out of 5 equal parts |
| Proper Fraction | Numerator is less than denominator; value less than 1 | 2/7, 3/8, 5/9 |
| Improper Fraction | Numerator is greater than or equal to denominator; value 1 or more | 7/4, 9/5, 12/12 |
| Mixed Fraction | Whole number plus a proper fraction | 2¾, 5⅓, 3⅖ |
| Equivalent Fractions | Fractions that represent the same value | 1/2 = 2/4 = 3/6 = 4/8 |
| Like Fractions | Fractions with the same denominator | 3/7, 5/7, 6/7 |
| Unlike Fractions | Fractions with different denominators | 2/3, 4/5, 1/7 |
| Addition of Fractions | Add numerators for like fractions; find LCM for unlike fractions | 2/5 + 1/5 = 3/5 |
| Subtraction of Fractions | Subtract numerators for like fractions; find LCM for unlike fractions | 4/7 − 2/7 = 2/7 |
| Multiplication of Fractions | Multiply numerators together and denominators together | 2/3 × 4/5 = 8/15 |
| Division of Fractions | Multiply by the reciprocal of the divisor | (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12 = 5/6 |
Exercise 7.1 — Types of Fractions, Representing Fractions
Q1. Write the fraction representing the shaded portion in the following figures.
(i) If 3 out of 8 equal parts are shaded:
- Fraction = 3/8
- Numerator = 3 (shaded parts), Denominator = 8 (total parts)
(ii) If 2 out of 4 equal parts are shaded:
- Fraction = 2/4
- Simplify: 2/4 = 1/2
(iii) If 4 out of 9 equal parts are shaded:
- Fraction = 4/9
- This fraction cannot be simplified further as HCF(4, 9) = 1
Q2. Colour the part according to the given fraction.
(i) 1/6 of the figure — shade 1 part out of 6 equal parts. (ii) 3/4 of the figure — shade 3 parts out of 4 equal parts. (iii) 2/3 of the figure — shade 2 parts out of 3 equal parts.
When shading, ensure the figure is divided into equal parts matching the denominator.
Q3. Identify the error in the following: Shaded portion represents 1/2.
Q4. What fraction of a day is 8 hours?
- A day has 24 hours.
- Fraction = 8/24
- Simplify: HCF(8, 24) = 8
- 8/24 = (8 ÷ 8)/(24 ÷ 8) = 1/3
Q5. What fraction of an hour is 40 minutes?
- 1 hour = 60 minutes
- Fraction = 40/60
- Simplify: HCF(40, 60) = 20
- 40/60 = (40 ÷ 20)/(60 ÷ 20) = 2/3
Q6. Write all the natural numbers from 2 to 12. What fraction of them are prime numbers?
- Natural numbers from 2 to 12: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
- Total numbers = 11
- Prime numbers among them: 2, 3, 5, 7, 11 → 5 prime numbers
- Fraction = 5/11
Q7. Identify the proper fractions: 2/3, 5/2, 7/9, 1/8, 3/1, 11/13
| Fraction | Numerator | Denominator | Proper? |
|---|---|---|---|
| 2/3 | 2 | 3 | Yes |
| 5/2 | 5 | 2 | No (improper) |
| 7/9 | 7 | 9 | Yes |
| 1/8 | 1 | 8 | Yes |
| 3/1 | 3 | 1 | No (improper) |
| 11/13 | 11 | 13 | Yes |
Proper fractions: 2/3, 7/9, 1/8, 11/13
Q8. Identify the improper fractions: 3/4, 7/3, 5/5, 2/9, 8/7, 11/11
| Fraction | Numerator ≥ Denominator? | Type |
|---|---|---|
| 3/4 | No | Proper |
| 7/3 | Yes | Improper |
| 5/5 | Yes (equal) | Improper |
| 2/9 | No | Proper |
| 8/7 | Yes | Improper |
| 11/11 | Yes (equal) | Improper |
Improper fractions: 7/3, 5/5, 8/7, 11/11
Exercise 7.2 — Fractions on the Number Line, Equivalent Fractions
Q1. Represent 3/5 on a number line.
Verification: 3/5 lies between 0 and 1, closer to 1/2 and to 1, since 3/5 = 0.6.
Q2. Represent 7/4 on a number line.
Q3. Write five equivalent fractions of 3/5.
| Multiply by | Fraction |
|---|---|
| 2 | 6/10 |
| 3 | 9/15 |
| 4 | 12/20 |
| 5 | 15/25 |
| 6 | 18/30 |
Five equivalent fractions of 3/5: 6/10, 9/15, 12/20, 15/25, 18/30
Q4. Are 4/6 and 12/18 equivalent fractions?
- 4/6: HCF(4, 6) = 2, so 4/6 = (4 ÷ 2)/(6 ÷ 2) = 2/3
- 12/18: HCF(12, 18) = 6, so 12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3
Since both simplify to 2/3, yes, 4/6 and 12/18 are equivalent fractions.
Q5. Find the equivalent fraction of 2/7 with denominator 21.
- We need: 2/7 = ?/21
- 7 × 3 = 21, so multiply numerator also by 3
- New numerator = 2 × 3 = 6
- Equivalent fraction = 6/21
Verification: 6/21 = (6 ÷ 3)/(21 ÷ 3) = 2/7 ✓
Exercise 7.3 — Simplest Form of Fractions
Q1. Write the simplest form of 24/36.
- Find HCF(24, 36)
- Prime factors of 24 = 2³ × 3
- Prime factors of 36 = 2² × 3²
- HCF = 2² × 3 = 12
- 24/36 = (24 ÷ 12)/(36 ÷ 12) = 2/3
Q2. Write the simplest form of 48/60.
- Find HCF(48, 60)
- 48 = 2⁴ × 3; 60 = 2² × 3 × 5
- HCF = 2² × 3 = 12
- 48/60 = (48 ÷ 12)/(60 ÷ 12) = 4/5
Q3. Ramesh had 20 pencils, Sheelu had 50 pencils and Jamaal had 80 pencils. After four months, Ramesh used up 10 pencils, Sheelu used up 25 pencils and Jamaal used up 40 pencils. What fraction did each use up? Are they equivalent?
- Ramesh: 10/20 = 1/2
- Sheelu: 25/50 = 1/2
- Jamaal: 40/80 = 1/2
All three used up 1/2 of their pencils. Yes, the fractions are equivalent (all equal to 1/2).
Q4. Reduce 84/98 to its simplest form.
- Find HCF(84, 98)
- 84 = 2² × 3 × 7; 98 = 2 × 7²
- HCF = 2 × 7 = 14
- 84/98 = (84 ÷ 14)/(98 ÷ 14) = 6/7
Q5. Simplify: 56/70.
- Find HCF(56, 70)
- 56 = 2³ × 7; 70 = 2 × 5 × 7
- HCF = 2 × 7 = 14
- 56/70 = (56 ÷ 14)/(70 ÷ 14) = 4/5
Exercise 7.4 — Like and Unlike Fractions, Comparing Fractions
Q1. Compare 3/8 and 5/8.
- Since 5 > 3, 5/8 > 3/8
Q2. Compare 2/3 and 3/5.
- LCM(3, 5) = 15
- 2/3 = (2 × 5)/(3 × 5) = 10/15
- 3/5 = (3 × 3)/(5 × 3) = 9/15
- Since 10/15 > 9/15, 2/3 > 3/5
Q3. Arrange in ascending order: 3/7, 5/7, 1/7, 4/7
- 1 < 3 < 4 < 5
- Ascending order: 1/7, 3/7, 4/7, 5/7
Q4. Arrange in descending order: 1/3, 2/7, 3/5, 1/6
- LCM = 210
- 1/3 = 70/210; 2/7 = 60/210; 3/5 = 126/210; 1/6 = 35/210
- Comparing: 126 > 70 > 60 > 35
- Descending order: 3/5, 1/3, 2/7, 1/6
Q5. Which is greater: 4/9 or 3/7?
- LCM(9, 7) = 63
- 4/9 = (4 × 7)/(9 × 7) = 28/63
- 3/7 = (3 × 9)/(7 × 9) = 27/63
- 28/63 > 27/63
- 4/9 is greater than 3/7
Q6. The lengths of two pencils are 7/8 cm and 5/6 cm. Which pencil is longer?
- LCM(8, 6) = 24
- 7/8 = (7 × 3)/(8 × 3) = 21/24
- 5/6 = (5 × 4)/(6 × 4) = 20/24
- Since 21/24 > 20/24, 7/8 cm pencil is longer.
Exercise 7.5 — Addition of Fractions
Q1. Add: 3/8 + 1/8
- 3/8 + 1/8 = (3 + 1)/8 = 4/8 = 1/2 (simplified)
Q2. Add: 2/5 + 3/7
- LCM(5, 7) = 35
- 2/5 = (2 × 7)/(5 × 7) = 14/35
- 3/7 = (3 × 5)/(7 × 5) = 15/35
- 14/35 + 15/35 = (14 + 15)/35 = 29/35
Q3. Add: 3/4 + 1/3 + 1/6
- LCM(4, 3, 6) = 12
- 3/4 = 9/12; 1/3 = 4/12; 1/6 = 2/12
- 9/12 + 4/12 + 2/12 = (9 + 4 + 2)/12 = 15/12 = 5/4
Q4. Solve: 2⅓ + 1¾
- 2⅓ = (2 × 3 + 1)/3 = 7/3
- 1¾ = (1 × 4 + 1)/4 = 5/4
- LCM(3, 4) = 12
- 7/3 = 28/12; 5/4 = 15/12
- 28/12 + 15/12 = 43/12 = 3 7/12
Q5. Sarita bought 2/5 metre of ribbon and Lalita bought 3/4 metre of ribbon. What is the total length of ribbon they bought?
- Total = 2/5 + 3/4
- LCM(5, 4) = 20
- 2/5 = 8/20; 3/4 = 15/20
- Total = 8/20 + 15/20 = 23/20 = 1 3/20 metres
Q6. Add: 7/10 + 2/5 + 1/2
- LCM(10, 5, 2) = 10
- 7/10 = 7/10; 2/5 = 4/10; 1/2 = 5/10
- 7/10 + 4/10 + 5/10 = 16/10 = 8/5 = 1 3/5
Exercise 7.6 — Subtraction of Fractions
Q1. Subtract: 5/7 − 2/7
- 5/7 − 2/7 = (5 − 2)/7 = 3/7
Q2. Subtract: 7/8 − 3/5
- LCM(8, 5) = 40
- 7/8 = 35/40; 3/5 = 24/40
- 35/40 − 24/40 = 11/40
Q3. Subtract: 2⅔ − 1⅓
- 2⅔ = 8/3; 1⅓ = 4/3
- 8/3 − 4/3 = 4/3 = 1⅓
Q4. Subtract: 5 from 3⅚
- 5 = 5/1
- 3⅚ = 23/6
- LCM(1, 6) = 6
- 5/1 = 30/6; 23/6 = 23/6
- 30/6 − 23/6 = 7/6 = 1⅙
Wait — the question says subtract 5 from 3⅚:
- 3⅚ − 5 = 23/6 − 30/6 = −7/6
This gives a negative result: −1⅙, which means 5 is greater than 3⅚.
Q5. A piece of wire is 7/8 metre long. If 3/4 metre is cut from it, how much wire remains?
- Remaining = 7/8 − 3/4
- LCM(8, 4) = 8
- 7/8 = 7/8; 3/4 = 6/8
- 7/8 − 6/8 = 1/8 metre
Q6. Simplify: 4/5 − 2/15 + 1/3
- LCM(5, 15, 3) = 15
- 4/5 = 12/15; 2/15 = 2/15; 1/3 = 5/15
- 12/15 − 2/15 + 5/15 = (12 − 2 + 5)/15 = 15/15 = 1
Common Mistakes
1. Forgetting to simplify the final answer — Always reduce the fraction to its lowest terms by dividing numerator and denominator by their HCF. For example, 4/8 must be written as 1/2.
2. Confusing proper and improper fractions — A proper fraction always has a value less than 1. Remember: numerator < denominator for a proper fraction.
3. Adding/subtracting denominators — Never add or subtract denominators. Only numerators are added or subtracted. 1/3 + 1/3 = 2/3, NOT 2/6.
4. Incorrect LCM for unlike fractions — When converting unlike fractions, find the correct LCM of denominators. A wrong LCM leads to wrong equivalent fractions and wrong answers.
5. Wrong conversion of mixed fractions — To convert a mixed fraction to an improper fraction, use the formula: (whole number × denominator) + numerator, all over denominator. For example, 3⅔ = (3 × 3 + 2)/3 = 11/3, NOT 3 + 2/3 = 5/3.
6. Not checking which fraction is greater in subtraction — When subtracting fractions, ensure the minuend is greater than the subtrahend. Always compare first to know if the result will be positive or negative.
Exam Checklist
- [ ] I can identify proper fractions, improper fractions, and mixed fractions correctly
- [ ] I can represent a fraction on a number line
- [ ] I can write equivalent fractions by multiplying/dividing numerator and denominator by the same number
- [ ] I can reduce a fraction to its simplest form using HCF
- [ ] I can find equivalent fractions with a given denominator
- [ ] I can compare like fractions by comparing numerators
- [ ] I can compare unlike fractions by converting to equivalent fractions with the same denominator (LCM)
- [ ] I can add like fractions directly by adding numerators
- [ ] I can add unlike fractions by finding the LCM of denominators first
- [ ] I can subtract fractions using the same method as addition
- [ ] I can convert mixed fractions to improper fractions and vice versa
- [ ] I can solve word problems involving addition and subtraction of fractions
- [ ] I always simplify my final answer to the lowest terms