Chapter 5 of 14
Chapter 5: Sharing and Measuring
Equal Sharing, Division Concepts, Quotients, Remainders & Real-Life Grouping | CBSE Session 2026–27
Core Pedagogical Competencies & Concepts
- Division as equal sharing and repeated subtraction.
- Terms of division: Dividend ÷ Divisor = Quotient with Remainder.
- The division check rule: Dividend = (Divisor × Quotient) + Remainder, where Remainder < Divisor.
- Solving real-world word problems involving distribution of seeds, sweets, and books.
All Textbook Exercises & In-Text Tasks (15 Questions)
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Q1
Division with Remainder & Verification
A farmer packs 148 oranges equally into boxes that hold 6 oranges each. (a) How many full boxes can he pack? (b) How many oranges are left over? (c) Verify your answer using the division verification rule.
[3 Marks]
✅ Step-by-Step Solution & Working
Step 1: Perform the division:
We need to calculate 148 ÷ 6.
• Divide 14 by 6: 6 × 2 = 12. Remainder = 14 - 12 = 2.
• Bring down 8 to make 28.
• Divide 28 by 6: 6 × 4 = 24. Remainder = 28 - 24 = 4.
• Quotient = 24, Remainder = 4.
(a) The farmer can pack 24 full boxes.
(b) 4 oranges are left over.
(c) Verification:
• Formula: Dividend = (Divisor × Quotient) + Remainder
• RHS = (6 × 24) + 4 = 144 + 4 = 148.
• LHS = Dividend = 148. Verified successfully!
We need to calculate 148 ÷ 6.
• Divide 14 by 6: 6 × 2 = 12. Remainder = 14 - 12 = 2.
• Bring down 8 to make 28.
• Divide 28 by 6: 6 × 4 = 24. Remainder = 28 - 24 = 4.
• Quotient = 24, Remainder = 4.
(a) The farmer can pack 24 full boxes.
(b) 4 oranges are left over.
(c) Verification:
• Formula: Dividend = (Divisor × Quotient) + Remainder
• RHS = (6 × 24) + 4 = 144 + 4 = 148.
• LHS = Dividend = 148. Verified successfully!
CBSE Step-by-Step Marking Scheme
Correct division steps showing Quotient 24 & Remainder 4
1.5 M
Accurate verification formula and arithmetic proof
1.5 M
Q2
Real-Life Grouping Word Problem
In a school library, there are 215 storybooks. The librarian wants to place an equal number of books on 7 empty shelves. How many books will be placed on each shelf, and how many additional books are needed to fill the last shelf completely?
[3 Marks]
✅ Step-by-Step Solution & Working
Step 1: Divide books equally:
Calculate 215 ÷ 7.
• 7 × 30 = 210.
• 215 - 210 = 5 remainder.
• So, each shelf gets 30 books.
• 5 books remain on the last shelf.
Step 2: Additional books needed for full shelf:
• Each shelf needs 31 books if all shelves are equally full.
• Currently, 5 books are left.
• Additional books needed = 7 - 5 = 2 books.
Calculate 215 ÷ 7.
• 7 × 30 = 210.
• 215 - 210 = 5 remainder.
• So, each shelf gets 30 books.
• 5 books remain on the last shelf.
Step 2: Additional books needed for full shelf:
• Each shelf needs 31 books if all shelves are equally full.
• Currently, 5 books are left.
• Additional books needed = 7 - 5 = 2 books.
CBSE Step-by-Step Marking Scheme
Correct quotient 30 books per shelf
1.5 M
Correct calculation of 2 additional books needed
1.5 M
Q3
Fraction of a Collection • Word Problem
A fruit vendor has a basket of 36 mangoes. He sells 1/4 of them in the morning and 1/2 of them in the afternoon. (a) How many mangoes did he sell in total? (b) How many mangoes are left?
[3 Marks]
✅ Step-by-Step Solution & Working
Step-by-Step Solution:
Total mangoes = 36
1. Morning Sales:
1/4 of 36 = 36 ÷ 4 = 9 mangoes.
2. Afternoon Sales:
1/2 of 36 = 36 ÷ 2 = 18 mangoes.
3. Total Sold:
9 + 18 = 27 mangoes.
4. Mangoes Left:
36 - 27 = 9 mangoes (which is 1/4 of the basket).
Total mangoes = 36
1. Morning Sales:
1/4 of 36 = 36 ÷ 4 = 9 mangoes.
2. Afternoon Sales:
1/2 of 36 = 36 ÷ 2 = 18 mangoes.
3. Total Sold:
9 + 18 = 27 mangoes.
4. Mangoes Left:
36 - 27 = 9 mangoes (which is 1/4 of the basket).
CBSE Step-by-Step Marking Scheme
Morning calculation (9) and afternoon (18)
1.5 M
Total sold (27) and remaining mangoes (9)
1.5 M
Q4
Equivalent Fractions • Shading Models
Show that 2/4 and 4/8 are equivalent to 1/2 using a rectangular strip model.
[2 Marks]
✅ Step-by-Step Solution & Working
Demonstration of Equivalent Fractions:
Take three identical strips of paper of length 12 cm:
1. Strip 1 (1/2): Divide into 2 equal parts of 6 cm each. Shade 1 part. Shaded length = 6 cm.
2. Strip 2 (2/4): Divide into 4 equal parts of 3 cm each. Shade 2 parts. Shaded length = 2 × 3 cm = 6 cm.
3. Strip 3 (4/8): Divide into 8 equal parts of 1.5 cm each. Shade 4 parts. Shaded length = 4 × 1.5 cm = 6 cm.
Since all three shaded portions cover the exact same physical length of 6 cm, 1/2 = 2/4 = 4/8. They represent the exact same fraction of the whole.
Take three identical strips of paper of length 12 cm:
1. Strip 1 (1/2): Divide into 2 equal parts of 6 cm each. Shade 1 part. Shaded length = 6 cm.
2. Strip 2 (2/4): Divide into 4 equal parts of 3 cm each. Shade 2 parts. Shaded length = 2 × 3 cm = 6 cm.
3. Strip 3 (4/8): Divide into 8 equal parts of 1.5 cm each. Shade 4 parts. Shaded length = 4 × 1.5 cm = 6 cm.
Since all three shaded portions cover the exact same physical length of 6 cm, 1/2 = 2/4 = 4/8. They represent the exact same fraction of the whole.
CBSE Step-by-Step Marking Scheme
Equal partition measurement of strips
1.0 M
Equal shaded length proving equivalence
1.0 M
Q5
Textbook Exercise • Real-World Context
In Chapter 5 ('Sharing and Measuring'), 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
School Van Seating Arrangement
A school organizes an excursion for 58 students. Each school cab can carry a maximum of 8 students safely.
Analytical Task: How many cabs must be booked so that EVERY student gets a seat?
Evaluation & Model Answer:
Answer: 8 cabs must be booked.
Explanation: 58 ÷ 8 = 7 with a remainder of 2 students.
7 cabs can only seat 7 × 8 = 56 students. The remaining 2 students cannot be left behind, so 1 more cab must be hired.
Total cabs required = 7 + 1 = 8 cabs.
Explanation: 58 ÷ 8 = 7 with a remainder of 2 students.
7 cabs can only seat 7 × 8 = 56 students. The remaining 2 students cannot be left behind, so 1 more cab must be hired.
Total cabs required = 7 + 1 = 8 cabs.
Rubric: 1 Mark for division quotient 7 with remainder 2; 1 Mark for practical round-up to 8 cabs.