Addition and Subtraction — Large Numbers

Mathematics

Class 5 | CBSE, NCERT & ICSE

Learning Objectives

Add numbers up to 8 digits (crores) using the column method
Subtract numbers up to 8 digits with and without borrowing (regrouping)
Understand and apply properties of addition (Commutative, Associative, Identity)
Check answers using the inverse operation (addition checks subtraction and vice versa)
Estimate sums and differences by rounding to the nearest lakh or crore
Solve real-life word problems involving addition and subtraction of large numbers

Introduction — When Do We Add and Subtract Large Numbers?

In Class 4, you learned to add and subtract numbers up to 5-6 digits. Now in Class 5, we work with numbers in lakhs and crores — numbers with 6, 7, and 8 digits! But the method stays the same: the column method (writing numbers one below the other and working from right to left).

Think about these real-life situations where we need to add or subtract large numbers:

The population of Delhi is 1,90,00,000 and Mumbai is 2,10,00,000. What is their combined population?
ISRO's budget for 2023 was ₹1,23,45,00,000. They spent ₹98,67,00,000. How much is left?
Indian Railways earned ₹2,40,15,00,000 from passenger fares and ₹1,56,80,00,000 from freight. What was the total earning?
A school collected ₹12,45,678 in Term 1 and ₹15,67,890 in Term 2. What is the total collection?

All of these need addition or subtraction of large numbers. Let us learn how!

Revision — Carrying and Borrowing

Before we work with large numbers, let us quickly revise the two key concepts:

Carrying (Regrouping in Addition)

When the sum of digits in any column is 10 or more, we write the ones digit and carry the tens digit to the next column on the left.
Example: 7 + 8 = 15. Write 5, carry 1 to the next column.

Borrowing (Regrouping in Subtraction)

When a digit in the top number is smaller than the digit below it, we borrow 1 from the next column on the left.
Borrowing 1 from the next column means adding 10 to the current digit.
Example: To subtract 8 from 3, borrow 1 from tens place. Now 3 becomes 13. 13 − 8 = 5.

Remember: Whether we are adding 2-digit numbers or 8-digit numbers, the process is exactly the same. We always start from the Ones column (rightmost) and move left, column by column.

Addition of Large Numbers (6-8 Digits)

To add large numbers, we use the column method:

Write the numbers one below the other, aligning digits by place value (ones under ones, tens under tens, etc.).
Start adding from the rightmost column (Ones place).
If the sum in any column is 10 or more, write the ones digit and carry the rest to the next column.
Continue until all columns (including any final carry) are done.

Example 1: Addition Without Carrying

Add: 34,52,100 + 21,34,678

TL	L	TTh	Th	H	T	O
3	4	5	2	1	0	0
2	1	3	4	6	7	8
5	5	8	6	7	7	8

Answer: 34,52,100 + 21,34,678 = 55,86,778

Here, no column sum exceeded 9, so there was no carrying needed.

Example 2: Addition With Carrying (7-Digit Numbers)

Add: 45,67,893 + 38,45,678

	TL	L	TTh	Th	H	T	O
Carry→	1	1	1	1	1	1
	4	5	6	7	8	9	3
+	3	8	4	5	6	7	8
	8	4	1	3	5	7	1

Step-by-step:

Ones: 3 + 8 = 11 → write 1, carry 1
Tens: 9 + 7 + 1(carry) = 17 → write 7, carry 1
Hundreds: 8 + 6 + 1 = 15 → write 5, carry 1
Thousands: 7 + 5 + 1 = 13 → write 3, carry 1
Ten Thousands: 6 + 4 + 1 = 11 → write 1, carry 1
Lakhs: 5 + 8 + 1 = 14 → write 4, carry 1
Ten Lakhs: 4 + 3 + 1 = 8

Answer: 45,67,893 + 38,45,678 = 84,13,571

Image needed: Step-by-step column addition of two 7-digit numbers (45,67,893 + 38,45,678) showing carries in red above each column. Each step highlighted with arrows showing the carry process from right to left.
Style: Clean educational diagram, hand-drawn chalk-on-blackboard style or neat notebook style, child-friendly
Size: 700×500px

Example 3: Addition of 8-Digit Numbers

Add: 4,56,78,345 + 3,87,65,789

Cr	TL	L	TTh	Th	H	T	O
4	5	6	7	8	3	4	5
3	8	7	6	5	7	8	9
8	4	4	4	4	1	3	4

Step-by-step:

O: 5 + 9 = 14 → write 4, carry 1
T: 4 + 8 + 1 = 13 → write 3, carry 1
H: 3 + 7 + 1 = 11 → write 1, carry 1
Th: 8 + 5 + 1 = 14 → write 4, carry 1
TTh: 7 + 6 + 1 = 14 → write 4, carry 1
L: 6 + 7 + 1 = 14 → write 4, carry 1
TL: 5 + 8 + 1 = 14 → write 4, carry 1
Cr: 4 + 3 + 1 = 8

Answer: 4,56,78,345 + 3,87,65,789 = 8,44,44,134

Example 4: Adding Three Numbers Together

Add: 12,34,567 + 23,45,678 + 34,56,789

Method: Add the first two numbers, then add the third to the result.

Step 1: 12,34,567 + 23,45,678 = 35,80,245

Step 2: 35,80,245 + 34,56,789 = 70,37,034

Answer: 12,34,567 + 23,45,678 + 34,56,789 = 70,37,034

Tip: You can also add all three numbers in one column — just add three digits at a time in each column.

Example 5: Adding Numbers with Different Number of Digits

Add: 5,43,21,000 + 89,45,673

When numbers have different number of digits, align them from the right. The shorter number will have empty spaces on the left (treat them as 0).

Cr	TL	L	TTh	Th	H	T	O
5	4	3	2	1	0	0	0
0	8	9	4	5	6	7	3
6	3	2	6	6	6	7	3

Answer: 5,43,21,000 + 89,45,673 = 6,32,66,673

Properties of Addition

Addition has some special properties that make calculations easier. These properties work for ALL numbers — small or large!

Three Properties of Addition

1. Commutative Property (Order Property): Changing the order of numbers does not change the sum.
a + b = b + a
Example: 34,56,789 + 12,34,567 = 12,34,567 + 34,56,789 = 46,91,356
2. Associative Property (Grouping Property): Changing the grouping of numbers does not change the sum.
(a + b) + c = a + (b + c)
Example: (10,00,000 + 25,00,000) + 15,00,000 = 10,00,000 + (25,00,000 + 15,00,000) = 50,00,000
3. Identity Property (Zero Property): Adding 0 to any number gives the same number.
a + 0 = a
Example: 5,67,89,012 + 0 = 5,67,89,012

Using Commutative Property to Check:
If 23,45,678 + 56,78,901 = 80,24,579, then 56,78,901 + 23,45,678 should also = 80,24,579. This is a quick way to verify your answer!

Think About It: Does subtraction have a Commutative Property? Try: Is 50,00,000 − 20,00,000 = 20,00,000 − 50,00,000? No! 30,00,000 ≠ −30,00,000. So subtraction is NOT commutative.

Subtraction of Large Numbers (6-8 Digits)

To subtract large numbers, we also use the column method:

Write the larger number on top and the smaller number below, aligning by place value.
Start subtracting from the rightmost column (Ones place).
If the top digit is smaller than the bottom digit, borrow 1 from the next column on the left.
Continue until all columns are done.

Example 1: Subtraction Without Borrowing

Subtract: 78,96,543 − 45,63,210

TL	L	TTh	Th	H	T	O
7	8	9	6	5	4	3
4	5	6	3	2	1	0
3	3	3	3	3	3	3

Answer: 78,96,543 − 45,63,210 = 33,33,333

Every top digit was larger than the bottom digit, so no borrowing was needed.

Example 2: Subtraction With Borrowing (7-Digit Numbers)

Subtract: 63,42,105 − 27,58,436

TL	L	TTh	Th	H	T	O
6	3	4	2	1	0	5
2	7	5	8	4	3	6
3	5	8	3	6	6	9

Step-by-step:

O: 5 − 6 → can't! Borrow from T. T becomes (0−1)→ can't! Borrow from H. H: 1→0, T: 0→10, then T: 10→9, O: 5→15. Now 15 − 6 = 9
T: 9 − 3 = 6
H: 0 − 4 → can't! Borrow from Th. Th: 2→1, H: 0→10. Now 10 − 4 = 6
Th: 1 − 8 → can't! Borrow from TTh. TTh: 4→3, Th: 1→11. Now 11 − 8 = 3
TTh: 3 − 5 → can't! Borrow from L. L: 3→2, TTh: 3→13. Now 13 − 5 = 8
L: 2 − 7 → can't! Borrow from TL. TL: 6→5, L: 2→12. Now 12 − 7 = 5
TL: 5 − 2 = 3

Answer: 63,42,105 − 27,58,436 = 35,83,669

Image needed: Step-by-step column subtraction of two 7-digit numbers (63,42,105 − 27,58,436) showing borrowing with crossed-out digits and new values written above. Each borrow shown with an arrow from left column to right. Use red for borrowed/changed digits.
Style: Neat notebook-style with lined paper background, child-friendly
Size: 700×500px

Example 3: Subtraction With Zeros (Borrowing Across Zeros)

Subtract: 5,00,00,000 − 23,45,678

This is tricky because the top number has many zeros! When we need to borrow but the next column is also 0, we keep going left until we find a non-zero digit.

Cr	TL	L	TTh	Th	H	T	O
5	0	0	0	0	0	0	0
0	2	3	4	5	6	7	8
4	7	6	5	4	3	2	2

How the borrowing works:

O: 0 − 8 → can't! Need to borrow. But T is 0, H is 0, Th is 0, TTh is 0, L is 0, TL is 0!
We go all the way to Cr (5). Borrow 1 from Cr: Cr becomes 4.
That 1 travels through: TL becomes 10→9 (gives 1 to L), L becomes 10→9, TTh becomes 10→9, Th becomes 10→9, H becomes 10→9, T becomes 10→9, O becomes 10.
Now: O: 10 − 8 = 2, T: 9 − 7 = 2, H: 9 − 6 = 3, Th: 9 − 5 = 4, TTh: 9 − 4 = 5, L: 9 − 3 = 6, TL: 9 − 2 = 7, Cr: 4 − 0 = 4

Answer: 5,00,00,000 − 23,45,678 = 4,76,54,322

Image needed: Visual showing subtraction from a number with zeros (5,00,00,000 − 23,45,678). Show the chain of borrowing across zeros — the 5 becomes 4, and each 0 becomes 9 (except the last which becomes 10). Use arrows and colour coding to show the borrowing chain.
Style: Colourful step-by-step diagram, blackboard or notebook style
Size: 750×450px

Quick Rule for Subtracting from Zeros

When subtracting from a number like 5,00,00,000 or 3,00,000:
The leftmost non-zero digit decreases by 1.
All the zeros in between become 9.
The last zero (Ones place) becomes 10, then you subtract from 10.
Example: 1,00,000 − 1 = 99,999 (the 1 becomes 0, all zeros become 9)

Example 4: Subtraction of 8-Digit Numbers

Subtract: 8,25,43,671 − 3,67,89,245

Cr	TL	L	TTh	Th	H	T	O
8	2	5	4	3	6	7	1
3	6	7	8	9	2	4	5
4	5	7	5	4	4	2	6

Answer: 8,25,43,671 − 3,67,89,245 = 4,57,54,426

Checking Your Answer — Using the Inverse Operation

Addition and subtraction are inverse operations — they undo each other. We can use this to check our answers!

How to Check

To check addition: Subtract one addend from the sum. You should get the other addend.
If A + B = C, then C − A = B and C − B = A.
To check subtraction: Add the difference to the smaller number. You should get the larger number.
If A − B = C, then C + B = A.

Example 1 — Checking Addition:
We found: 45,67,893 + 38,45,678 = 84,13,571
Check: 84,13,571 − 38,45,678 = 45,67,893 ✓ Correct!

Example 2 — Checking Subtraction:
We found: 5,00,00,000 − 23,45,678 = 4,76,54,322
Check: 4,76,54,322 + 23,45,678 = 5,00,00,000 ✓ Correct!

Good Habit: Always check your answer when working with large numbers. A small mistake in one column can give a completely wrong answer. The inverse operation check takes just 1-2 minutes and saves marks in exams!

Estimation — Rounding Before Adding or Subtracting

Sometimes we don't need the exact answer — we just need a rough idea. This is called estimation. We round the numbers first, then add or subtract the rounded numbers.

When Is Estimation Useful?

Quickly checking if your exact answer is reasonable
Reading numbers in newspapers (populations, budgets are often rounded)
Making quick decisions (Is the total over ₹1 crore? Is the difference more than 50 lakh?)
Mental maths — doing calculations in your head

Rounding Rules

To round to the nearest lakh: Look at the ten-thousands digit. If it is 5 or more, round up. If less than 5, round down.

To round to the nearest crore: Look at the ten-lakhs digit. If it is 5 or more, round up. If less than 5, round down.

Example 1: Estimate the sum of 43,67,892 + 28,34,561

Round to nearest lakh:
43,67,892 ≈ 44,00,000 (6 in TTh ≥ 5, so round up)
28,34,561 ≈ 28,00,000 (3 in TTh < 5, so round down)
Estimated sum ≈ 44,00,000 + 28,00,000 = 72,00,000
(Actual sum = 72,02,453 — very close!)

Example 2: Estimate the difference of 7,85,43,000 − 3,24,67,000

Round to nearest crore:
7,85,43,000 ≈ 8,00,00,000 (8 in TL ≥ 5, round up)
3,24,67,000 ≈ 3,00,00,000 (2 in TL < 5, round down)
Estimated difference ≈ 8,00,00,000 − 3,00,00,000 = 5,00,00,000
(Actual difference = 4,60,76,000)

Example 3: Arjun's family spent ₹3,45,678 on house repairs and ₹2,78,450 on furniture. Estimate the total spending.

Round to nearest lakh:
₹3,45,678 ≈ ₹3,00,000
₹2,78,450 ≈ ₹3,00,000
Estimated total ≈ ₹6,00,000
(Actual total = ₹6,24,128)

Word Problems — Indian Context

Let us solve some real-life problems using addition and subtraction of large numbers.

Problem 1: Population
The population of Rajasthan is 7,89,00,000 and the population of Gujarat is 6,38,00,000. What is the combined population of both states?

Solution:
Combined population = 7,89,00,000 + 6,38,00,000
= 14,27,00,000 (Fourteen crore twenty-seven lakh)

Answer: The combined population is 14,27,00,000.

Problem 2: ISRO Budget
ISRO received a budget of ₹1,34,80,00,000 for the year. They spent ₹89,56,00,000 on satellite missions. How much budget is remaining?

Solution:
Remaining budget = ₹1,34,80,00,000 − ₹89,56,00,000
= ₹45,24,00,000 (Forty-five crore twenty-four lakh rupees)

Answer: The remaining budget is ₹45,24,00,000.

Problem 3: Railway Distances
The distance from Delhi to Chennai by rail is 2,182 km. The distance from Chennai to Kanyakumari is 710 km. What is the total distance from Delhi to Kanyakumari via Chennai?
Also, how much farther is Delhi-Chennai than Chennai-Kanyakumari?

Solution:
Total distance = 2,182 + 710 = 2,892 km
Difference = 2,182 − 710 = 1,472 km

Answer: Total distance is 2,892 km. Delhi-Chennai is 1,472 km farther.

Problem 4: School Budget
Priya's school received a grant of ₹25,67,890. They spent ₹12,45,678 on books and ₹8,90,000 on sports equipment. How much money is left?

Solution:
Total spent = ₹12,45,678 + ₹8,90,000 = ₹21,35,678
Money left = ₹25,67,890 − ₹21,35,678 = ₹4,32,212

Answer: ₹4,32,212 is left.

Problem 5: IPL Cricket
In IPL 2023, Mumbai Indians scored a total of 24,56,789 runs across all seasons (combined). Chennai Super Kings scored 23,89,456 runs. How many more runs did Mumbai Indians score?

Solution:
Difference = 24,56,789 − 23,89,456 = 67,333 runs

Answer: Mumbai Indians scored 67,333 more runs.

Image needed: An infographic showing real-life addition and subtraction scenarios in India — a map of India with two state populations being added, an ISRO rocket with budget numbers, a train route showing Delhi-Chennai-Kanyakumari distances, a school building with money amounts.
Style: Colourful infographic, child-friendly icons and illustrations, bright background
Size: 800×500px

Key Words

Term	Meaning
Addend	A number that is being added (in 5 + 3 = 8, both 5 and 3 are addends)
Sum	The result of addition (in 5 + 3 = 8, the sum is 8)
Difference	The result of subtraction (in 8 − 3 = 5, the difference is 5)
Minuend	The number from which we subtract (the larger number on top)
Subtrahend	The number being subtracted (the smaller number below)
Carrying (Regrouping)	Moving an extra ten to the next column in addition
Borrowing (Regrouping)	Taking 10 from the next column in subtraction
Estimation	Finding an approximate answer by rounding
Inverse Operation	The opposite operation (addition undoes subtraction, and vice versa)
Commutative	Order doesn't matter (a + b = b + a)
Associative	Grouping doesn't matter ((a+b)+c = a+(b+c))
Column Method	Writing numbers vertically, aligned by place value, to add or subtract

Key Points to Remember

Always align numbers by place value (ones under ones, tens under tens) before adding or subtracting.
Start from the rightmost column (Ones) and work left.
In addition: if column sum ≥ 10, write ones digit and carry to next column.
In subtraction: if top digit < bottom digit, borrow 1 from the next column to the left.
When subtracting from zeros: the borrowing chain continues until a non-zero digit is found. Each zero becomes 9 (except the last one, which becomes 10).
Check addition using subtraction. Check subtraction using addition.
Addition is commutative (a + b = b + a) and associative ((a+b)+c = a+(b+c)).
Subtraction is NOT commutative and NOT associative.
Adding 0 to any number gives the same number (Identity Property).
Estimation = rounding numbers before calculating to get an approximate answer.
In word problems: "total", "combined", "altogether", "in all" → Addition. "Difference", "how much more", "how much less", "remaining", "left" → Subtraction.

Finished learning? Now test what you know!

🎯 Quick Quiz — Test Yourself!

Answer these questions right here. You get instant feedback — no paper needed!

Score: 0 / 8

1. 3,45,678 + 2,54,322 = ?

3,45,678 + 2,54,322 = 6,00,000.

2. 10,00,000 − 1 = ?

One less than ten lakh (1,000,000) is 9,99,999.

3. Estimate 4,876 + 3,192 by rounding to the nearest thousand:

4,876 ≈ 5,000 and 3,192 ≈ 3,000, so about 8,000.

4. Which is the additive identity (adding it changes nothing)?

Adding 0 to any number leaves it unchanged.

5. 6,42,105 + 2,33,316 = ? (digits only)

6,42,105 + 2,33,316 = 8,75,421.

6. 5,00,000 − 23,436 = ? (digits only)

5,00,000 − 23,436 = 4,76,564.

7. A school collected ₹4,25,000 and spent ₹2,75,000. How much is left? (digits only)

4,25,000 − 2,75,000 = 1,50,000.

8. What must be added to 3,10,000 to make 4,00,000? (digits only)

4,00,000 − 3,10,000 = 90,000.

Practice Questions

A. Add the following (Column Method):

1. 34,56,789 + 45,67,890

2. 5,67,89,012 + 2,45,67,890

3. 78,90,123 + 56,78,901

4. 4,00,00,000 + 3,99,99,999

5. 1,23,45,678 + 8,76,54,322

6. 6,54,32,100 + 89,67,900

B. Subtract the following (Column Method):

1. 89,45,678 − 34,67,890

2. 7,00,00,000 − 2,34,56,789

3. 9,12,34,567 − 4,56,78,901

4. 5,00,00,000 − 1

5. 6,05,04,030 − 2,78,56,891

6. 3,40,00,000 − 1,56,78,234

C. Estimate the sum or difference by rounding to nearest lakh:

1. 56,78,345 + 34,23,678 ≈

2. 7,89,45,000 − 3,12,67,000 ≈

3. 23,45,600 + 67,89,400 ≈

4. 9,00,00,000 − 4,56,78,000 ≈

D. Fill in the blanks using properties of addition:

1. 45,67,890 + 23,45,678 = 23,45,678 + (Property: )

2. (12,00,000 + 34,00,000) + 56,00,000 = 12,00,000 + ( + 56,00,000) (Property: )

3. 8,76,54,321 + 0 = (Property: )

4. If 34,56,789 + 56,78,901 = 91,35,690, then 56,78,901 + 34,56,789 =

E. Find the missing number:

1. + 34,56,789 = 78,90,123

2. 5,00,00,000 − = 2,67,89,345

3. 1,23,45,678 + = 5,00,00,000

4. − 45,67,890 = 54,32,110

F. True or False:

1. 56,78,900 + 43,21,100 = 1,00,00,000

2. Subtraction is commutative (a − b = b − a).

3. The sum of two 7-digit numbers is always an 8-digit number.

4. If we subtract a number from itself, the answer is always 0.

5. Estimation gives the exact answer.

G. Choose the correct answer (MCQ):

1. What is 4,56,78,000 + 3,43,22,000?

a) 7,00,00,000 b) 8,00,00,000 c) 7,99,00,000 d) 8,00,00,001

2. What is 9,00,00,000 − 1?

a) 8,99,99,999 b) 8,00,00,000 c) 9,99,99,999 d) 8,99,99,990

3. Which property is shown: 25,00,000 + 0 = 25,00,000?

a) Commutative b) Associative c) Identity d) Distributive

4. The estimated sum of 78,45,000 + 21,67,000 (rounded to nearest lakh) is:

a) 99,00,000 b) 1,00,00,000 c) 1,00,12,000 d) 99,12,000

H. Word Problems:

1. The population of Uttar Pradesh is 2,40,00,000 and the population of Bihar is 1,28,00,000. What is the total population of both states?

2. A factory produced 45,67,890 toys last year and 52,34,567 toys this year. How many more toys were produced this year?

3. Meena's father earns ₹18,50,000 per year. The family spends ₹14,25,600 per year. How much do they save?

4. ISRO spent ₹6,15,00,00,000 on Chandrayaan-3 and ₹4,50,00,00,000 on Mangalyaan. What was the total spending on both missions?

5. A library has 12,34,567 Hindi books, 8,90,345 English books, and 5,67,890 regional language books. How many books are there in total?

6. Ravi collected ₹3,45,678 for a charity drive. Sita collected ₹2,89,456. How much more did Ravi collect than Sita? What is the total collection?

Activity — Budget Planner

Arjun's family has a monthly income of ₹1,25,000. Help them plan their budget! Fill in the expenses and calculate the savings.

Expense	Amount (₹)
House Rent	35,000
Groceries & Food	22,500
School Fees (2 children)	18,000
Electricity & Water Bills	4,500
Transport (Petrol/Bus)	8,000
Medicines & Doctor	3,500
Clothing & Shopping	5,000
Entertainment	3,000

Questions:

What is the total monthly expense? (Add all expenses)
How much money is saved each month? (Income − Total Expense)
How much will the family save in one full year (12 months)?
If they need ₹6,00,000 for a family vacation, how many months will they need to save?

Bonus: Create a budget for your own family! Ask your parents about approximate monthly expenses and calculate the savings.

Want to review the lesson again?

Mathematics

Addition and Subtraction — Large Numbers

Class 5 | CBSE / NCERT / ICSE

Name: Class: Date:

Learning Objectives

Add numbers up to 8 digits using the column method
Subtract numbers up to 8 digits with and without borrowing
Apply properties of addition (Commutative, Associative, Identity)
Check answers using the inverse operation
Estimate sums and differences by rounding
Solve real-life word problems with large numbers

Introduction

In Class 5, we add and subtract numbers with 6, 7, and 8 digits (lakhs and crores). The method is the same as before — the column method — but the numbers are much larger! We use this in real life for populations, budgets, distances, and more.

Revision — Carrying and Borrowing

Carrying (Addition): When column sum ≥ 10, write ones digit, carry the rest to next column.
Borrowing (Subtraction): When top digit < bottom digit, borrow 1 from next column (adds 10 to current digit).

Addition of Large Numbers (6-8 Digits)

Steps: 1) Align by place value. 2) Start from Ones (right). 3) Carry if sum ≥ 10. 4) Continue left.

Example 1 (No Carrying): 34,52,100 + 21,34,678 = 55,86,778

[Image: Column addition of 45,67,893 + 38,45,678 showing carries in red above each column]

Example 3 (8-digit): 4,56,78,345 + 3,87,65,789 = 8,44,44,134

Example 4 (Three numbers): 12,34,567 + 23,45,678 + 34,56,789
Step 1: 12,34,567 + 23,45,678 = 35,80,245
Step 2: 35,80,245 + 34,56,789 = 70,37,034

Example 5 (Different digits): 5,43,21,000 + 89,45,673 = 6,32,66,673
(Align from right; treat missing left digits as 0)

Properties of Addition

1. Commutative: a + b = b + a (order doesn't matter)
2. Associative: (a+b)+c = a+(b+c) (grouping doesn't matter)
3. Identity: a + 0 = a (zero changes nothing)

Note: Subtraction is NOT commutative and NOT associative.

Subtraction of Large Numbers (6-8 Digits)

Steps: 1) Write larger number on top, align by place value. 2) Start from Ones. 3) Borrow if top < bottom. 4) Continue left.

Example 1 (No Borrowing): 78,96,543 − 45,63,210 = 33,33,333

[Image: Column subtraction of 63,42,105 − 27,58,436 showing borrowing with crossed-out digits]

Example 3 (Zeros — Borrowing Chain): 5,00,00,000 − 23,45,678
The 5 becomes 4. All zeros become 9 (chain borrowing). Last position: 10 − 8 = 2.
Answer: 4,76,54,322

[Image: Subtraction from zeros — 5,00,00,000 − 23,45,678 showing the chain of borrowing across zeros]

Quick Rule for Zeros: When subtracting from a number like 5,00,00,000:
• Leftmost non-zero digit decreases by 1.
• All zeros in between become 9.
• Last zero becomes 10, then subtract.
Example: 1,00,000 − 1 = 99,999

Example 4 (8-digit): 8,25,43,671 − 3,67,89,245 = 4,57,54,426

Checking Your Answer

Check Addition: Sum − one addend = other addend
Check Subtraction: Difference + subtrahend = minuend

Example: 45,67,893 + 38,45,678 = 84,13,571 → Check: 84,13,571 − 38,45,678 = 45,67,893 ✓
Example: 5,00,00,000 − 23,45,678 = 4,76,54,322 → Check: 4,76,54,322 + 23,45,678 = 5,00,00,000 ✓

Estimation

Round numbers before calculating. Useful for quick checks and mental maths.

Round to nearest lakh: Look at TTh digit. ≥5 → round up. <5 → round down.
Round to nearest crore: Look at TL digit. ≥5 → round up. <5 → round down.

Example 1: 43,67,892 + 28,34,561 ≈ 44,00,000 + 28,00,000 = 72,00,000 (Actual: 72,02,453)
Example 2: 7,85,43,000 − 3,24,67,000 ≈ 8,00,00,000 − 3,00,00,000 = 5,00,00,000

Word Problems (Indian Context)

1. Population of Rajasthan (7,89,00,000) + Gujarat (6,38,00,000) = 14,27,00,000
2. ISRO budget ₹1,34,80,00,000 − spent ₹89,56,00,000 = ₹45,24,00,000 remaining
3. Delhi to Chennai (2,182 km) + Chennai to Kanyakumari (710 km) = 2,892 km total
4. School grant ₹25,67,890 − books ₹12,45,678 − sports ₹8,90,000 = ₹4,32,212 left
5. MI runs (24,56,789) − CSK runs (23,89,456) = 67,333 more runs

[Image: Real-life addition/subtraction infographic — India map with populations, ISRO rocket, train routes, school budget]

Key Words

Term	Meaning
Addend	Number being added
Sum	Result of addition
Difference	Result of subtraction
Minuend	Number from which we subtract
Subtrahend	Number being subtracted
Carrying	Moving extra ten to next column (addition)
Borrowing	Taking 10 from next column (subtraction)
Estimation	Approximate answer by rounding
Inverse Operation	Opposite operation (add ↔ subtract)
Commutative	a + b = b + a
Associative	(a+b)+c = a+(b+c)

Key Points:
• Align by place value. Start from Ones (right). Carry in addition, borrow in subtraction.
• Zeros: borrowing chain — each 0 becomes 9, leftmost non-zero digit decreases by 1.
• Check: Addition → verify with subtraction. Subtraction → verify with addition.
• Addition: commutative + associative. Subtraction: neither.
• Word clues: "total/combined/altogether" = add. "Difference/remaining/left/more" = subtract.
• Estimation: round to nearest lakh or crore before calculating.

Practice Questions

A. Add the following (Column Method):

1. 34,56,789 + 45,67,890 =

2. 5,67,89,012 + 2,45,67,890 =

3. 78,90,123 + 56,78,901 =

4. 4,00,00,000 + 3,99,99,999 =

5. 1,23,45,678 + 8,76,54,322 =

6. 6,54,32,100 + 89,67,900 =

B. Subtract the following (Column Method):

1. 89,45,678 − 34,67,890 =

2. 7,00,00,000 − 2,34,56,789 =

3. 9,12,34,567 − 4,56,78,901 =

4. 5,00,00,000 − 1 =

5. 6,05,04,030 − 2,78,56,891 =

6. 3,40,00,000 − 1,56,78,234 =

C. Estimate the sum or difference (round to nearest lakh):

1. 56,78,345 + 34,23,678 ≈

2. 7,89,45,000 − 3,12,67,000 ≈

3. 23,45,600 + 67,89,400 ≈

4. 9,00,00,000 − 4,56,78,000 ≈

D. Fill in the blanks (Properties of Addition):

1. 45,67,890 + 23,45,678 = 23,45,678 + (Property: )

2. (12,00,000 + 34,00,000) + 56,00,000 = 12,00,000 + ( + 56,00,000)

3. 8,76,54,321 + 0 = (Property: )

4. If 34,56,789 + 56,78,901 = 91,35,690, then 56,78,901 + 34,56,789 =

E. Find the missing number:

1. + 34,56,789 = 78,90,123

2. 5,00,00,000 − = 2,67,89,345

3. 1,23,45,678 + = 5,00,00,000

4. − 45,67,890 = 54,32,110

F. True or False:

1. 56,78,900 + 43,21,100 = 1,00,00,000

2. Subtraction is commutative.

3. The sum of two 7-digit numbers is always 8-digit.

4. A number minus itself is always 0.

5. Estimation gives the exact answer.

G. Choose the correct answer (MCQ):

1. What is 4,56,78,000 + 3,43,22,000?

a) 7,00,00,000 b) 8,00,00,000 c) 7,99,00,000 d) 8,00,00,001

2. What is 9,00,00,000 − 1?

a) 8,99,99,999 b) 8,00,00,000 c) 9,99,99,999 d) 8,99,99,990

3. Which property: 25,00,000 + 0 = 25,00,000?

a) Commutative b) Associative c) Identity d) Distributive

4. Estimated sum of 78,45,000 + 21,67,000 (nearest lakh):

a) 99,00,000 b) 1,00,00,000 c) 1,00,12,000 d) 99,12,000

H. Word Problems:

1. Population of UP is 2,40,00,000 and Bihar is 1,28,00,000. What is the total population?

2. A factory produced 45,67,890 toys last year and 52,34,567 this year. How many more toys this year?

3. Meena's father earns ₹18,50,000/year and spends ₹14,25,600/year. How much is saved?

4. ISRO: Chandrayaan-3 (₹6,15,00,00,000) + Mangalyaan (₹4,50,00,00,000). Total spending?

5. Library: 12,34,567 Hindi + 8,90,345 English + 5,67,890 regional books. Total?

6. Ravi collected ₹3,45,678 and Sita collected ₹2,89,456. How much more did Ravi collect? What is the total?

Activity — Budget Planner

Arjun's family income: ₹1,25,000/month. Expenses:

Expense	Amount (₹)
House Rent	35,000
Groceries & Food	22,500
School Fees	18,000
Bills (Electricity/Water)	4,500
Transport	8,000
Medicines	3,500
Clothing	5,000
Entertainment	3,000

Questions:

1. Total monthly expense?

2. Monthly savings (Income − Expense)?

3. Savings in 1 year (12 months)?

4. Months needed to save ₹6,00,000?

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