Complete Summary, Explanations, and Solutions for Fractions – Ganita Prakash Class VI, Chapter 7 – Numerator, Denominator, Equivalent Fractions, Operations, Questions, Answers

Chapter at a Glance – Fractions

This chapter explains fractions as equal shares of a whole, using examples like rotis and chikkis. It covers fractional units, measuring with fractions, marking on number lines, and mixed fractions.

Main Topics Covered

Introduction to fractions as shares (e.g., 1 roti shared among 2 or 4 children: \(\frac{1}{2}\), \(\frac{1}{4}\))
Fractional units and equal shares: Comparing \(\frac{1}{5}\) and \(\frac{1}{9}\)
Knowledge from the past: Ancient Indian terms for fractions
Fractions as parts of a whole (using chikki examples)
Measuring using fractional units (paper strips, rotis)
Marking fractions on the number line
Mixed fractions: Fractions greater than 1

Key Takeaways for Exams

Fraction Basics

Equal shares; numerator (parts), denominator (whole).

Unit Fractions

\(\frac{1}{n}\); smaller as n increases.

Proper/Improper

Less than 1: numerator < denominator; greater: numerator > denominator.

Mixed Fractions

Whole + fraction, e.g., \(1 \frac{1}{2}\).

Number Line

Mark fractions between integers.

Applications

Sharing food, weights, lengths.

Key Rules & Properties – Fractions

Important rules for understanding and working with fractions.

Fraction Definitions

Fractions represent equal parts of a whole.

Type	Definition	Example
Unit Fraction	1 part of n equal parts	\(\frac{1}{2}\), \(\frac{1}{3}\)
Proper Fraction	Numerator < Denominator	\(\frac{3}{4}\)
Improper Fraction	Numerator > Denominator	\(\frac{5}{2}\)
Mixed Fraction	Whole + Proper Fraction	\(2 \frac{1}{2}\)

Comparison Rules

Rule	Example
Larger denominator = Smaller unit fraction	\(\frac{1}{9} < \frac{1}{5}\)
Fractions >1: Numerator > Denominator	\(\frac{3}{2} > 1\)
Convert improper to mixed	\(\frac{3}{2} = 1 + \frac{1}{2}\)

Properties

Equal Parts: Fractions must be from equal divisions.
Number Line: Divide intervals into equal parts.
Measurement: Use unit fractions to build larger ones.

Concept Cards – Quick Explanations

Fraction as Share

Equal division of whole, e.g., 1 roti / 2 = \(\frac{1}{2}\).

Unit Fraction

\(\frac{1}{n}\); smaller with larger n.

Parts of Whole

Chikki divided into equal pieces, e.g., \(\frac{3}{4}\).

Measuring Fractions

Paper strips folded for halves, quarters.

Number Line

Mark \(\frac{1}{2}\), \(\frac{3}{5}\) between 0 and 1.

Mixed Fractions

Greater than 1, e.g., \(\frac{3}{2} = 1 \frac{1}{2}\).

Numerator/Denominator

Top: parts; Bottom: total parts.

Ancient Terms

\(\frac{3}{4}\) as 'tri-pada' in Rig Veda.

Proper vs Improper

Less or more than 1.

Reading Fractions

\(\frac{3}{4}\) as '3 times \(\frac{1}{4}\)'.

Examples + Solutions

Example 1: Sharing 1 Roti Among 4 Children

Solution: Each gets \(\frac{1}{4}\) roti. \(\frac{1}{2} > \frac{1}{4}\).

Example 2: Comparing \(\frac{1}{5}\) and \(\frac{1}{9}\)

Solution: \(\frac{1}{9} < \frac{1}{5}\) as more shares mean smaller piece.

Example 3: Chikki Broken into 2 Pieces

Solution: Bigger piece = \(\frac{3}{4}\), smaller = \(\frac{1}{4}\).

Example 4: Paper Strip Folded into 4

Solution: Each part \(\frac{1}{4}\); 2 parts = \(\frac{2}{4} = \frac{1}{2}\).

Example 5: Marking \(\frac{1}{3}\) on Number Line

Solution: Divide 0-1 into 3 equal parts.

Example 6: Mixed Fraction \(\frac{5}{2}\)

Solution: \(2 + \frac{1}{2}\).

Example 7: Sum of Weights \(\frac{1}{2} + \frac{1}{4}\)

Solution: \(\frac{3}{4}\) kg.

Figure it Out Solutions (All Solved)

Section 7.1: Fractional Units and Equal Shares

1. Three guavas weigh 1 kg equally.

Ans. Each \(\frac{1}{3}\) kg.

2. 1 kg rice in 4 equal packets.

Ans. Each \(\frac{1}{4}\) kg.

3. 3 glasses juice shared by 4 friends.

Ans. Each \(\frac{3}{4}\) glass.

4. Big fish \(\frac{1}{2}\) kg, small \(\frac{1}{4}\) kg, together.

Ans. \(\frac{3}{4}\) kg.

5. Arrange fraction words: quarter, half, three quarters, one and a quarter, one and a half, two and a half.

Ans. Quarter, half, three quarters, one and a quarter, one and a half, two and a half.

Section 7.2: Fractional Units as Parts of a Whole

Figures a-h: How much of whole chikki?

Ans. a. \(\frac{1}{2}\), b. \(\frac{1}{3}\), c. \(\frac{1}{4}\), d. \(\frac{1}{5}\), e. \(\frac{1}{6}\), f. \(\frac{1}{7}\), g. \(\frac{1}{8}\), h. \(\frac{1}{9}\). (Based on divisions shown in PDF.)

Section 7.3: Measuring Using Fractional Units

Do it once more! Fill blanks for \(\frac{1}{8}\).

Ans. 2 times \(\frac{1}{8} = \frac{1}{4}\), 4 times = \(\frac{1}{2}\), 6 times = \(\frac{3}{4}\), 8 times = 1.

Section 7.4: Marking Fraction Lengths on the Number Line

1. Draw \(\frac{1}{10}\), \(\frac{3}{10}\), \(\frac{4}{5}\).

Ans. Divide 0-1 into 10 parts for tenths; \(\frac{4}{5} = \frac{8}{10}\).

2. Write five more fractions and mark.

Ans. E.g., \(\frac{1}{4}\), \(\frac{2}{3}\), \(\frac{3}{5}\), \(\frac{5}{6}\), \(\frac{7}{8}\).

3. How many fractions between 0 and 1?

Ans. Infinitely many.

4. Length of black line (beyond 1).

Ans. \(\frac{3}{2}\).

5. Lengths of black lines (multiple).

Ans. \(\frac{6}{5}\), \(\frac{7}{5}\), \(\frac{8}{5}\), \(\frac{9}{5}\).

Section 7.3 Figure it Out (Additional)

1. Continue table of \(\frac{1}{2}\) for 2 steps.

Ans. 6 times = 3, 7 times = 3.5.

2. Table for \(\frac{1}{4}\).

Ans. 1=\(\frac{1}{4}\), 2=\(\frac{1}{2}\), 3=\(\frac{3}{4}\), 4=1, etc.

3. Make \(\frac{1}{3}\) and \(\frac{1}{6}\) with strip.

Ans. Fold into 3, then each into 2 for sixths.

4a. 5 times \(\frac{1}{4}\) roti.

Ans. \(\frac{5}{4} = 1 + \frac{1}{4}\).

4b. 9 times \(\frac{1}{4}\) roti.

Ans. \(\frac{9}{4} = 2 + \frac{1}{4}\).

5. Match fractional units with pictures.

Ans. \(\frac{1}{3}\) to third pie, \(\frac{1}{5}\) to fifth, etc.

Extra Practice Questions (Exam-Ready) – Chapter 7

25+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags

1-Mark Questions (Very Short Answer)

1. 1 roti shared by 3: each gets?

2. Numerator in \(\frac{3}{4}\).

3. \(\frac{1}{4}\) + \(\frac{1}{4}\).

4. Compare \(\frac{1}{6}\) and \(\frac{1}{3}\).

5. Mixed form of \(\frac{7}{4}\).

2-Mark Questions (Short Answer)

6. Explain unit fraction.

7. Mark \(\frac{2}{5}\) on number line.

8. 2 kg shared by 5: each?

9. \(\frac{5}{3}\) as mixed.

10. Ancient term for \(\frac{3}{4}\).

3-Mark Questions (Reasoning)

11. Why \(\frac{1}{100} > \frac{1}{200}\)?

12. Convert \(\frac{4}{3}\) to mixed and explain.

13. Fractions between 0 and 1?

14. Paper strip for \(\frac{1}{8}\).

15. \(\frac{3}{2}\) on number line.

4–5 Mark Questions (Application)

16. 5 glasses juice for 4 friends.

17. Chikki: 3/6 + 2/6.

18. Compare \(\frac{2}{3}\) and \(\frac{3}{4}\).

19. Mark \(\frac{7}{5}\) on line.

20. 1 kg in 6 packets.

Challenge Questions (6+ Marks)

21. Explain mixed fractions with roti.

22. Fractions words in languages.

23. Sum \(\frac{1}{2} + \frac{3}{4} + \frac{1}{4}\).

24. Paper strip for thirds and sixths.

25. Improper fractions greater than 2.

Common Mistakes & How to Avoid

Mistake 1: Larger Denominator = Larger Fraction

Thinking \(\frac{1}{9} > \frac{1}{5}\).

Avoid: Remember more shares = smaller piece.

Mistake 2: Unequal Parts

Not ensuring divisions are equal.

Avoid: Always divide into equal sizes.

Mistake 3: Confusing Numerator/Denominator

Swapping top and bottom.

Avoid: Top: parts taken; Bottom: total parts.

Mistake 4: Improper Fractions

Not converting to mixed.

Avoid: Divide numerator by denominator.

Mistake 5: Number Line Errors

Unequal intervals.

Avoid: Count equal segments.

Mistake 6: Adding Fractions Wrongly

Adding numerators/denominators directly.

Avoid: Common denominator first (though not in chapter, preview).

History & Fun Facts

Ancient Origins

Fractions used in India since Rig Veda; \(\frac{3}{4}\) as 'tri-pada' or 'teen paav' in Hindi, 'mukkaal' in Tamil.

Words for fractions in Indian languages date back centuries.

Real-Life Applications

Daily Life: Sharing food, measuring weights.
Cooking: Recipes use halves, quarters.
Time: Half-hour, quarter-hour.
Money: Dividing bills equally.

Fun Facts

Egyptians used fractions without numerators >1, except \(\frac{2}{3}\).
\(\frac{1}{2}\) is the most common fraction in daily use.
Fractions help in music: Notes as quarters, halves.
Infinite fractions between any two numbers.
Pizza sharing teaches fractions visually.

Did You Know?

Virahanka numbers relate to fractions in poetry meters.

Quick Revision One-Pager

Key Concepts and Examples

Concept	Example
Unit Fraction	\(\frac{1}{4}\): 1/4 roti
Proper Fraction	\(\frac{3}{4} < 1\)
Improper Fraction	\(\frac{5}{4} > 1\)
Mixed Fraction	\(1 \frac{1}{4}\)
Number Line	Mark \(\frac{2}{3}\) between 0-1

Quick Rules

✓ Fractions: Equal parts of whole.
✓ Larger denominator: Smaller value for unit fractions.
✓ Mixed: Whole + fraction part.
✓ Number Line: Divide equally.
✓ Reading: 'Times' unit fraction.

Mind Map

Central: Fractions

Types: Unit, proper, improper, mixed
Representations:
- Shares (roti/chikki)
- Lengths (strips/number line)
Applications: Sharing, measuring

Exam Tips

Before Solving

Identify whole and parts

During Solving

Draw visuals

After Solving

Check if equal

Time-Savers

Use unit fractions

Interactive Quiz – 15 Questions

Test Your Fractions Knowledge!