1. Chapter at a Glance
India's population is more than 140 crore. A cricket stadium can hold about 1 lakh people. The distance from the Earth to the Sun is about 15 crore kilometres. To understand such facts we need to read, write, compare and estimate large numbers confidently. This chapter takes you on a journey through lakhs and crores, millions and billions, and shows how powers of ten make large numbers easy to handle.
- Place value up to crores (Indian system) and billions (International system).
- Reading, writing and placing commas in large numbers.
- Converting between lakh–crore and million–billion.
- Comparing and ordering large numbers.
- Rounding and estimating large quantities sensibly.
- Patterns in multiplying by 10, 100, 1000 and beyond.
2. Place Value: Indian System
In our number system each place is 10 times the place on its right. Starting from ones: ones, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores, ten crores, arabs …
| Place | Value | Power of 10 |
|---|---|---|
| One | 1 | 10⁰ |
| Ten | 10 | 10¹ |
| Hundred | 100 | 10² |
| Thousand | 1,000 | 10³ |
| Ten thousand | 10,000 | 10⁴ |
| Lakh | 1,00,000 | 10⁵ |
| Ten lakh | 10,00,000 | 10⁶ |
| Crore | 1,00,00,000 | 10⁷ |
| Ten crore | 10,00,00,000 | 10⁸ |
| Arab (100 crore) | 1,00,00,00,000 | 10⁹ |
Commas in the Indian system
The first comma comes after the hundreds (three digits from the right); after that, commas come after every two digits. Example: 7,45,08,312 is read as seven crore forty-five lakh eight thousand three hundred twelve.
Write in figures: twelve crore five lakh sixty thousand nine.
Solution: Crores: 12; lakhs: 05; thousands: 60; hundreds-tens-ones: 009. Number = 12,05,60,009.
3. Place Value: International System
The International system groups digits in threes: ones, thousands, millions, billions. Commas come after every three digits from the right.
- 1 million = 1,000,000 = 10 lakh.
- 10 million = 1 crore.
- 100 million = 10 crore.
- 1 billion = 1,000,000,000 = 100 crore.
Write 3,46,50,000 in the International system and read it.
Solution: Remove commas: 34650000. Group in threes from the right: 34,650,000 — thirty-four million six hundred fifty thousand.
4. Converting between the Systems
| Indian | International |
|---|---|
| 1 lakh | 100 thousand |
| 10 lakh | 1 million |
| 1 crore | 10 million |
| 10 crore | 100 million |
| 100 crore | 1 billion |
| 1 lakh crore | 1 trillion |
A country's population is 1.4 billion. Express it in crores.
Solution: 1 billion = 100 crore, so 1.4 billion = 1.4 × 100 = 140 crore.
5. Comparing and Ordering Large Numbers
- The number with more digits is greater.
- If the number of digits is the same, compare the digits from the left; the first place where they differ decides.
Example: 4,58,72,310 and 4,58,27,999 have the same number of digits and match up to "4,58,". In the next place 7 > 2, so 4,58,72,310 is greater.
Largest and smallest numbers with given digits
- Largest: arrange digits in descending order. Using 3, 0, 8, 5, 1 → 85,310.
- Smallest: arrange in ascending order, but zero cannot be first → 10,358.
6. Rounding and Estimation
For very large quantities, exact values are often not needed or not known. We round to a convenient place.
- Find the place you want to round to (for example, the lakhs place).
- Look at the digit just to its right.
- If that digit is 5 or more, increase the rounding digit by 1; otherwise keep it.
- Replace all digits to the right with zeros.
Round 6,47,82,950 to the nearest lakh and to the nearest crore.
Solution: Nearest lakh: lakhs digit 7, next digit 8 ≥ 5 → 6,48,00,000. Nearest crore: crore digit 6, next digit 4 < 5 → 6,00,00,000.
Estimating sums and products
Round each number first, then calculate. 38,76,421 + 21,03,850 ≈ 39 lakh + 21 lakh = 60 lakh. Estimation is useful to check exact answers quickly.
Getting a feel for large numbers
- If you count one number per second without stopping, counting to 1 lakh takes about 28 hours; to 1 crore, about 116 days.
- A 500-page book contains roughly 1.5 to 2 lakh words.
- One crore rupees in ₹500 notes is 20,000 notes.
7. Patterns with Powers of Ten
- Multiplying by 10 adds one zero; by 100 adds two zeros; by 1000 adds three zeros.
- 1000 × 1000 = 10,00,000 = 10 lakh = 1 million.
- 10,000 × 10,000 = 10,00,00,000 = 10 crore = 100 million.
- 1 lakh × 1 lakh = 10⁵ × 10⁵ = 10¹⁰ = 1000 crore = 10 billion.
How many lakhs make 1 crore? How many thousands make 1 million?
Solution: 1 crore ÷ 1 lakh = 10⁷ ÷ 10⁵ = 100 lakhs. 1 million ÷ 1 thousand = 10⁶ ÷ 10³ = 1000 thousands.
8. Solved Problems (NCERT Style)
Read 50,08,03,027 in the Indian system and rewrite it in the International system.
Solution: Indian groups (crore | lakh | thousand | ones): 50 | 08 | 03 | 027 → fifty crore eight lakh three thousand twenty-seven. Removing commas gives 500803027; grouping in threes from the right gives 500,803,027 — five hundred million eight hundred three thousand twenty-seven.
A city has a population of 1,24,56,000. If on average 4 people live in a household, roughly how many households are there?
Solution: 1,24,56,000 ÷ 4 = 31,14,000. So there are about 31 lakh households. Dividing by 4 is the same as halving twice: 1,24,56,000 → 62,28,000 → 31,14,000.
The Moon is about 3,84,400 km from the Earth and the Sun is about 14,96,00,000 km away. Roughly how many times farther is the Sun?
Solution: Round: Moon ≈ 4 lakh km, Sun ≈ 15 crore km = 1500 lakh km. 1500 ÷ 4 = 375, so the Sun is roughly 400 times farther (a more exact division gives about 389). Estimation quickly shows the size of the difference.
In 7,32,45,618 the digit 3 is replaced by 8. By how much does the number increase?
Solution: The digit 3 is in the ten lakhs place (value 30,00,000). Changing it to 8 makes that value 80,00,000. Increase = 80,00,000 − 30,00,000 = 50,00,000 (50 lakh).
How many digits does the product 1 crore × 1 lakh have?
Solution: 10⁷ × 10⁵ = 10¹², which is 1 followed by 12 zeros, so it has 13 digits. It equals 1 lakh crore, or 1 trillion.
A news report says "about 2.5 crore people visited the fair". Which of these exact counts could it be describing: 2,47,30,000 or 2,58,90,000?
Solution: Rounded to the nearest ten lakh, 2,47,30,000 → 2,50,00,000 (2.5 crore) and 2,58,90,000 → 2,60,00,000 (2.6 crore). So the report fits 2,47,30,000.
These problems show the three big skills of this chapter: reading numbers in both systems, moving between them using powers of ten, and estimating sensibly before calculating exactly.
Write the number just after 9,99,99,999 and the number just before 1,00,00,000. What do you notice?
Solution: 9,99,99,999 + 1 = 10,00,00,000 (ten crore) — every 9 turns into 0 and the carry creates a new place. 1,00,00,000 − 1 = 99,99,999, the largest 7-digit number. Adding 1 to a number made only of 9s always gives a power of ten.
9. Master Formula Table
| Relation | Value |
|---|---|
| 1 lakh | 1,00,000 = 10⁵ = 100 thousand |
| 1 crore | 1,00,00,000 = 10⁷ = 100 lakh |
| 1 million | 1,000,000 = 10⁶ = 10 lakh |
| 1 billion | 1,000,000,000 = 10⁹ = 100 crore |
| 1 trillion | 10¹² = 1 lakh crore |
| Indian commas | 3 digits, then every 2 |
| International commas | every 3 digits |
| Rounding rule | next digit ≥ 5 → round up |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Place value | Each place is 10 × the next | Value of a digit |
| Indian system | Lakh, crore; 3-2-2 commas | Reading and writing |
| International system | Million, billion; 3-3-3 commas | Conversions |
| Comparison | Digits, then left-to-right | Ordering lists |
| Rounding | Look at the next digit | Nearest lakh/crore |
| Estimation | Round, then compute | Checking answers |
Chart: how many of each unit make 1 crore
(Bar lengths show the pattern, not exact scale — each step down multiplies the count by 10.)
11. Mind Map
- Lakh, crore, arab
- Commas 3-2-2
- Million, billion
- Commas 3-3-3
- Compare, order
- Round, estimate
- Adding zeros
- 10⁵, 10⁷, 10⁹
12. Tips and Tricks
- Count zeros: lakh has 5, crore has 7, million has 6, billion has 9.
- To convert crore to million, multiply by 10; million to lakh, multiply by 10.
- When reading a number, first put the commas, then read group by group.
- Use rounding to check whether a long calculation is reasonable.
- Putting Indian commas every three digits (that is the International style).
- Forgetting zeros in empty places: "five crore six" is 5,00,00,006.
- Thinking 1 billion = 1 crore (it is 100 crore).
- Starting the smallest number with zero.
13. Important One-liners
- Each place value is ten times the place to its right.
- 1 lakh = 1,00,000 = 10⁵.
- 1 crore = 100 lakh = 10⁷.
- 1 million = 10 lakh.
- 1 crore = 10 million.
- 1 billion = 100 crore.
- The Indian system uses lakhs and crores; the International system uses millions and billions.
- A number with more digits is greater.
- Rounding up happens when the next digit is 5 or more.
- 10,000 × 10,000 = 10 crore.
14. Exam-pattern Questions with Detailed Answers
How many zeros are there in one crore?
- 5
- 6
- 7
- 8
Show answer
1 crore = 1,00,00,000 = 10⁷, which has 7 zeros.
Answer: (C) 7
Write 9,08,40,506 in words (Indian system).
Show answer
Crores 9, lakhs 08, thousands 40, then 506.
Answer: Nine crore eight lakh forty thousand five hundred six
Express 5 million in lakhs.
Show answer
1 million = 10 lakh, so 5 million = 50 lakh.
Answer: 50 lakh
Write the place value of 6 in 3,62,58,104 and in 36,258,104.
Show answer
Both represent the same number 3,62,58,104 = 36,258,104. The digit 6 is in the ten lakhs (= millions) place: value 60,00,000 = 6,000,000.
Answer: 60 lakh = 6 million
Round 2,74,96,500 to the nearest ten lakh.
- 2,70,00,000
- 2,75,00,000
- 2,80,00,000
- 3,00,00,000
Show answer
Ten-lakh digit is 7; the next digit (lakh) is 4 < 5, so keep 7 and replace the rest with zeros.
Answer: (A) 2,70,00,000
Form the largest and smallest 7-digit numbers using the digits 4, 0, 9, 2, 7, 1, 5 once each. Write them with Indian commas.
Show answer
Largest: digits in descending order → 97,54,210. Smallest: ascending order with 0 not first → 10,24,579.
Answer: 97,54,210 and 10,24,579
A district has 18,45,300 voters. If 72 lakh is the population of the state's capital region, which is larger, and by about how many lakh?
Show answer
18,45,300 ≈ 18 lakh. 72 lakh − 18 lakh ≈ 54 lakh.
Answer: The capital region, by about 54 lakh
How many ₹2,000-worth bundles are needed to make ₹1 crore?
Show answer
1,00,00,000 ÷ 2,000 = 5,000.
Answer: 5,000 bundles
If you count one number every second without stopping, roughly how many days will you take to count up to 1 crore?
Show answer
Seconds in a day = 24 × 60 × 60 = 86,400. Days = 1,00,00,000 ÷ 86,400 ≈ 115.7.
Answer: About 116 days
A company earned 3.75 billion rupees. Write this in crores and in figures with Indian commas.
Show answer
1 billion = 100 crore, so 3.75 billion = 375 crore = 3,75,00,00,000 rupees.
Answer: ₹375 crore = ₹3,75,00,00,000
Estimate 4,87,312 × 196 by rounding each number to its leading place, then compare with the exact value.
Show answer
Estimate: 5,00,000 × 200 = 10,00,00,000 (10 crore). Exact: 4,87,312 × 196 = 9,55,13,152.
The estimate is close (both about 9–10 crore), which confirms the exact answer is reasonable.
Answer: Estimate ≈ 10 crore; exact 9,55,13,152
How many 6-digit numbers are there in all? Express your answer in lakhs.
Show answer
Six-digit numbers run from 1,00,000 to 9,99,999. Count = 9,99,999 − 1,00,000 + 1 = 9,00,000.
Answer: 9,00,000 = 9 lakh
15. Quick Revision Summary
- Indian: ones, thousands, lakhs (10⁵), crores (10⁷); commas 3-2-2.
- International: thousands, millions (10⁶), billions (10⁹); commas 3-3-3.
- 10 lakh = 1 million; 1 crore = 10 million; 100 crore = 1 billion.
- Compare by number of digits, then from the left.
- Round using the next digit (≥ 5 → up); estimate by rounding first.