Class 7 Maths Ch 8: Working with Fractions – learn to multiply fractions with whole numbers and other fractions with notes, solved sums, extra questions and quiz for CBSE Exam

Complete Chapter 8 guide: multiplying a whole number by a fraction (Aaron’s walking distance), multiplying a fraction by a fraction (tortoise speed example), using unit squares to visualize fraction multiplication, solving word problems involving land distribution, internet costs, milk consumption, and moon setting times, plus clear steps, solved examples and practice questions for CBSE Class 7 Maths

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Class 7 Mathematics Chapter 8: Working with Fractions – Complete Notes, Solutions, Questions & Answers 2025

Chapter at a Glance – Working with Fractions

This chapter introduces multiplication and division of fractions, including operations with whole numbers and fractions, reciprocals, Brahmagupta's formulas, and practical applications.

Main Topics Covered

Multiplication of fractions with whole numbers
Multiplication of two fractions
Connection between area and fraction multiplication
Simplifying fractions by cancelling common factors
Division of fractions using reciprocals
Brahmagupta's formulas for multiplication and division
Product and quotient relationships
Real-world problem solving with fractions

Key Takeaways for Exams

Fraction × Whole

\(\frac{a}{b} \times c = \frac{a \times c}{b}\)

Fraction × Fraction

\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)

Reciprocal

Reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\)

Division Formula

\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

Cancel Common Factors

Simplify before multiplying

Area Connection

Rectangle area = product of sides

Key Formulas & Rules – Working with Fractions

Important formulas and rules for fraction operations.

Multiplication Rules

Operation	Formula	Example
Whole × Fraction	\(n \times \frac{a}{b} = \frac{n \times a}{b}\)	\(5 \times \frac{2}{3} = \frac{10}{3}\)
Fraction × Fraction	\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)	\(\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}\)
Unit Fraction × Unit Fraction	\(\frac{1}{b} \times \frac{1}{d} = \frac{1}{b \times d}\)	\(\frac{1}{12} \times \frac{1}{18} = \frac{1}{216}\)

Division Rules

Operation	Formula	Example
Fraction ÷ Fraction	\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)	\(\frac{2}{3} \div \frac{3}{5} = \frac{2}{3} \times \frac{5}{3} = \frac{10}{9}\)
Whole ÷ Fraction	\(n \div \frac{a}{b} = n \times \frac{b}{a}\)	\(6 \div \frac{1}{4} = 6 \times 4 = 24\)
Fraction ÷ Whole	\(\frac{a}{b} \div n = \frac{a}{b \times n}\)	\(\frac{1}{4} \div 5 = \frac{1}{20}\)

Important Properties

Reciprocal Property: \(\frac{a}{b} \times \frac{b}{a} = 1\)
Commutative: \(\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}\)
Area Formula: Area of rectangle with fractional sides = product of length and breadth
Cancellation: Divide numerator and denominator by common factors before multiplying

Product & Quotient Relationships

For Multiplication:

If both numbers > 1: Product > both numbers
If both numbers between 0 and 1: Product < both numbers
If one between 0 and 1, one > 1: Product is in between

For Division:

If divisor between 0 and 1: Quotient > dividend
If divisor > 1: Quotient < dividend

Concept Cards – Quick Explanations

Multiplying Whole × Fraction

Divide multiplicand by denominator, then multiply by numerator.

\(5 \times \frac{2}{3} = 5 \times 2 \div 3 = \frac{10}{3}\)

Multiplying Fractions

Multiply numerators, multiply denominators.

\(\frac{3}{4} \times \frac{2}{5} = \frac{6}{20}\)

Unit Square Method

Use unit square to visualize fraction multiplication geometrically.

Area Connection

Rectangle area with sides \(\frac{1}{2}\) and \(\frac{1}{4}\) is \(\frac{1}{8}\) sq units.

Cancelling Common Factors

\(\frac{12}{7} \times \frac{5}{24} = \frac{1 \times 5}{7 \times 2} = \frac{5}{14}\)

Reciprocal

Flip numerator and denominator. Reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\).

Division Method

Multiply by reciprocal of divisor.

\(\frac{2}{3} \div \frac{3}{5} = \frac{2}{3} \times \frac{5}{3}\)

Brahmagupta's Formulas

First stated in 628 CE for general fractions.

Product < 1

When both fractions < 1, product < both.

Quotient > Dividend

When divisor < 1, quotient > dividend.

Order Doesn't Matter

\(\frac{1}{2} \times \frac{1}{4} = \frac{1}{4} \times \frac{1}{2}\)

Examples + Solutions

Example 1: Distance Covered by Tortoise

Problem: A tortoise walks \(\frac{1}{4}\) km in 1 hour. How far in 3 hours?

Solution:

Distance = \(3 \times \frac{1}{4} = \frac{3}{4}\) km

Answer: \(\frac{3}{4}\) km

Example 2: Aaron Walking

Problem: Aaron walks 3 km in 1 hour. How far in \(\frac{2}{5}\) hours?

Solution:

In \(\frac{1}{5}\) hour: \(3 \div 5 = \frac{3}{5}\) km
In \(\frac{2}{5}\) hours: \(2 \times \frac{3}{5} = \frac{6}{5}\) km

Answer: \(\frac{6}{5}\) km = 1.2 km

Example 3: Farmer's Land Distribution

Problem: A farmer gives \(\frac{2}{3}\) acre to each of 5 grandchildren. Total land?

Solution:

\(5 \times \frac{2}{3} = \frac{10}{3}\) acres = \(3\frac{1}{3}\) acres

Answer: \(\frac{10}{3}\) acres

Example 4: Internet Time Cost

Problem: 1 hour costs ₹8. Cost for \(1\frac{1}{4}\) hours?

Solution:

\(1\frac{1}{4} = \frac{5}{4}\) hours
Cost = \(\frac{5}{4} \times 8 = 5 \times 2 = 10\)

Answer: ₹10

Example 5: Tortoise in Half Hour

Problem: Tortoise walks \(\frac{1}{4}\) km in 1 hour. Distance in \(\frac{1}{2}\) hour?

Solution:

\(\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\) km

Answer: \(\frac{1}{8}\) km

Example 6: Two Fractions Multiplication

Problem: Find \(\frac{3}{4} \times \frac{2}{5}\)

Solution:

Using unit square: Whole divided into 4×5=20 parts, 3×2=6 shaded.

\(\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}\)

Answer: \(\frac{3}{10}\)

Example 7: Leena's Tea

Problem: 5 cups use \(\frac{1}{4}\) litre milk. Milk per cup?

Solution:

\(\frac{1}{4} \div 5 = \frac{1}{4} \times \frac{1}{5} = \frac{1}{20}\) litre

Answer: \(\frac{1}{20}\) litre per cup

Example 8: Baudhāyana's Problem

Problem: Cover \(7\frac{1}{2}\) sq units with square bricks of side \(\frac{1}{5}\) units. How many bricks?

Solution:

Each brick area = \(\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}\) sq units
Total area = \(\frac{15}{2}\) sq units
Number = \(\frac{15}{2} \div \frac{1}{25} = \frac{15}{2} \times 25 = \frac{375}{2}\)

Answer: \(\frac{375}{2}\) = 187.5 bricks

Example 9: Four Fountains Problem

Problem: Four fountains fill cistern in 1 day, \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{5}\) days. Time together?

Solution:

First: 1 cistern/day
Second: 2 cisterns/day
Third: 4 cisterns/day
Fourth: 5 cisterns/day
Together: 1+2+4+5 = 12 cisterns/day
Time = \(\frac{1}{12}\) day

Answer: \(\frac{1}{12}\) day

Figure it Out Solutions (All Solved)

Section 8.1 - Page 176

1. Tenzin drinks \(\frac{1}{2}\) glass milk daily. In a week? In January?

Week: \(7 \times \frac{1}{2} = \frac{7}{2} = 3\frac{1}{2}\) glasses

January (31 days): \(31 \times \frac{1}{2} = \frac{31}{2} = 15\frac{1}{2}\) glasses

2. Team makes 1 km canal in 8 days. In one day? In a week?

One day: \(\frac{1}{8}\) km

One week (5 days): \(5 \times \frac{1}{8} = \frac{5}{8}\) km

3. Manju and 2 neighbors share 5 litres oil weekly. Each family gets? In 4 weeks?

Per week: \(5 \div 3 = \frac{5}{3}\) litres

In 4 weeks: \(4 \times \frac{5}{3} = \frac{20}{3} = 6\frac{2}{3}\) litres

4. Moon sets \(\frac{5}{6}\) hour later daily. Monday 10pm, Thursday time?

Days later: 3 days (Tue, Wed, Thu)

Delay: \(3 \times \frac{5}{6} = \frac{15}{6} = 2\frac{1}{2}\) hours

Answer: 12:30 am (Thursday night)

5. Multiply and convert to mixed fraction:

(a) \(7 \times \frac{3}{5} = \frac{21}{5} = 4\frac{1}{5}\)
(b) \(4 \times \frac{1}{3} = \frac{4}{3} = 1\frac{1}{3}\)
(c) \(\frac{9}{7} \times 6 = \frac{54}{7} = 7\frac{5}{7}\)
(d) \(\frac{13}{11} \times 6 = \frac{78}{11} = 7\frac{1}{11}\)

Section 8.1 - Page 180-181

1. Find products using unit square:

(a) \(\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}\)
(b) \(\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}\)
(c) \(\frac{1}{5} \times \frac{1}{2} = \frac{1}{10}\)
(d) \(\frac{1}{6} \times \frac{1}{5} = \frac{1}{30}\)

\(\frac{1}{12} \times \frac{1}{18} = \frac{1}{216}\)

2. Find products using unit square:

(a) \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)
(b) \(\frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}\)
(c) \(\frac{3}{5} \times \frac{1}{2} = \frac{3}{10}\)
(d) \(\frac{4}{6} \times \frac{3}{5} = \frac{12}{30} = \frac{2}{5}\)

Section 8.1 - Page 183

1. Water tank fills \(\frac{7}{10}\) in 1 hour. How much in:

(a) \(\frac{1}{3}\) hour: \(\frac{1}{3} \times \frac{7}{10} = \frac{7}{30}\)
(b) \(\frac{2}{3}\) hour: \(\frac{2}{3} \times \frac{7}{10} = \frac{14}{30} = \frac{7}{15}\)
(c) \(\frac{3}{4}\) hour: \(\frac{3}{4} \times \frac{7}{10} = \frac{21}{40}\)
(d) \(\frac{7}{10}\) hour: \(\frac{7}{10} \times \frac{7}{10} = \frac{49}{100}\)
(e) For full tank: \(\frac{10}{7}\) hours = \(1\frac{3}{7}\) hours

2. Somu's land: \(\frac{1}{6}\) for road, half remaining to Krishna, \(\frac{1}{3}\) to Bora.

Remaining after road: \(1 - \frac{1}{6} = \frac{5}{6}\)

(a) Krishna: \(\frac{1}{2} \times \frac{5}{6} = \frac{5}{12}\)
(b) Bora: \(\frac{1}{3} \times \frac{5}{6} = \frac{5}{18}\)
(c) Somu: \(\frac{5}{6} - \frac{5}{12} - \frac{5}{18} = \frac{15-7.5-4.17}{18} = \frac{5}{36}\) (approximately)

3. Rectangle area: sides \(3\frac{3}{4}\) ft and \(9\frac{3}{5}\) ft

\(3\frac{3}{4} = \frac{15}{4}\), \(9\frac{3}{5} = \frac{48}{5}\)

Area = \(\frac{15}{4} \times \frac{48}{5} = \frac{15 \times 48}{20} = \frac{720}{20} = 36\) sq ft

4. Four saplings, \(\frac{3}{4}\) m apart. Distance first to last?

3 gaps between 4 saplings

Distance = \(3 \times \frac{3}{4} = \frac{9}{4} = 2\frac{1}{4}\) m

5. Heavier: \(\frac{12}{15}\) of 500g or \(\frac{3}{20}\) of 4kg?

First: \(\frac{12}{15} \times 500 = \frac{4}{5} \times 500 = 400\)g

Second: \(\frac{3}{20} \times 4000 = 600\)g

Answer: \(\frac{3}{20}\) of 4kg is heavier

Section 8.3 - Page 196-198

1. Evaluate divisions:

\(3 \div \frac{7}{9} = 3 \times \frac{9}{7} = \frac{27}{7}\)
\(\frac{14}{4} \div 2 = \frac{14}{4} \times \frac{1}{2} = \frac{14}{8} = \frac{7}{4}\)
\(\frac{2}{3} \div \frac{14}{6} = \frac{2}{3} \times \frac{6}{14} = \frac{12}{42} = \frac{2}{7}\)
\(\frac{4}{3} \div \frac{3}{4} = \frac{4}{3} \times \frac{4}{3} = \frac{16}{9}\)
\(\frac{7}{4} \div \frac{1}{7} = \frac{7}{4} \times 7 = \frac{49}{4}\)
\(\frac{8}{2} \div \frac{4}{15} = 4 \times \frac{15}{4} = 15\)
\(\frac{1}{5} \div \frac{1}{9} = \frac{1}{5} \times 9 = \frac{9}{5}\)
\(\frac{1}{6} \div \frac{11}{12} = \frac{1}{6} \times \frac{12}{11} = \frac{2}{11}\)
\(3\frac{2}{3} \div 1\frac{3}{8} = \frac{11}{3} \div \frac{11}{8} = \frac{11}{3} \times \frac{8}{11} = \frac{8}{3}\)

2. Choose correct expression:

(a) 8m lace, \(\frac{1}{4}\)m per bag: (iii) \(8 \div \frac{1}{4} = 32\) bags
(b) \(\frac{1}{2}\)m ribbon for 8 badges: (iv) \(\frac{1}{2} \div 8 = \frac{1}{16}\)m
(c) \(\frac{1}{6}\)kg flour per loaf, 5kg total: (iii) \(5 \div \frac{1}{6} = 30\) loaves

3. \(\frac{1}{4}\)kg flour for 12 rotis. For 6 rotis?

Per roti: \(\frac{1}{4} \div 12 = \frac{1}{48}\)kg

For 6: \(6 \times \frac{1}{48} = \frac{6}{48} = \frac{1}{8}\)kg

4. Sridharacharya problem: Sum of \(1 \div \frac{1}{6}\), \(1 \div \frac{1}{10}\), etc.

\(1 \div \frac{1}{6} = 6\), \(1 \div \frac{1}{10} = 10\), \(1 \div \frac{1}{13} = 13\)

\(1 \div \frac{1}{9} = 9\), \(1 \div \frac{1}{2} = 2\)

Sum = 6+10+13+9+2 = 40

5. Mira's novel: 400 pages, read \(\frac{1}{5}\) yesterday, \(\frac{3}{10}\) today. Pages left?

Yesterday: \(\frac{1}{5} \times 400 = 80\) pages

Today: \(\frac{3}{10} \times 400 = 120\) pages

Left: 400-80-120 = 200 pages

6. Car runs 16km per litre. Distance with \(2\frac{3}{4}\) litres?

\(2\frac{3}{4} = \frac{11}{4}\) litres

Distance = \(16 \times \frac{11}{4} = 4 \times 11 = 44\)km

7. Train takes \(5\frac{1}{6}\) hours, plane \(\frac{1}{2}\) hour. Time saved?

\(5\frac{1}{6} - \frac{1}{2} = \frac{31}{6} - \frac{3}{6} = \frac{28}{6} = \frac{14}{3} = 4\frac{2}{3}\) hours

8. \(\frac{4}{5}\) cake finished, remaining shared by 3 friends. Each gets?

Remaining: \(1 - \frac{4}{5} = \frac{1}{5}\)

Each friend: \(\frac{1}{5} \div 3 = \frac{1}{15}\)

9. Product \(\frac{565}{465} \times \frac{707}{676}\) is:

Both fractions > 1, so product > both: (a), (c), (e)

10-12. Fraction puzzles with diagrams

Solve by breaking down geometrically or using patterns

(12) Pattern: Each term = \(\frac{n}{n+1}\), product telescopes to \(\frac{1}{10}\)

Extra Practice Questions (Exam-Ready) – Chapter 8

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

1-Mark Questions (Very Short Answer)

1. What is reciprocal of \(\frac{7}{9}\)?

2. \(\frac{1}{4} \times \frac{1}{5} = ?\)

3. What is \(5 \times \frac{1}{3}\)?

4. Brahmagupta's multiplication formula?

5. What is \(1 \div \frac{1}{2}\)?

2-Mark Questions (Short Answer)

6. Find \(\frac{3}{4} \times \frac{8}{9}\)

7. Simplify: \(\frac{12}{5} \div \frac{3}{10}\)

8. Convert \(2\frac{3}{4}\) to improper and multiply by 6

9. Area of rectangle: \(\frac{2}{3}\)m × \(\frac{3}{5}\)m

10. If \(\frac{3}{4}\)kg for 6 bags, how much per bag?

3-Mark Questions (Reasoning)

11. When is product of two fractions less than both?

12. Explain cancelling common factors with example

13. Why does dividing by fraction < 1 increase the number?

14. Find \(\frac{5}{12} \times \frac{7}{18}\) using Brahmagupta

15. Compare: \(\frac{3}{4} \times 5\) and \(\frac{3}{4} \div 5\)

4–5 Mark Questions (Application)

16. \(\frac{2}{5}\) of water tank fills in 1 hour. Time to fill completely?

17. A baker uses \(\frac{3}{8}\)kg flour per cake. Flour for 12 cakes?

18. Runner covers \(\frac{5}{6}\)km in \(\frac{1}{4}\) hour. Speed in km/h?

19. Miser's dramma problem: Find \(\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{1}{4} \times \frac{1}{5} \times \frac{1}{16}\)

20. Field \(\frac{3}{5}\) completed, \(\frac{2}{3}\) of remaining tomorrow. Fraction left?

Challenge Questions (6+ Marks)

21. Three pipes fill tank in \(\frac{2}{3}\), \(\frac{3}{4}\), \(\frac{4}{5}\) hours. Time together?

22. Rectangle sides \(3\frac{1}{4}\) and \(2\frac{2}{5}\). Find area and perimeter.

23. Pattern: \((1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{n})\). Find general form.

24. Worker completes \(\frac{2}{7}\) work in 4 days. Days for full work?

25. Fraction of square shaded (complex geometric)

History & Fun Facts

Brahmagupta (628 CE) - Father of Fraction Arithmetic

Brahmagupta first codified general rules for fraction operations in his book Brāhmasphuṭasiddhānta.

His Multiplication Rule: "Product of numerators divided by product of denominators"

His Division Rule: "Interchange numerator and denominator of divisor, then multiply"

Ancient Indian Contributions

Śhulbasūtra (800 BCE): Used non-unit fractions in geometry for fire altar construction
Umasvati (150 BCE): Mentioned fraction reduction in philosophical work
Bhāskara I (629 CE): Gave geometric interpretation using unit squares
Bhāskara II (1150 CE): Explained reciprocals clearly in Līlāvatī

Fun Problems from History

Bhāskarāchārya's Miser Problem

A miser gave \(\frac{1}{5}\) of \(\frac{1}{16}\) of \(\frac{1}{4}\) of \(\frac{1}{2}\) of \(\frac{2}{3}\) of \(\frac{3}{4}\) of a dramma.

Product = \(\frac{1}{1280}\) dramma = 1 cowrie shell (lowest coin!)

Shows mathematician's humor about miserly behavior!

Currency in Ancient India

Coin Type	Material	Value
Dinar/Gadyana	Gold	Highest
Dramma/Tanka	Silver	Medium
Kasu/Pana	Copper	Low
Cowrie Shell	Shell	Lowest

Typical conversions: 1 dramma = 1280 cowrie shells

Global Spread

Indian fraction theory spread through:

Arab mathematicians like al-Hassâr (c. 1192 CE) of Morocco
Europe via Arabs in medieval period
Worldwide adoption by 17th century

Did You Know?

Unit squares for visualization date back to Bhāskara I (629 CE)
Egyptian fractions (sum of unit fractions) were different from Indian approach
Modern fraction bar (÷ line) came much later; ancients wrote fractions differently
Chess problem (non-attacking queens) connects to fraction patterns

Quick Revision One-Pager

Key Formulas Summary

Operation	Formula	Key Point
Multiplication	\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)	Multiply across
Division	\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)	Multiply by reciprocal
Reciprocal	Of \(\frac{a}{b}\) is \(\frac{b}{a}\)	Flip fraction
Cancellation	Divide common factors first	Simplifies calculation
Area	Rectangle = l × b	Works for fractions too

Quick Rules

✓ Product of two fractions < 1: Both must be < 1
✓ Product > both numbers: Both must be > 1
✓ Quotient > dividend: Divisor < 1
✓ Quotient < dividend: Divisor > 1
✓ Always simplify to lowest terms
✓ Convert mixed fractions before operations
✓ Cancellation saves time and reduces errors

Mind Map

Central: Working with Fractions

Multiplication:
- Whole × Fraction
- Fraction × Fraction
- Brahmagupta's formula
- Unit square visualization
- Cancel before multiply
Division:
- Reciprocal method
- Convert to multiplication
- Brahmagupta's division rule
Applications:
- Distance problems
- Area calculations
- Work/time problems
- Sharing problems
Properties:
- Product relationships
- Quotient relationships
- Commutative property

Exam Tips

Before Solving

Convert mixed fractions, identify operation needed

During Solving

Cancel common factors, show all steps clearly

After Solving

Simplify answer, check if it makes sense

Time-Savers

Use cancellation, memorize reciprocals of common fractions

This article has been published in CBSE Class 7 Annual Assessment. Explore everything related to it here.

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