Class 7 : Maths – Lesson 8. Working with Fractions

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On December 11, 2025

EXPLANATION AND ANALYSIS

🔵 Introduction: Fractions in Everyday Life

🧠 In daily life, we often come across situations where a whole object is divided into parts. For example, half a roti, one-fourth of a cake, or three-quarters of an hour. Such parts of a whole are represented using fractions.

🌿 Fractions help us express quantities that are not whole numbers and allow us to perform calculations accurately in real-life situations like sharing, measuring, and dividing.

This chapter focuses on working with fractions, that is, understanding them better and learning how to perform operations on them.

🟢 Meaning of a Fraction

🧠 A fraction represents a part of a whole.

🔹 It has two parts: numerator and denominator
🔹 The numerator shows the number of parts taken
🔹 The denominator shows the total number of equal parts

📌 Example
In the fraction 3/5
🔹 3 is the numerator
🔹 5 is the denominator

💡 Concept:
Fractions are meaningful only when the whole is divided into equal parts.

🔵 Types of Fractions

🧠 Fractions can be classified based on their values.

🔹 Proper fraction: numerator < denominator
🔹 Improper fraction: numerator ≥ denominator
🔹 Mixed fraction: a whole number with a fraction

📌 Examples
🔹 2/7 is a proper fraction
🔹 9/4 is an improper fraction
🔹 2 1/3 is a mixed fraction

✏️ Note:
Improper fractions can be converted into mixed fractions and vice versa.

🟢 Equivalent Fractions

🧠 Equivalent fractions are fractions that represent the same value.

🔹 They look different but mean the same
🔹 Obtained by multiplying or dividing numerator and denominator by the same number

📌 Example
1/2 = 2/4 = 3/6

💡 Concept:
Multiplying or dividing both parts of a fraction by the same number does not change its value.

🔵 Simplest Form of a Fraction

🧠 A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.

📌 Example
6/12 can be simplified to
3/6 → 1/2

✏️ Note:
Simplifying makes fractions easier to understand and compare.

🟢 Comparing Fractions

🧠 To compare fractions, we must bring them to a common denominator.

🔹 Convert fractions into equivalent fractions
🔹 Compare numerators

📌 Example
Compare 3/4 and 2/3

🔹 LCM of 4 and 3 is 12
🔹 3/4 = 9/12
🔹 2/3 = 8/12
🔹 9/12 > 8/12

💡 Concept:
The fraction with the greater numerator is greater when denominators are the same.

🔵 Addition of Fractions

🧠 Fractions can be added by following these steps.

🔹 Find a common denominator
🔹 Convert fractions to equivalent fractions
🔹 Add the numerators

📌 Example
1/4 + 2/4 = 3/4

✏️ Note:
Fractions with the same denominator can be added directly.

🟢 Subtraction of Fractions

🧠 Subtraction of fractions is similar to addition.

🔹 Use a common denominator
🔹 Subtract the numerators

📌 Example
5/6 − 1/6 = 4/6 = 2/3

💡 Concept:
Always simplify the result if possible.

🔵 Multiplication of Fractions

🧠 To multiply fractions, multiply numerators together and denominators together.

📌 Example
2/3 × 4/5

🔹 Numerator: 2 × 4 = 8
🔹 Denominator: 3 × 5 = 15
🔹 Result = 8/15

✏️ Note:
Cross-cancellation can make multiplication easier.

🟢 Division of Fractions

🧠 Dividing by a fraction means multiplying by its reciprocal.

🔹 Reciprocal of a/b is b/a

📌 Example
3/4 ÷ 2/5

🔹 Reciprocal of 2/5 = 5/2
🔹 3/4 × 5/2 = 15/8

💡 Concept:
Division of fractions always involves multiplication by the reciprocal.

🔴 Common Mistakes to Avoid

🔴 Adding denominators directly
🔴 Forgetting to simplify fractions
🔴 Not finding a common denominator
🔴 Mixing up reciprocal in division

✏️ Note:
Step-by-step work reduces errors while working with fractions.

🟢 Importance of Fractions

🧠 Learning to work with fractions helps students to:

🔹 Share quantities fairly
🔹 Measure accurately
🔹 Understand ratios and percentages
🔹 Prepare for algebra

Fractions form a strong base for higher mathematical concepts.

📘 Summary

🔵 Fractions represent parts of a whole
🟢 They have numerator and denominator
🟡 Equivalent fractions represent the same value
🔴 Fractions can be simplified
🔵 Fractions can be added, subtracted, multiplied, and divided
🟢 Correct steps ensure correct answers

📝 Quick Recap

📝 Quick Recap
🔵 Fractions show parts of a whole
🟢 Equivalent fractions have the same value
🟡 Use common denominators to add or subtract
🔴 Multiply numerators and denominators to multiply fractions
🔵 Divide fractions by multiplying with reciprocal

——————————————————————————————————————————————————————————————————————————–

TEXTBOOK QUESTIONS

🔵 MULTIPLICATION OF FRACTIONS

🔒 ❓ 1. Tenzin drinks 1/2 glass of milk every day. How many glasses of milk does he drink in a week? How many glasses of milk did he drink in the month of January?

📌 ✅ Answer:

🔹 Milk per day = 1/2 glass

🔹 In one week (7 days):
7 × 1/2 = 7/2

🔹 7/2 = 3 1/2 glasses

🔹 In January (31 days):
31 × 1/2 = 31/2

🔹 31/2 = 15 1/2 glasses

🔒 ❓ 2. A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make ____ km of the water canal. If they work 5 days a week, they can make ____ km of the water canal in a week.

📌 ✅ Answer:

🔹 Work in 8 days = 1 km

🔹 Work in 1 day = 1/8 km

🔹 Work in 5 days = 5 × 1/8

🔹 = 5/8 km

🔒 ❓ 3. Manju and two of her neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in a week? How much oil will one family get in 4 weeks?

📌 ✅ Answer:

🔹 Total oil = 5 litres

🔹 Number of families = 3

🔹 Oil per family per week = 5/3 litres

🔹 5/3 = 1 2/3 litres

🔹 In 4 weeks:
4 × 5/3 = 20/3

🔹 20/3 = 6 2/3 litres

🔒 ❓ 4. Safa saw the Moon setting on Monday at 10 pm. Her mother told her that every day the Moon sets 5/6 hour later than the previous day. How many hours after 10 pm will the moon set on Thursday?

📌 ✅ Answer:

🔹 Increase per day = 5/6 hour

🔹 From Monday to Thursday = 3 days

🔹 Total increase = 3 × 5/6

🔹 = 15/6

🔹 = 5/2

🔹 5/2 = 2 1/2 hours

🔹 So, moon will set 2 1/2 hours after 10 pm

🔹 That is 12:30 am

🔒 ❓ 5. Multiply and then convert it into a mixed fraction:

🔒 ❓ (a) 7 × 3/5

📌 ✅ Answer:

🔹 7 × 3/5 = 21/5

🔹 21/5 = 4 1/5

🔒 ❓ (b) 4 × 1/3

📌 ✅ Answer:

🔹 4 × 1/3 = 4/3

🔹 4/3 = 1 1/3

🔒 ❓ (c) 9/7 × 6

📌 ✅ Answer:

🔹 9/7 × 6 = 54/7

🔹 54/7 = 7 5/7

🔒 ❓ (d) 13/11 × 6

📌 ✅ Answer:

🔹 13/11 × 6 = 78/11

🔹 78/11 = 7 1/11

🔵CONNECTION BETWEEN THE AREA OF A RECTANGLE AND FRACTION MULTIPLICATION

🔒 ❓ 1. Find the following products. Use a unit square as a whole for representing the fractions:

(a) 1/3 × 1/5

📌 ✅ Answer:
🔹 Multiply numerators: 1 × 1 = 1
🔹 Multiply denominators: 3 × 5 = 15
🔹 Product = 1/15

(b) 1/4 × 1/3

📌 ✅ Answer:
🔹 Multiply numerators: 1 × 1 = 1
🔹 Multiply denominators: 4 × 3 = 12
🔹 Product = 1/12

📌 ✅ Answer:
🔹 Multiply numerators: 1 × 1 = 1
🔹 Multiply denominators: 5 × 2 = 10
🔹 Product = 1/10

(d) 1/6 × 1/5

📌 ✅ Answer:
🔹 Multiply numerators: 1 × 1 = 1
🔹 Multiply denominators: 6 × 5 = 30
🔹 Product = 1/30

Now, find 1/12 × 1/18

📌 ✅ Answer:
🔹 Multiply numerators: 1 × 1 = 1
🔹 Multiply denominators: 12 × 18 = 216
🔹 Product = 1/216

🔹 General Rule Observed:
🔸 When two unit fractions are multiplied,
🔸 The result is 1/(product of denominators).

🔒 ❓ 2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations.

(a) 2/3 × 4/5

📌 ✅ Answer:
🔹 Multiply numerators: 2 × 4 = 8
🔹 Multiply denominators: 3 × 5 = 15
🔹 Product = 8/15

(b) 1/4 × 2/3

📌 ✅ Answer:
🔹 Multiply numerators: 1 × 2 = 2
🔹 Multiply denominators: 4 × 3 = 12
🔹 Simplify 2/12 by dividing by 2
🔹 Product = 1/6

📌 ✅ Answer:
🔹 Multiply numerators: 3 × 1 = 3
🔹 Multiply denominators: 5 × 2 = 10
🔹 Product = 3/10

(d) 4/6 × 3/5

📌 ✅ Answer:
🔹 Simplify 4/6 = 2/3
🔹 Multiply numerators: 2 × 3 = 6
🔹 Multiply denominators: 3 × 5 = 15
🔹 Simplify 6/15 by dividing by 3
🔹 Product = 2/5

🔵A PINCH OF HISTORY

🔒 ❓ 1. A water tank is filled from a tap. If the tap is open for 1 hour, 7/10 of the tank gets filled. How much of the tank is filled if the tap is open for

(a) 1/3 hour ________
(b) 2/3 hour ________
(c) 3/4 hour ________
(d) 7/10 hour ________
(e) For the tank to be full, how long should the tap be running?

📌 ✅ Answer:

🔹 In 1 hour, filled = 7/10 of tank
🔹 So, in 1 hour → rate = 7/10 tank

(a) 1/3 hour

🔹 Filled = 7/10 × 1/3
🔸 = 7/30

(b) 2/3 hour

🔹 Filled = 7/10 × 2/3
🔸 = 14/30
🔸 = 7/15

🔹 Filled = 7/10 × 3/4
🔸 = 21/40

(d) 7/10 hour

🔹 Filled = 7/10 × 7/10
🔸 = 49/100

(e) For full tank

🔹 Let required time = t hours
🔹 7/10 × t = 1
🔸 t = 1 ÷ (7/10)
🔸 = 1 × 10/7
🔸 = 10/7 hours

🔹 10/7 = 1 3/7 hours

🔒 ❓ 2. The government has taken 1/6 of Somu’s land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and 1/3 of it to her son Bora. After giving them their shares, she keeps the remaining land for herself.

(a) What part of the original land did Krishna get?
(b) What part of the original land did Bora get?
(c) What part of the original land did Somu keep for herself?

📌 ✅ Answer:

🔹 Land taken = 1/6
🔹 Remaining land = 1 − 1/6
🔸 = 5/6

(a) Krishna gets half of remaining

🔹 = 1/2 × 5/6
🔸 = 5/12

(b) Bora gets 1/3 of remaining

🔹 = 1/3 × 5/6
🔸 = 5/18

🔹 Total given = 5/12 + 5/18

🔹 LCM of 12 and 18 = 36

🔹 5/12 = 15/36
🔹 5/18 = 10/36

🔹 Total given = 25/36

🔹 Remaining from 5/6

🔹 5/6 = 30/36

🔹 Somu keeps = 30/36 − 25/36
🔸 = 5/36

🔒 ❓ 3. Find the area of a rectangle of sides 3 3/4 ft and 9 3/5 ft.

📌 ✅ Answer:

🔹 3 3/4 = 15/4
🔹 9 3/5 = 48/5

🔹 Area = 15/4 × 48/5

🔹 Cancel 15 and 5

🔸 = 3/4 × 48

🔹 Cancel 48 and 4

🔸 = 3 × 12
🔸 = 36

🔹 Area = 36 sq ft

🔒 ❓ 4. Tsewang plants four saplings in a row in his garden. The distance between two saplings is 3/4 m. Find the distance between the first and last sapling.

📌 ✅ Answer:

🔹 Four saplings create 3 gaps

🔹 Distance = 3 × 3/4
🔸 = 9/4 m
🔸 = 2 1/4 m

🔒 ❓ 5. Which is heavier: 12/15 of 500 grams or 3/20 of 4 kg?

📌 ✅ Answer:

🔹 12/15 of 500 g

🔸 = 4/5 × 500
🔸 = 400 g

🔹 3/20 of 4 kg

🔹 4 kg = 4000 g

🔸 = 3/20 × 4000
🔸 = 600 g

🔹 600 g > 400 g

🔹 3/20 of 4 kg is heavier.

🔵 A PINCH OF HISTORY

🔒 ❓ 1. Evaluate the following:

📌 ✅ Answer:

🔹 3 ÷ 7/9
🔸 = 3 × 9/7
🔸 = 27/7
🔸 = 3 6/7

🔹 14/4 ÷ 2
🔸 = 7/2 ÷ 2
🔸 = 7/2 × 1/2
🔸 = 7/4
🔸 = 1 3/4

🔹 2/3 ÷ 2/3
🔸 = 2/3 × 3/2
🔸 = 1

🔹 14/6 ÷ 7/3
🔸 = 7/3 ÷ 7/3
🔸 = 7/3 × 3/7
🔸 = 1

🔹 4/3 ÷ 3/4
🔸 = 4/3 × 4/3
🔸 = 16/9
🔸 = 1 7/9

🔹 7/4 ÷ 1/7
🔸 = 7/4 × 7/1
🔸 = 49/4
🔸 = 12 1/4

🔹 8/2 ÷ 4/15
🔸 = 4 ÷ 4/15
🔸 = 4 × 15/4
🔸 = 15

🔹 1/5 ÷ 1/9
🔸 = 1/5 × 9/1
🔸 = 9/5
🔸 = 1 4/5

🔹 1/6 ÷ 11/12
🔸 = 1/6 × 12/11
🔸 = 12/66
🔸 = 2/11

🔹 3 2/3 ÷ 1 3/8
🔸 = 11/3 ÷ 11/8
🔸 = 11/3 × 8/11
🔸 = 8/3
🔸 = 2 2/3

🔒 ❓ 2. For each of the questions below, choose the expression that describes the solution. Then simplify it.

(a) Maria bought 8 m of lace. She used 1/4 m for each bag. How many bags did she decorate?
(i) 8 × 1/4
(ii) 1/8 × 1/4
(iii) 8 ÷ 1/4
(iv) 1/4 ÷ 8

📌 ✅ Answer:

🔹 Correct expression: (iii) 8 ÷ 1/4
🔸 = 8 × 4/1
🔸 = 32
🔹 Bags decorated = 32

(b) 1/2 meter of ribbon is used to make 8 badges. What is the length used for each badge?
(i) 8 × 1/2
(ii) 1/2 ÷ 1/8
(iii) 8 ÷ 1/2
(iv) 1/2 ÷ 8

📌 ✅ Answer:

🔹 Correct expression: (iv) 1/2 ÷ 8
🔸 = 1/2 × 1/8
🔸 = 1/16 m
🔹 Ribbon per badge = 1/16 m

(c) A baker needs 1/6 kg of flour for one loaf. He has 5 kg. How many loaves can he make?
(i) 5 × 1/6
(ii) 1/6 ÷ 5
(iii) 5 ÷ 1/6
(iv) 5 × 6

📌 ✅ Answer:

🔹 Correct expression: (iii) 5 ÷ 1/6
🔸 = 5 × 6/1
🔸 = 30
🔹 Loaves of bread = 30

🔒 ❓ 3. If 1/4 kg of flour is used to make 12 rotis, how much flour is used to make 6 rotis?

📌 ✅ Answer:

🔹 Flour for 1 roti
🔸 = (1/4) ÷ 12
🔸 = 1/4 × 1/12
🔸 = 1/48 kg

🔹 Flour for 6 rotis
🔸 = 6 × 1/48
🔸 = 6/48
🔸 = 1/8 kg

🔒 ❓ 4. Friend, after thinking, what sum will be obtained by adding together 1 ÷ 1/6 , 1 ÷ 1/10 , 1 ÷ 1/13 , 1 ÷ 1/9 , and 1 ÷ 1/2 ? What should the friend say?
📌 ✅ Answer:
🔹 1 ÷ (1/6) = 1 × 6 = 6
🔹 1 ÷ (1/10) = 1 × 10 = 10
🔹 1 ÷ (1/13) = 1 × 13 = 13
🔹 1 ÷ (1/9) = 1 × 9 = 9
🔹 1 ÷ (1/2) = 1 × 2 = 2
🔹 Sum = 6 + 10 + 13 + 9 + 2
🔹 Sum = 40

🔒 ❓ 5. Mira is reading a novel that has 400 pages. She read 1/5 of the pages yesterday and 3/10 of the pages today. How many more pages does she need to read to finish the novel?
📌 ✅ Answer:
🔹 Fraction read = 1/5 + 3/10
🔹 1/5 = 2/10
🔹 Fraction read = 2/10 + 3/10 = 5/10
🔹 5/10 = 1/2
🔹 Pages read = (1/2) × 400 = 200
🔹 Pages remaining = 400 − 200 = 200
🔹 Final: Mira needs to read 200 pages more.

🔒 ❓ 6. A car runs 16 km using 1 litre of petrol. How far will it go using 2 3/4 litres of petrol?
📌 ✅ Answer:
🔹 2 3/4 = 11/4 litres
🔹 Distance = 16 × (11/4) km
🔹 16 ÷ 4 = 4
🔹 Distance = 4 × 11 = 44 km
🔹 Final: The car will go 44 km.

🔒 ❓ 7. Amritpal decides on a destination for his vacation. If he takes a train, it will take him 5 1/6 hours to get there. If he takes a plane, it will take him 1/2 hour. How many hours does the plane save?
📌 ✅ Answer:
🔹 Train time = 5 1/6 = 31/6 hours
🔹 Plane time = 1/2 = 3/6 hours
🔹 Time saved = 31/6 − 3/6
🔹 Time saved = 28/6
🔹 28/6 = 14/3
🔹 14/3 = 4 2/3 hours
🔹 Final: The plane saves 4 2/3 hours.

🔒 ❓ 8. Mariam’s grandmother baked a cake. Mariam and her cousins finished 4/5 of the cake. The remaining cake was shared equally by Mariam’s three friends. How much of the cake did each friend get?
📌 ✅ Answer:
🔹 Fraction remaining = 1 − 4/5 = 1/5
🔹 Each friend’s share = (1/5) ÷ 3
🔹 (1/5) ÷ 3 = (1/5) × (1/3)
🔹 Each friend’s share = 1/15
🔹 Final: Each friend got 1/15 of the cake.

🔒 ❓ 9. Choose the option(s) describing the product of (565/465) × (707/676):
📌 ✅ Answer:
🔹 565/465 > 1 (numerator > denominator)
🔹 707/676 > 1 (numerator > denominator)
🔹 Product = (565/465) × (707/676) > 1
🔹 Since we multiply 565/465 by a number > 1, product > 565/465
🔹 Since we multiply 707/676 by a number > 1, product > 707/676
🔹 Final true options: (a), (c), (e)

🔒 ❓ Question 10
What fraction of the whole square is shaded?

📌 ✅ Answer:

🔹 The big square is divided into 4 equal small squares.
🔹 So, each small square = 1/4 of the whole square.

🔹 The shaded region lies inside the bottom-right small square.

🔹 That small square is divided into 2 equal parts by a diagonal line.
🔹 So, each triangular half = 1/2 of 1/4.

🔹 Therefore, shaded part
= 1/2 × 1/4
= 1/8

🔹 Hence, the shaded fraction of the whole square is

✔️ 1/8

🔒 ❓ Question 11
A colony of ants set out in search of food. As they search, they keep splitting equally at each red point (as shown in the figure) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source?

📌 ✅ Answer:

🔹 At every red point, the ants split into 2 equal groups.
Each split multiplies the fraction by 1/2.

🔹 Follow the path to the Mango Tree:
From the starting point to the mango tree, there are 3 equal splits.

🔹 Fraction reaching mango tree
= 1/2 × 1/2 × 1/2
= 1/8

🔹 Now follow the path to the Sugarcane Field:
From the starting point to the sugarcane field, there are also 3 equal splits.

🔹 Fraction reaching sugarcane field
= 1/2 × 1/2 × 1/2
= 1/8

🔹 Therefore,

Fraction reaching Mango Tree = 1/8
Fraction reaching Sugarcane Field = 1/8

✔️ Final Answer: Each food source receives 1/8 of the original group.

🔒 ❓ Question 12
What is 1 − 1/2 ?

(1 − 1/2) × (1 − 1/3) ?

(1 − 1/2) × (1 − 1/3) × (1 − 1/4) × (1 − 1/5) ?

(1 − 1/2) × (1 − 1/3) × (1 − 1/4) × (1 − 1/5) × (1 − 1/6) × (1 − 1/7) × (1 − 1/8) × (1 − 1/9) × (1 − 1/10) ?

Make a general statement and explain.

📌 ✅ Answer:

🔹 1 − 1/2 = 1/2

🔹 1 − 1/3 = 2/3

So,
(1 − 1/2)(1 − 1/3)
= 1/2 × 2/3
= 1/3

🔹 1 − 1/4 = 3/4
🔹 1 − 1/5 = 4/5

So,
(1 − 1/2)(1 − 1/3)(1 − 1/4)(1 − 1/5)
= 1/2 × 2/3 × 3/4 × 4/5
= 1/5

🔹 Similarly,

(1 − 1/2)(1 − 1/3)(1 − 1/4)…(1 − 1/10)
= 1/2 × 2/3 × 3/4 × 4/5 × 5/6 × 6/7 × 7/8 × 8/9 × 9/10

🔹 All intermediate numbers cancel.

🔹 Only 1 at the top and 10 at the bottom remain.

🔹 Therefore,

(1 − 1/2)(1 − 1/3)…(1 − 1/10) = 1/10

🔹 General statement:

(1 − 1/2)(1 − 1/3)(1 − 1/4)…(1 − 1/n) = 1/n

✔️ Final Answer:

Important Notice : The lessons and study materials up to July–August 2025 have been sent. Soon, after receiving feedback from teachers and students, we will deliver the latest and more innovative study materials for the upcoming months.

Important Notice : The lessons and study materials up to July–August 2025 have been sent. Soon, after receiving feedback from teachers and students, we will deliver the latest and more innovative study materials for the upcoming months.

Class 7 : Maths – Lesson 8. Working with Fractions

EXPLANATION AND ANALYSIS

🔵 Introduction: Fractions in Everyday Life

🟢 Meaning of a Fraction

🔵 Types of Fractions

🟢 Equivalent Fractions

🔵 Simplest Form of a Fraction

🟢 Comparing Fractions

🔵 Addition of Fractions

🟢 Subtraction of Fractions

🔵 Multiplication of Fractions

🟢 Division of Fractions

🔴 Common Mistakes to Avoid

🟢 Importance of Fractions

📘 Summary

📝 Quick Recap

TEXTBOOK QUESTIONS

OTHER IMPORTANT QUESTIONS

🔵 Section A – Very Short Answer (1 × 6 = 6 marks)

🟢 Section B – Short Answer I (2 × 6 = 12 marks)

🟡 Section C – Short Answer II (3 × 10 = 30 marks)

🔴 Section D – Long Answer (4 × 8 = 32 marks)

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