Operations on Numbers

1. Addition — Combining Numbers

Addition of Large Numbers

Align digits by place value. Add column by column from the RIGHT. Carry when sum exceeds 9.

   2 3 4 5 6 7 8
 + 1 9 8 7 6 5 4
 ----------------
   4 3 3 3 3 3 2

'Start from the ones place. If the sum is 10 or more, CARRY the extra digit to the next column on the LEFT.'

Properties of Addition

Property	Explanation	Example
Commutative	Changing order does NOT change sum	25 + 30 = 30 + 25
Associative	Changing grouping does NOT change sum	(5 + 9) + 6 = 5 + (9 + 6)
Identity	Zero added to a number gives the same number	437 + 0 = 437
Property of 10	Adding 10 increases the tens digit by 1	456 + 10 = 466

Estimation in Addition

Round each number to the nearest thousand, then add.

4,279 + 3,612 ≈ 4,000 + 4,000 = 8,000 (Actual sum = 7,891 — close!)

2. Subtraction — Finding the Difference

Subtraction of Large Numbers

Align digits by place value. Subtract column by column from the RIGHT. Borrow when needed.

   7 8 9 3 4 5
 − 4 5 6 7 8 9
 -------------
   3 3 2 5 5 6

'When you borrow from the next column, that column's digit decreases by ONE, and the current column gets TEN added to it.'

Properties of Subtraction

Property	Explanation	Example
NOT commutative	7 − 5 ≠ 5 − 7	Order matters!
Subtracting zero	Any number minus zero = itself	834 − 0 = 834
Subtracting itself	Any number minus itself = zero	834 − 834 = 0
Relation to addition	Difference + subtrahend = minuend	345 − 123 = 222, so 222 + 123 = 345

Checking Subtraction Using Addition

'Always check your subtraction by ADDING the answer to the number you subtracted. You should get the original number back.'

3. Multiplication — Repeated Addition

Multiplication of Large Numbers

2-digit × 2-digit:

     4 5
   × 3 6
   -----
   2 7 0  (45 × 6)
 1 3 5 0  (45 × 30)
 --------
 1 6 2 0

'Multiply by the ones digit first. Then multiply by the tens digit — remember to PLACE A ZERO in the ones column. Add the partial products.'

Properties of Multiplication

Property	Explanation	Example
Commutative	5 × 7 = 7 × 5	Order does not matter
Associative	(2 × 3) × 4 = 2 × (3 × 4)	Grouping does not matter
Identity	Multiplying by 1 gives the same number	89 × 1 = 89
Zero property	Multiplying by 0 gives 0	56 × 0 = 0
Distributive	a × (b + c) = a × b + a × c	4 × (10 + 2) = 4 × 10 + 4 × 2 = 48

Multiplying by 10, 100, 1000

Multiply By	How To	Example
10	Add ONE zero	45 × 10 = 450
100	Add TWO zeros	45 × 100 = 4,500
1000	Add THREE zeros	45 × 1000 = 45,000

'To multiply by 10, 100, or 1000 — just COUNT the zeros and add them to the end of the number.'

Terms of Division

Dividend ÷ Divisor = Quotient + Remainder

'Check: Dividend = Divisor × Quotient + Remainder. The remainder must ALWAYS be less than the divisor.'

Long Division

       3 4 2 R 1
     ───────────
 5 ) 1 7 1 1
    − 1 5  ↓
      ────
        2 1
      − 2 0
        ───
          1 1
        − 1 0
          ───
            1

'Bring down digits ONE at a time. Divide, multiply, subtract, bring down — repeat until no digits remain.'

Properties of Division

Property	Explanation	Example
NOT commutative	10 ÷ 2 ≠ 2 ÷ 10	Order matters
Dividing by 1	Any number ÷ 1 = itself	56 ÷ 1 = 56
Dividing by itself	Any number ÷ itself = 1 (except 0)	25 ÷ 25 = 1
Dividing zero	0 ÷ any number = 0	0 ÷ 7 = 0
Cannot divide by zero	Division by zero is UNDEFINED	8 ÷ 0 has no meaning

5. BODMAS — Order of Operations

When an expression has MULTIPLE operations, follow this order:

Letter	Stands For	Example
B	Brackets (solve inside FIRST)	(4 + 3) × 2 = 7 × 2 = 14
O	Of (multiplication with 'of')	Half of 20 = 10
D	Division (left to right)	12 ÷ 3 × 2 = 4 × 2 = 8
M	Multiplication (left to right)	4 × 3 + 5 = 12 + 5 = 17
A	Addition (left to right)	10 + 5 − 3 = 12
S	Subtraction (left to right)	15 − 4 + 2 = 13

'BRACKETS FIRST — ALWAYS. If there are nested brackets, solve the INNERMOST bracket first.'

Example: 15 − [2 + (6 − 4) × 3]

Step 1: Solve innermost bracket (6 − 4) = 2 Step 2: Multiply 2 × 3 = 6 Step 3: Add 2 + 6 = 8 Step 4: Subtract 15 − 8 = 7

6. Word Problems — Step by Step

Strategy

Step	Action
1	Read the problem CAREFULLY — twice
2	Identify what is GIVEN and what is ASKED
3	Decide which OPERATION to use
4	ESTIMATE the answer first
5	SOLVE accurately
6	CHECK if the answer makes sense

Example Problem

A school has 2,456 students. Each student needs 5 notebooks. How many notebooks are needed?

Given: 2,456 students. 5 notebooks each.
Operation: Multiplication.
Estimate: 2,500 × 5 = 12,500
Solve: 2,456 × 5 = 12,280 notebooks.
Check: 12,280 ÷ 2,456 = 5. Correct.

Key Facts to Remember

'Addition and subtraction are INVERSE operations. Multiplication and division are INVERSE operations.'
The product of any number and ZERO is ZERO.
The quotient of any number divided by ONE is the number itself.
Always ESTIMATE before solving — it helps you catch big mistakes.
In BODMAS, division and multiplication have EQUAL priority (left to right).

Common Mistakes

Mistake	Why It Is Wrong	Correct Approach
Subtracting a larger digit from a smaller without borrowing	423 − 156: in tens column, 2 − 5 — you MUST borrow	Borrow 1 hundred: 12 tens − 5 tens = 7 tens
Forgetting the zero in partial products	45 × 36: while multiplying by 30, write 1350 NOT 135	The zero represents the ones place of 30
BODMAS — adding before dividing	12 ÷ 3 × 2 + 1 = ? Many solve 3 × 2 first	Division and multiplication have equal priority — go LEFT to RIGHT
Remainder larger than divisor	17 ÷ 5 = 3 R 2 (not 2 R 7)	Remainder must ALWAYS be smaller than divisor

Exam Focus (ICSE Class 5)

Topic	Marks (Typical)	Question Type
Addition/Subtraction of large numbers	4-5 marks	Direct computation
Multiplication and division	4-5 marks	Word problems and computation
BODMAS	3-4 marks	Simplify expressions
Word problems (multi-step)	4-5 marks	Application-based
Properties of operations	2-3 marks	Fill in the blanks / True-False

Self-Test: 5 Questions

Q1. Simplify using BODMAS: 24 − [12 + (8 − 3) × 2] ÷ 6

Q2. A factory produces 1,245 toys per day. How many toys does it produce in 3 months (90 days)?

Q3. Find the product: 4,567 × 208

Q4. Divide and check: 8,945 ÷ 27

Q5. Riya has 500. She buys 3 books at 85 each and 2 pens at 12 each. How much money is left with her?

Answers

A1. 24 − [12 + (8 − 3) × 2] ÷ 6 = 24 − [12 + 5 × 2] ÷ 6 = 24 − [12 + 10] ÷ 6 = 24 − 22 ÷ 6 = 24 − 3.67 = 20.33

A2. 1,245 × 90 = 1,12,050 toys.

A3. 4,567 × 208 = 9,49,936.

A4. 8,945 ÷ 27 = 331 R 8. Check: 331 × 27 + 8 = 8,937 + 8 = 8,945.

A5. Spent = 3 × 85 + 2 × 12 = 255 + 24 = 279. Left = 500 − 279 = 221.

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Practice more, ask doubts, find a tutor

ICSE Class 5 mock tests

ICSE Class 5 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Operations on Numbers usually read these next.

1

Large Numbers

PLACE VALUE systems — Indian and International. READING and WRITING large numbers. COMPARISON and ORDERING. ROUNDING off. ROMAN NUMERALS up to M (1000). ICSE Class 5 Mathematics.

1

Numbers, Operations, Fractions, Decimals & Geometry

Building MATHEMATICAL FLUENCY: Large numbers (lakhs, crores), the four operations, FRACTIONS and DECIMALS, and the GEOMETRY of angles, triangles, and circles. ICSE Class 5 Mathematics.

2

Measurement, Money, Percentage, Data & Patterns

MEASUREMENT — length, weight, capacity. MONEY — rupees, profit and loss. PERCENTAGE. Reading DATA from graphs. And the BEAUTY of number PATTERNS. ICSE Class 5 Mathematics.

3

Factors and Multiples

Explore FACTORS and MULTIPLES. Identify PRIME and COMPOSITE numbers. Use DIVISIBILITY TESTS. Find HCF and LCM. Understand CO-PRIME numbers and their real-world uses. ICSE Class 5 Mathematics.

🧑‍🏫