Chapter 9 of 14
Chapter 9: Equal Groups
Multiplication as Arrays, Distributive Break-up, Box Method & Tables up to 15 | CBSE Session 2026–27
Core Pedagogical Competencies & Concepts
- Multiplication concepts: multiplicand × multiplier = product.
- Building multiplication tables using known facts (e.g., table of 7 = table of 5 + table of 2).
- Using the box / grid multiplication method for 2-digit by 1-digit and 2-digit numbers.
- Solving multi-step real-world problems involving rows and columns of objects.
All Textbook Exercises & In-Text Tasks (15 Questions)
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Q1
Multiplication Strategy & Break-up
Show how to construct the multiplication table of 8 using the tables of 5 and 3. Write out the values for 8 × 1 up to 8 × 6.
[3 Marks]
✅ Step-by-Step Solution & Working
By the distributive property of multiplication:
8 × n = (5 + 3) × n = (5 × n) + (3 × n).
• 8 × 1 = (5 × 1) + (3 × 1) = 5 + 3 = 8
• 8 × 2 = (5 × 2) + (3 × 2) = 10 + 6 = 16
• 8 × 3 = (5 × 3) + (3 × 3) = 15 + 9 = 24
• 8 × 4 = (5 × 4) + (3 × 4) = 20 + 12 = 32
• 8 × 5 = (5 × 5) + (3 × 5) = 25 + 15 = 40
• 8 × 6 = (5 × 6) + (3 × 6) = 30 + 18 = 48
8 × n = (5 + 3) × n = (5 × n) + (3 × n).
• 8 × 1 = (5 × 1) + (3 × 1) = 5 + 3 = 8
• 8 × 2 = (5 × 2) + (3 × 2) = 10 + 6 = 16
• 8 × 3 = (5 × 3) + (3 × 3) = 15 + 9 = 24
• 8 × 4 = (5 × 4) + (3 × 4) = 20 + 12 = 32
• 8 × 5 = (5 × 5) + (3 × 5) = 25 + 15 = 40
• 8 × 6 = (5 × 6) + (3 × 6) = 30 + 18 = 48
CBSE Step-by-Step Marking Scheme
Explains distributive break-up (5 + 3)
1.0 M
Correct step-by-step table construction
2.0 M
Q2
Two-Digit Multiplication Word Problem
A cinema hall has 28 rows of seats, and each row has 15 seats. During a premiere show, all seats are occupied except 12 seats. How many people are watching the movie?
[3 Marks]
✅ Step-by-Step Solution & Working
Step 1: Calculate total seating capacity:
Total seats = 28 rows × 15 seats/row.
• Using break-up: 28 × 15 = 28 × (10 + 5) = (28 × 10) + (28 × 5)
• 28 × 10 = 280
• 28 × 5 = 140
• Total seats = 280 + 140 = 420 seats.
Step 2: Subtract empty seats:
• Empty seats = 12
• People watching = 420 - 12 = 408 people.
Total seats = 28 rows × 15 seats/row.
• Using break-up: 28 × 15 = 28 × (10 + 5) = (28 × 10) + (28 × 5)
• 28 × 10 = 280
• 28 × 5 = 140
• Total seats = 280 + 140 = 420 seats.
Step 2: Subtract empty seats:
• Empty seats = 12
• People watching = 420 - 12 = 408 people.
CBSE Step-by-Step Marking Scheme
Accurate multiplication 28 × 15 = 420
2.0 M
Correct subtraction giving 408 people
1.0 M
Q3
Multiplication Algorithm • 3-Digit by 2-Digit
A school library buys 24 copies of an encyclopedia. If each encyclopedia costs ₹345, calculate the total bill amount using the standard multiplication method.
[3 Marks]
✅ Step-by-Step Solution & Working
Step-by-Step Multiplication:
3 4 5
× 2 4
-----------
1 3 8 0 (345 × 4)
+ 6 9 0 0 (345 × 20)
-----------
8 2 8 0
The total bill amount is ₹8,280.
3 4 5
× 2 4
-----------
1 3 8 0 (345 × 4)
+ 6 9 0 0 (345 × 20)
-----------
8 2 8 0
The total bill amount is ₹8,280.
CBSE Step-by-Step Marking Scheme
Partial product 1 (345 × 4 = 1380)
1.0 M
Partial product 2 (345 × 20 = 6900)
1.0 M
Final total: ₹8,280
1.0 M
Q4
Mental Math • Multiples of 10 and 100
Calculate mentally using place value patterns:
(a) 70 × 60
(b) 125 × 40
(c) 48 × 25
[3 Marks]
(a) 70 × 60
(b) 125 × 40
(c) 48 × 25
✅ Step-by-Step Solution & Working
Mental Calculation Strategies:
• (a) 70 × 60:
Multiply the non-zero digits: 7 × 6 = 42, then attach two zeroes: 4,200.
• (b) 125 × 40:
Think of (125 × 4) × 10 = 500 × 10 = 5,000.
• (c) 48 × 25:
Since 25 = 100 ÷ 4, we divide 48 by 4 and multiply by 100: (48 ÷ 4) × 100 = 12 × 100 = 1,200.
• (a) 70 × 60:
Multiply the non-zero digits: 7 × 6 = 42, then attach two zeroes: 4,200.
• (b) 125 × 40:
Think of (125 × 4) × 10 = 500 × 10 = 5,000.
• (c) 48 × 25:
Since 25 = 100 ÷ 4, we divide 48 by 4 and multiply by 100: (48 ÷ 4) × 100 = 12 × 100 = 1,200.
CBSE Step-by-Step Marking Scheme
(a) 4,200
1.0 M
(b) 5,000
1.0 M
(c) 1,200
1.0 M
Q5
Textbook Exercise • Real-World Context
In Chapter 9 ('Equal Groups'), a classroom project investigates real-world mathematics: If a school bus travels 48 kilometres on 4 litres of diesel, what is the mileage (kilometres per litre) of the bus? How much diesel is needed for a 120-kilometre journey?
[3 Marks]
✅ Step-by-Step Solution & Working
Step-by-Step Solution:
1. Mileage calculation:
• Distance = 48 km, Diesel used = 4 L.
• Mileage = 48 ÷ 4 = 12 km per litre.
2. Diesel for 120 km:
• Required diesel = Total distance ÷ Mileage = 120 ÷ 12 = 10 litres.
1. Mileage calculation:
• Distance = 48 km, Diesel used = 4 L.
• Mileage = 48 ÷ 4 = 12 km per litre.
2. Diesel for 120 km:
• Required diesel = Total distance ÷ Mileage = 120 ÷ 12 = 10 litres.
CBSE Step-by-Step Marking Scheme
Mileage = 12 km/L
1.5 M
Diesel needed = 10 L
1.5 M
Q6
Practice Time • Measurement & Conversions
Convert the following measurements into standard units:
(a) 4 metres 65 centimetres into centimetres.
(b) 3500 millilitres into litres and millilitres.
(c) 2 kilograms 750 grams into grams.
[3 Marks]
(a) 4 metres 65 centimetres into centimetres.
(b) 3500 millilitres into litres and millilitres.
(c) 2 kilograms 750 grams into grams.
✅ Step-by-Step Solution & Working
Conversions:
• (a) 1 m = 100 cm → (4 × 100) + 65 = 465 cm.
• (b) 1000 mL = 1 L → 3500 mL = 3 L 500 mL.
• (c) 1 kg = 1000 g → (2 × 1000) + 750 = 2750 g.
• (a) 1 m = 100 cm → (4 × 100) + 65 = 465 cm.
• (b) 1000 mL = 1 L → 3500 mL = 3 L 500 mL.
• (c) 1 kg = 1000 g → (2 × 1000) + 750 = 2750 g.
CBSE Step-by-Step Marking Scheme
465 cm
1 M
3 L 500 mL
1 M
2750 g
1 M
Q7
Try These • Geometry and Shapes
Look at a solid brick:
(a) How many faces does a brick have?
(b) What is the shape of each face?
(c) How many straight edges and how many corners (vertices) does a brick have?
[3 Marks]
(a) How many faces does a brick have?
(b) What is the shape of each face?
(c) How many straight edges and how many corners (vertices) does a brick have?
✅ Step-by-Step Solution & Working
Properties of a Brick (Cuboid):
• (a) Faces: A brick has 6 flat faces.
• (b) Shape of faces: Each face is a rectangle.
• (c) Edges and Corners: It has 12 straight edges and 8 corners (vertices).
• (a) Faces: A brick has 6 flat faces.
• (b) Shape of faces: Each face is a rectangle.
• (c) Edges and Corners: It has 12 straight edges and 8 corners (vertices).
CBSE Step-by-Step Marking Scheme
6 faces
1 M
Rectangular faces
1 M
12 edges and 8 corners
1 M
Q8
Money & Accounts • Rate List Calculation
At a recycling shop, the rate list shows: Waste paper = ₹12 per kg, Plastic = ₹18 per kg, Iron = ₹25 per kg.
Ramesh sells 10 kg of waste paper, 5 kg of plastic, and 4 kg of iron.
(a) Find the cost of each item sold.
(b) What is the total money Ramesh earns?
[3 Marks]
Ramesh sells 10 kg of waste paper, 5 kg of plastic, and 4 kg of iron.
(a) Find the cost of each item sold.
(b) What is the total money Ramesh earns?
✅ Step-by-Step Solution & Working
Calculations:
• Waste paper: 10 × ₹12 = ₹120.
• Plastic: 5 × ₹18 = ₹90.
• Iron: 4 × ₹25 = ₹100.
• Total Earnings: ₹120 + ₹90 + ₹100 = ₹310.
• Waste paper: 10 × ₹12 = ₹120.
• Plastic: 5 × ₹18 = ₹90.
• Iron: 4 × ₹25 = ₹100.
• Total Earnings: ₹120 + ₹90 + ₹100 = ₹310.
CBSE Step-by-Step Marking Scheme
Individual costs (₹120, ₹90, ₹100)
1.5 M
Total = ₹310
1.5 M
Q9
Fractions Exploration • Parts of a Whole
A rectangular chocolate bar has 16 equal pieces. Reena gives 1/4 of the chocolate to her brother and 1/2 of it to her friend.
(a) How many pieces does her brother receive?
(b) How many pieces does her friend receive?
(c) How many pieces are left for Reena?
[3 Marks]
(a) How many pieces does her brother receive?
(b) How many pieces does her friend receive?
(c) How many pieces are left for Reena?
✅ Step-by-Step Solution & Working
Fractional Breakdown:
• Total pieces = 16.
• Brother's share (1/4): 16 ÷ 4 = 4 pieces.
• Friend's share (1/2): 16 ÷ 2 = 8 pieces.
• Remaining for Reena: 16 - (4 + 8) = 16 - 12 = 4 pieces (which is 1/4 of the bar).
• Total pieces = 16.
• Brother's share (1/4): 16 ÷ 4 = 4 pieces.
• Friend's share (1/2): 16 ÷ 2 = 8 pieces.
• Remaining for Reena: 16 - (4 + 8) = 16 - 12 = 4 pieces (which is 1/4 of the bar).
CBSE Step-by-Step Marking Scheme
Brother = 4 pieces
1 M
Friend = 8 pieces
1 M
Reena = 4 pieces
1 M
Q10
Time & Clocks • Elapsed Time
A school cricket match began at 9:30 AM and concluded at 1:15 PM.
(a) Convert 1:15 PM into 24-hour railway clock time.
(b) How long did the cricket match last?
[3 Marks]
(a) Convert 1:15 PM into 24-hour railway clock time.
(b) How long did the cricket match last?
✅ Step-by-Step Solution & Working
Time Calculations:
• (a) 24-hour time: 1:15 PM = 12:00 + 1:15 = 13:15 hours.
• (b) Duration:
• From 9:30 AM to 12:30 PM = 3 hours.
• From 12:30 PM to 1:15 PM = 45 minutes.
• Total duration = 3 hours and 45 minutes.
• (a) 24-hour time: 1:15 PM = 12:00 + 1:15 = 13:15 hours.
• (b) Duration:
• From 9:30 AM to 12:30 PM = 3 hours.
• From 12:30 PM to 1:15 PM = 45 minutes.
• Total duration = 3 hours and 45 minutes.
CBSE Step-by-Step Marking Scheme
13:15 hours in 24-hr clock
1 M
Duration = 3 hours 45 minutes with steps
2 M
Q11
Perimeter & Area • Fencing a Field
A square vegetable garden has a side length of 15 metres.
(a) What is the total length of wire fence needed to go around the garden once?
(b) If fencing costs ₹20 per metre, find the total cost of fencing.
[3 Marks]
(a) What is the total length of wire fence needed to go around the garden once?
(b) If fencing costs ₹20 per metre, find the total cost of fencing.
✅ Step-by-Step Solution & Working
Perimeter & Cost:
• (a) Perimeter of square: 4 × Side = 4 × 15 m = 60 metres.
• (b) Total Cost: Perimeter × Rate = 60 m × ₹20/m = ₹1,200.
• (a) Perimeter of square: 4 × Side = 4 × 15 m = 60 metres.
• (b) Total Cost: Perimeter × Rate = 60 m × ₹20/m = ₹1,200.
CBSE Step-by-Step Marking Scheme
Perimeter = 60 m
1.5 M
Cost = ₹1,200
1.5 M
Q12
Mental Math • Division and Multiplication
Solve mentally using number breakdown:
(a) 72 ÷ 6 = ?
(b) 15 × 8 = ?
(c) 25 × 40 = ?
(d) 96 ÷ 8 = ?
[2 Marks]
(a) 72 ÷ 6 = ?
(b) 15 × 8 = ?
(c) 25 × 40 = ?
(d) 96 ÷ 8 = ?
✅ Step-by-Step Solution & Working
Solutions:
• (a) 72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6) = 10 + 2 = 12.
• (b) 15 × 8 = (10 × 8) + (5 × 8) = 80 + 40 = 120.
• (c) 25 × 40 = (25 × 4) × 10 = 100 × 10 = 1000.
• (d) 96 ÷ 8 = (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12.
• (a) 72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6) = 10 + 2 = 12.
• (b) 15 × 8 = (10 × 8) + (5 × 8) = 80 + 40 = 120.
• (c) 25 × 40 = (25 × 4) × 10 = 100 × 10 = 1000.
• (d) 96 ÷ 8 = (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12.
CBSE Step-by-Step Marking Scheme
All 4 mental math facts correct
2 M
Q13
Patterns & Symmetries • Rotational Turns
Look at the letter 'H' and the letter 'N':
(a) What does each look like after a half-turn (180° rotation)?
(b) What does each look like after a quarter-turn (90° rotation)?
[2 Marks]
(a) What does each look like after a half-turn (180° rotation)?
(b) What does each look like after a quarter-turn (90° rotation)?
✅ Step-by-Step Solution & Working
Rotational Symmetry:
• (a) Half-turn (180°): Both 'H' and 'N' look exactly the same after a half-turn.
• (b) Quarter-turn (90°): 'H' rotates to look like 'I' with sleeping bars, and 'N' rotates to look like 'Z'.
• (a) Half-turn (180°): Both 'H' and 'N' look exactly the same after a half-turn.
• (b) Quarter-turn (90°): 'H' rotates to look like 'I' with sleeping bars, and 'N' rotates to look like 'Z'.
CBSE Step-by-Step Marking Scheme
Half-turn explanation (same)
1 M
Quarter-turn explanation
1 M
Q14
Smart Charts • Data Interpretation
In a Class 4 survey of 40 children, 20 like Cricket, 10 like Football, and 10 like Badminton.
(a) What fraction of children like Cricket?
(b) Draw a simple circular pie chart (Chapati Chart) representing this data.
[3 Marks]
(a) What fraction of children like Cricket?
(b) Draw a simple circular pie chart (Chapati Chart) representing this data.
✅ Step-by-Step Solution & Working
Data & Chapati Chart:
• (a) Fraction: 20 out of 40 = 20/40 = 1/2 (half of the class).
• (b) Chapati Chart: A circle divided in half represents Cricket (1/2). The other half is split equally into two quarters (1/4 for Football and 1/4 for Badminton).
• (a) Fraction: 20 out of 40 = 20/40 = 1/2 (half of the class).
• (b) Chapati Chart: A circle divided in half represents Cricket (1/2). The other half is split equally into two quarters (1/4 for Football and 1/4 for Badminton).
CBSE Step-by-Step Marking Scheme
Fraction = 1/2 for cricket
1.5 M
Pie chart sectors explained (1/2, 1/4, 1/4)
1.5 M
Q15
Chapter Review • True or False
State True or False:
(a) A circle has no corners and no straight edges.
(b) 1 litre is equal to 100 millilitres.
(c) The perimeter of any rectangle is 2 × (Length + Breadth).
[3 Marks]
(a) A circle has no corners and no straight edges.
(b) 1 litre is equal to 100 millilitres.
(c) The perimeter of any rectangle is 2 × (Length + Breadth).
✅ Step-by-Step Solution & Working
Answers:
• (a) True: A circle is a smooth curve with 0 corners.
• (b) False: 1 Litre is equal to 1000 millilitres.
• (c) True: Perimeter = 2 × (L + B).
• (a) True: A circle is a smooth curve with 0 corners.
• (b) False: 1 Litre is equal to 1000 millilitres.
• (c) True: Perimeter = 2 × (L + B).
CBSE Step-by-Step Marking Scheme
True, False, True with explanations
3 M
🌟 NEP 2020 Competency-Based Case Study
Tree Plantation Drive Rows
The Forest Department plants 36 saplings in each row across 25 parallel strips. The cost of protecting each sapling with a bamboo guard is ₹50.
Analytical Task: How many total saplings are planted, and what is the total cost of bamboo guards for all saplings?
Evaluation & Model Answer:
Step 1: Total saplings:
36 × 25 = 36 × (100 ÷ 4) = 3,600 ÷ 4 = 900 saplings.
Step 2: Total cost:
900 × ₹50 = ₹45,000.
36 × 25 = 36 × (100 ÷ 4) = 3,600 ÷ 4 = 900 saplings.
Step 2: Total cost:
900 × ₹50 = ₹45,000.
Rubric: 1 Mark for 900 saplings; 1 Mark for ₹45,000.