Addition and Subtraction — Large Numbers
1. Addition of Large Numbers
Steps for Addition:
- Write numbers in COLUMNS — align by place value (ones under ones, tens under tens, etc.).
- Start from the RIGHT (ones place) and add each column.
- If the sum is 10 or more, CARRY OVER to the next column.
Example 1:
Add 34,567 and 23,456
| TTh | Th | H | T | O | |
|---|---|---|---|---|---|
| 3 | 4 | 5 | 6 | 7 | |
| + | 2 | 3 | 4 | 5 | 6 |
| Carry | 1 | 1 | 1 | ||
| Total | 5 | 8 | 0 | 2 | 3 |
Answer: 58,023
Example 2:
Add 1,23,456 + 2,34,567
Step 1 — Align by place value: 1,23,456 + 2,34,567 Step 2 — Add from right: 6 + 7 = 13 → write 3, carry 1 Step 3 — 5 + 6 + 1 (carry) = 12 → write 2, carry 1 Step 4 — 4 + 5 + 1 (carry) = 10 → write 0, carry 1 Step 5 — 3 + 4 + 1 (carry) = 8 Step 6 — 2 + 3 = 5 Step 7 — 1 + 2 = 3
Answer: 3,58,023
'Always add from the RIGHT — ones first, then tens, then hundreds. Never start from the left!'
2. Properties of Addition
| Property | Meaning | Example |
|---|---|---|
| Commutative | Order does NOT matter. | 15 + 25 = 25 + 15 |
| Associative | Grouping does NOT matter. | (10 + 20) + 30 = 10 + (20 + 30) |
| Identity | Adding ZERO gives the same number. | 567 + 0 = 567 |
Key Fact:
'Addition is COMMUTATIVE — you can add numbers in ANY order and get the SAME sum.'
3. Subtraction of Large Numbers
Steps for Subtraction:
- Write numbers in COLUMNS — larger number on TOP.
- Start from the RIGHT (ones place).
- If the top digit is SMALLER than the bottom digit, BORROW from the next column.
Example 1:
Subtract 23,456 from 56,789
| TTh | Th | H | T | O | |
|---|---|---|---|---|---|
| 5 | 6 | 7 | 8 | 9 | |
| - | 2 | 3 | 4 | 5 | 6 |
| Total | 3 | 3 | 3 | 3 | 3 |
Answer: 33,333
Example 2 (with borrowing):
Subtract 28,567 from 54,234
| TTh | Th | H | T | O | |
|---|---|---|---|---|---|
| 5 | 4 | 2 | 3 | 4 | |
| - | 2 | 8 | 5 | 6 | 7 |
| Borrow | 13→3 | 11→1 | 12→2 | 14 | |
| Total | 2 | 5 | 6 | 6 | 7 |
Answer: 25,667
'When borrowing, the number you borrow from DECREASES by 1. The number you borrow to INCREASES by 10.'
4. Properties of Subtraction
| Property | Meaning | Example |
|---|---|---|
| Subtracting zero | Number stays the same. | 789 - 0 = 789 |
| Subtracting itself | Result is ALWAYS zero. | 567 - 567 = 0 |
| NOT commutative | Order MATTERS! | 20 - 10 ≠ 10 - 20 |
Important:
'Subtraction is NOT commutative. 20 - 10 is NOT the same as 10 - 20. You cannot swap the numbers!'
5. Checking Your Answers
Check Addition with Subtraction:
If A + B = C, then C - B = A and C - A = B.
Example: 34,567 + 23,456 = 58,023 Check: 58,023 - 23,456 = 34,567 ✓
Check Subtraction with Addition:
If A - B = C, then C + B = A.
Example: 54,234 - 28,567 = 25,667 Check: 25,667 + 28,567 = 54,234 ✓
6. Word Problems
Problem 1 — Money:
Riya has Rs. 45,670 in her bank account. She deposits Rs. 23,890 more. How much money does she have now?
Solution: 45,670 + 23,890 = Rs. 69,560
Problem 2 — Distance:
A train travels 2,34,567 meters on Day 1 and 1,89,432 meters on Day 2. What is the total distance travelled?
Solution: 2,34,567 + 1,89,432 = 4,23,999 meters
Problem 3 — Comparison:
A book has 3,45,678 words. Another book has 1,23,456 words. How many more words does the first book have?
Solution: 3,45,678 - 1,23,456 = 2,22,222 words
Problem 4 — Quantity:
A shopkeeper had 1,00,000 eggs. He sold 67,890 eggs. How many eggs are left?
Solution: 1,00,000 - 67,890 = 32,110 eggs
7. Common Mistakes
- Not aligning by place value: 'Always write ones under ones, tens under tens. Misalignment is the NUMBER ONE cause of wrong answers!'
- Forgetting to carry or borrow: 'If a column adds to 10 or more, you MUST carry. If the top digit is smaller, you MUST borrow.'
- Borrowing incorrectly: 'When you borrow 1 from a column, the digit there decreases by 1. The digit you borrowed for becomes 10 more.'
- Subtracting larger from smaller: 'Always check: the top number should be BIGGER when subtracting. If not, double-check your problem.'
8. Key Facts to Remember
- 'Always add/subtract from the RIGHT — ones place first.'
- 'Addition can be CHECKED by subtraction, and subtraction by addition.'
- 'The SUM of any number and ZERO is the number itself.'
- 'When you subtract a number from ITSELF, the answer is ALWAYS ZERO.'
- 'Carry when the sum is 10 or more. Borrow when the top digit is smaller than the bottom digit.'
9. Self-Test
Q1: Add: 56,789 + 34,567
Q2: Subtract: 78,901 - 45,678
Q3: A farmer grows 1,23,456 kg of wheat in one season and 98,765 kg in the next. How much wheat does he grow in total?
Q4: A school has 2,500 students. If 1,875 students are boys, how many are girls?
Q5: Check if this is correct: 67,890 - 23,456 = 44,434. Use addition to verify.
Q6: Which property? 345 + 678 = 678 + 345
Q7: The population of Town A is 1,45,670 and Town B is 2,34,890. Which town has more people? How many more?
Answers:
A1: 91,356 A2: 33,223 A3: 1,23,456 + 98,765 = 2,22,221 kg A4: 2,500 - 1,875 = 625 girls A5: Check: 44,434 + 23,456 = 67,890 ✓ Correct. A6: Commutative Property of Addition. A7: Town B has more. 2,34,890 - 1,45,670 = 89,220 more people.
