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Large Numbers Around Us

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces students to large numbers beyond thousands, specifically lakhs, crores, and arabs in the Indian system, and millions and billions in the American system. It helps students understand the magnitude of these numbers through real-life examples like population, distance, and everyday objects. The big idea is that large numbers are not just abstract symbols—they represent real quantities that we can compare, estimate, and manipulate using place value, rounding, and multiplication shortcuts.
Key Concepts (11)
Lakh
1 lakh is 1 followed by 5 zeroes, written as 1,00,000 in the Indian system.
e.g. There were about one lakh varieties of rice in India earlier.
Crore
1 crore is 1 followed by 7 zeroes, written as 1,00,00,000 in the Indian system. 100 lakhs make 1 crore.
e.g. The population of Mumbai is more than 1 crore 24 lakhs.
Arab
1 arab is 1 followed by 9 zeroes, written as 1,00,00,00,000 in the Indian system. 100 crores make 1 arab.
e.g. The number 9,87,65,01,234 is read as 9 arab 87 crore 65 lakh 1 thousand 234.
Million
1 million is 1 followed by 6 zeroes, written as 1,000,000 in the American system. It is equal to 10 lakhs.
e.g. The distance between Earth and Sun is about 2100 × 70,000 = 14,70,00,000 km, which is 147 million km.
Billion
1 billion is 1 followed by 9 zeroes, written as 1,000,000,000 in the American system. It is equal to 100 crores or 1 arab.
e.g. The number 9,876,501,234 is read as 9 billion 876 million 501 thousand 234.
Indian Place Value System
In the Indian system, commas are placed in a 3-2-2-2 pattern from right to left: thousands, lakhs, crores, arabs.
e.g. The number 12,78,830 is read as twelve lakh seventy-eight thousand eight hundred thirty.
American (International) Place Value System
In the American system, commas are placed uniformly in a 3-3-3 pattern from right to left: thousands, millions, billions.
e.g. The number 4,050,678 is read as four million fifty thousand six hundred seventy-eight.
Rounding (Approximation)
Rounding means replacing a number with a nearby number that is easier to work with, either by rounding up (making it larger) or rounding down (making it smaller).
e.g. The population of Chintamani in 2011 was 76,068, but we often say 'about 75,000' to give an idea of its size.
Nearest Neighbour (Rounding to nearest thousand, lakh, crore)
For a given number, we can find its nearest thousand, ten thousand, lakh, ten lakh, or crore by looking at the digits to the right of that place.
e.g. For 6,72,85,183, the nearest lakh is 6,73,00,000 and the nearest crore is 7,00,00,000.
Multiplication Shortcut (using factors like 5, 25, 125)
Multiplying by 5 is the same as dividing by 2 and then multiplying by 10. Multiplying by 25 is the same as dividing by 4 and multiplying by 100. Multiplying by 125 is the same as dividing by 8 and multiplying by 1000.
e.g. 116 × 5 = 116 × (10 ÷ 2) = (116 ÷ 2) × 10 = 58 × 10 = 580.
Number of digits in a product
The product of two numbers can have at most the sum of the number of digits of the two numbers, and at least one less than that sum.
e.g. A 2-digit × 2-digit product can be 3-digit (e.g., 10×10=100) or 4-digit (e.g., 99×99=9801).
Formulas & Equations (8)
1 lakh = 1,00,000
Reads as: One lakh equals one followed by five zeroes.
Variables: No variables; it is a fixed value.
Use when: When converting between lakhs and smaller units or when reading/writing large numbers in Indian system.
⚠ Students often write 1,00,000 as 100,000 (American style) or forget the comma after the first digit.
1 crore = 1,00,00,000
Reads as: One crore equals one followed by seven zeroes.
Variables: No variables; it is a fixed value.
Use when: When converting between crores and lakhs or when reading/writing numbers in Indian system.
⚠ Students often confuse crore with million (1,000,000) and write 1,00,00,000 incorrectly.
1 million = 1,000,000
Reads as: One million equals one followed by six zeroes.
Variables: No variables; it is a fixed value.
Use when: When reading/writing numbers in American system or converting between lakhs and millions.
⚠ Students think 1 million = 1 crore, but 1 million = 10 lakhs.
1 billion = 1,000,000,000
Reads as: One billion equals one followed by nine zeroes.
Variables: No variables; it is a fixed value.
Use when: When reading/writing very large numbers in American system.
⚠ Students confuse billion with arab (both have 9 zeroes, but different names in different systems).
1 arab = 1,00,00,00,000
Reads as: One arab equals one followed by nine zeroes in Indian system.
Variables: No variables; it is a fixed value.
Use when: When reading/writing numbers in Indian system beyond crores.
⚠ Students forget that 100 crores = 1 arab.
Multiplying by 5: a × 5 = (a ÷ 2) × 10
Reads as: To multiply a number by 5, divide it by 2 and then multiply by 10.
Variables: a = any number
Use when: When you need to quickly multiply a number by 5, especially in mental math or estimation.
⚠ Students forget to multiply by 10 after dividing by 2, or they divide the wrong number.
Multiplying by 25: a × 25 = (a ÷ 4) × 100
Reads as: To multiply a number by 25, divide it by 4 and then multiply by 100.
Variables: a = any number
Use when: When you need to quickly multiply a number by 25.
⚠ Students forget to multiply by 100 after dividing by 4.
Multiplying by 125: a × 125 = (a ÷ 8) × 1000
Reads as: To multiply a number by 125, divide it by 8 and then multiply by 1000.
Variables: a = any number
Use when: When you need to quickly multiply a number by 125.
⚠ Students forget to multiply by 1000 after dividing by 8.
How It Works
Reading and Writing Large Numbers in Indian System
  1. Step 1: Write the number with commas in the 3-2-2 pattern from right to left (thousands, lakhs, crores).
  2. Step 2: Read the leftmost group as 'x crore' if it has 2 digits, or 'x arab' if it has 3 digits, etc.
  3. Step 3: Read the next group as 'x lakh'.
  4. Step 4: Read the next group as 'x thousand'.
  5. Step 5: Read the last group as 'x hundred and y' (if any).
💡 Always place commas from the right: first comma after 3 digits (thousands), then after every 2 digits (lakhs, crores).
Reading and Writing Large Numbers in American System
  1. Step 1: Write the number with commas every 3 digits from right to left.
  2. Step 2: Read the leftmost group as 'x billion' (if 3 digits), then 'x million', then 'x thousand', then the last group as is.
💡 In American system, commas are placed every 3 digits, not after 2 digits like in Indian system.
Rounding Numbers to Nearest Lakh, Crore, etc.
  1. Step 1: Identify the place value you are rounding to (e.g., nearest lakh).
  2. Step 2: Look at the digit immediately to the right of that place.
  3. Step 3: If that digit is 5 or more, increase the digit in the rounding place by 1 and replace all digits to the right with zeroes.
  4. Step 4: If that digit is less than 5, keep the digit in the rounding place the same and replace all digits to the right with zeroes.
💡 For nearest lakh, look at the ten-thousands digit. For nearest crore, look at the ten-lakh digit.
Using Multiplication Shortcuts (5, 25, 125)
  1. Step 1: Identify the multiplier (5, 25, or 125).
  2. Step 2: Replace the multiplier with its equivalent fraction: 5 = 10/2, 25 = 100/4, 125 = 1000/8.
  3. Step 3: Multiply the given number by the numerator (10, 100, or 1000) and divide by the denominator (2, 4, or 8).
  4. Step 4: Perform the division first if it makes the calculation easier.
💡 Always do the division first (if possible) to keep numbers smaller and avoid errors.
Determining Number of Digits in a Product
  1. Step 1: Find the smallest possible product by multiplying the smallest numbers in the given range (e.g., 10×10 for 2-digit numbers).
  2. Step 2: Find the largest possible product by multiplying the largest numbers in the given range (e.g., 99×99 for 2-digit numbers).
  3. Step 3: The product will have a number of digits between the number of digits of the smallest product and the largest product.
💡 The product of two m-digit and n-digit numbers can have either (m+n) or (m+n-1) digits.
Exam Traps — Don't Make These Mistakes
⚠ Confusing Lakh and Crore with Million and Billion
✗ 1 lakh = 1,000,000 (one million) or 1 crore = 1,000,000 (one million)
→ 1 lakh is 1,00,000 (5 zeroes) and 1 crore is 1,00,00,000 (7 zeroes). 1 million is 1,000,000 (6 zeroes) which is 10 lakhs, not 1 lakh.
✓ 1 lakh = 1,00,000 (Indian) = 100,000 (American). 1 crore = 1,00,00,000 (Indian) = 10,000,000 (American).
⚠ Wrong Comma Placement in Indian System
✗ Writing 1,00,000 as 100,000 (American style) or writing 12,78,830 as 1,27,8830
→ In Indian system, commas are placed after 3 digits from right (thousands), then after every 2 digits (lakhs, crores). 100,000 is American style.
✓ Always place commas from right: first comma after 3 digits, then after every 2 digits. Example: 12,78,830.
⚠ Rounding to Nearest Lakh Instead of Nearest Crore
✗ For 6,72,85,183, they write nearest crore as 6,00,00,000 instead of 7,00,00,000
→ When rounding to nearest crore, we look at the ten-lakh digit (the digit in the 7th place from right). Here it is 7, which is ≥5, so we round up the crore digit from 6 to 7.
✓ Nearest crore of 6,72,85,183 is 7,00,00,000 because the ten-lakh digit (7) is ≥5.
⚠ Forgetting to Multiply by 10/100/1000 in Multiplication Shortcuts
✗ 116 × 5 = 58 (only dividing by 2, forgetting to multiply by 10)
→ The shortcut is: a × 5 = (a ÷ 2) × 10. Dividing by 2 gives 58, but we must then multiply by 10 to get 580.
✓ 116 × 5 = (116 ÷ 2) × 10 = 58 × 10 = 580.
⚠ Incorrect Number of Digits in Product
✗ Product of two 2-digit numbers can be 2-digit or 3-digit
→ The smallest 2-digit number is 10, and 10×10=100 (3-digit). The largest is 99×99=9801 (4-digit). So product can be 3-digit or 4-digit, never 2-digit.
✓ Product of two 2-digit numbers can be 3-digit (e.g., 10×10=100) or 4-digit (e.g., 99×99=9801).
Compare & Contrast
Placement of commas and naming of large numbers
Indian Place Value System
American (International) Place Value System
Indian system uses a 3-2-2 pattern (thousands, lakhs, crores) while American system uses a uniform 3-3-3 pattern (thousands, millions, billions). For example, 1,00,000 is one lakh in Indian but one hundred thousand in American.
Whether the approximated number is greater or less than the actual number
Rounding Up
Rounding Down
Rounding up gives an approximated number that is more than the actual number (e.g., ordering 750 sweets for 732 people). Rounding down gives an approximated number that is less than the actual number (e.g., saying ₹470 is around ₹450).
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA-I (2 marks), 1 SA-II (3 marks), 1 LA (5 marks) — total 11 marks
Most asked: Reading and writing numbers in Indian place value system (lakhs, crores) · Rounding numbers to nearest lakh, crore, etc. · Multiplication shortcuts using 5, 25, 125 · Comparing and converting between Indian and American systems
Don't skip: Word problems involving population data and approximation (e.g., 'By how much did the population increase?'); Writing expressions for numbers using place value (like Chitti's calculator problems); Multiplication shortcuts and patterns in products
One-Line Revision (read aloud before the exam)
  1. 1 lakh = 1,00,000 (5 zeroes); 1 crore = 1,00,00,000 (7 zeroes); 1 arab = 1,00,00,00,000 (9 zeroes).
  2. 1 million = 1,000,000 (6 zeroes) = 10 lakhs; 1 billion = 1,000,000,000 (9 zeroes) = 100 crores.
  3. In Indian system, commas are placed in a 3-2-2 pattern from right: thousands, lakhs, crores.
  4. In American system, commas are placed every 3 digits from right: thousands, millions, billions.
  5. To round to nearest lakh, look at the ten-thousands digit; if ≥5, round up the lakh digit.
  6. To round to nearest crore, look at the ten-lakh digit; if ≥5, round up the crore digit.
  7. Multiplying by 5 is same as dividing by 2 and multiplying by 10: a × 5 = (a ÷ 2) × 10.
  8. Multiplying by 25 is same as dividing by 4 and multiplying by 100: a × 25 = (a ÷ 4) × 100.
  9. Multiplying by 125 is same as dividing by 8 and multiplying by 1000: a × 125 = (a ÷ 8) × 1000.
  10. Product of two 2-digit numbers can be 3-digit or 4-digit; product of two 3-digit numbers can be 5-digit or 6-digit.
  11. Approximation is useful when exact numbers are not needed, e.g., population is 'about 75,000' instead of 76,068.
  12. 1 lakh days = about 274 years; 1 lakh people shoulder-to-shoulder stretch about 38 km.
  13. The Statue of Unity is about 180 m tall; Kunchikal waterfall is about 450 m tall.
  14. The distance between Earth and Sun is about 14,70,00,000 km (147 million km).
  15. The Amazon river discharges about 6400 × 62,500 = 40,00,00,000 litres of water per second.

MCQs — 10 Questions with Answers

Q1. What is the number name of 3,00,600 in the Indian system?

  • A. Three lakh six hundred ✓
  • B. Three hundred thousand six hundred
  • C. Thirty lakh six hundred
  • D. Three lakh and six hundred

Answer: A — In Indian system, 3,00,600 reads as three lakh (3 in lakh place) and six hundred (600 ones).

Q2. According to the text, how much less than one lakh is 75,000?

  • A. 20,000
  • B. 25,000 ✓
  • C. 15,000
  • D. 30,000

Answer: B — 1,00,000 − 75,000 = 25,000, so 75,000 is 25,000 less than one lakh.

Q3. Which of the following is the correct way to write one crore in numbers?

  • A. 1,000,000
  • B. 10,00,000
  • C. 1,00,00,000 ✓
  • D. 1,00,000

Answer: C — One crore is 1,00,00,000 in the Indian system (1 followed by 7 zeroes).

Q4. How many times should the +100 button be pressed to show 10,000?

  • A. 100 times ✓
  • B. 10 times
  • C. 1,000 times
  • D. 50 times

Answer: A — 10,000 ÷ 100 = 100, so the button must be pressed 100 times.

Q5. Roxie and Estu live in Chintamani, Karnataka. If they ate 2 varieties of rice daily, how many days would it take to taste 1 lakh varieties?

  • A. 50,000 days ✓
  • B. 1,00,000 days
  • C. 2,00,000 days
  • D. 2,00,000 years

Answer: A — 1,00,000 ÷ 2 = 50,000 days needed to taste all varieties at the rate of 2 per day.

Q6. The height of the Statue of Unity in Gujarat is about 180 metres. Kunchikal waterfall drops from 450 metres. How much taller is the waterfall than the statue?

  • A. 270 metres ✓
  • B. 250 metres
  • C. 630 metres
  • D. 450 metres

Answer: A — 450 − 180 = 270 metres, so the waterfall is 270 metres taller than the Statue of Unity.

Q7. According to the American system, what is 1,00,00,000 called?

  • A. One billion
  • B. Hundred million
  • C. Ten million ✓
  • D. One million

Answer: C — In the American system, 1,00,00,000 (1 crore) equals 10,000,000, which is ten million.

Q8. If 1 lakh people stood shoulder to shoulder in a line, how far would the line stretch?

  • A. 38 metres
  • B. 38 kilometres ✓
  • C. 380 kilometres
  • D. 3.8 kilometres

Answer: B — According to the text, 1 lakh people in a shoulder-to-shoulder line would stretch 38 kilometres.

Q9. Which number system (Indian or American) groups digits in a 3-2-2-2 pattern from right to left?

  • A. American system only
  • B. Indian system only ✓
  • C. Both systems use the same pattern
  • D. Neither system uses this pattern

Answer: B — The Indian system uses 3-2-2-2 pattern (thousands, lakhs, crores, etc.), while American uses 3-3-3-3 pattern.

Q10. Creative Chitti wants to make 5072 using button presses. One method is (5 × 1000) + (0 × 100) + (7 × 10) + (2 × 1). How many total button presses does this method require?

  • A. 14 presses ✓
  • B. 12 presses
  • C. 16 presses
  • D. 5,072 presses

Answer: A — 5 presses of +1000 + 0 presses of +100 + 7 presses of +10 + 2 presses of +1 = 5 + 0 + 7 + 2 = 14 presses.

Flashcards

What is one lakh written in numbers?

One lakh is written as 1,00,000 (1 followed by 5 zeroes).

How many lakhs make one crore?

One hundred lakhs (100 lakhs) make one crore.

What is the number name of 1,00,00,000 in Indian system?

1,00,00,000 is called one crore in the Indian system.

How many zeroes does one arab have?

One arab (1,00,00,00,000) has 9 zeroes.

What is the difference between 75,000 and 1 lakh?

The difference is 25,000 (since 1,00,000 − 75,000 = 25,000).

In Indian place value system, how are commas placed?

Commas are placed in a 3-2-2-2 pattern from right to left (thousands, lakhs, crores, etc.).

How many days are in 100 years (ignoring leap years)?

There are 36,500 days in 100 years (365 × 100 = 36,500).

What is the relationship between thousands and lakhs?

One lakh is 100 times a thousand (1,00,000 ÷ 1,000 = 100).

Write 12,78,830 in words using Indian system.

12,78,830 is written as twelve lakh seventy eight thousand eight hundred thirty.

How many button presses on a +1000 calculator to show 1 lakh?

100 button presses are needed since 1,00,000 ÷ 1,000 = 100.

Important Board Questions

Write the number name of 27,30,000 in the Indian place value system. [1 mark]

Identify the crore place first (27), then the lakh place (30), then the thousands place. Read from left to right.

According to the 2011 Census, Chintamani had a population of 75,000. By 2024, it increased to 1,06,000. By how much did the population increase? Express your answer in words and numbers. [2 marks]

Subtract 2011 population from 2024 population: 1,06,000 − 75,000. The answer will be in the form 'thirty one thousand' and 31,000.

If a person tasted 3 varieties of rice every day, how many days would it take them to taste 1 lakh varieties? If they live for 100 years (365 days per year), would they be able to taste all varieties? Show your working and explain. [3 marks]

First divide 1,00,000 by 3 to find number of days. Then calculate days in 100 years as 365 × 100 = 36,500. Compare the two answers. Show: (100,000 ÷ 3 ≈ 33,333 days) and (100 × 365 = 36,500 days).

Study the Indian and American numbering systems shown in the textbook. (a) How many zeroes does 1 arab have? (b) Write 10,87,65,432 in both Indian and American systems with commas. (c) Explain why the Indian system places commas in a 3-2-2-2 pattern whereas the American system uses a 3-3-3-3 pattern. Write your observation about which system you find easier to use and why. [5 marks]

For (a): arab = 100 crores, crore has 7 zeroes, so arab has 9 zeroes. For (b): Indian = 10,87,65,432 (ten arab eighty-seven crore sixty-five lakh forty-two thousand four hundred thirty-two); American = 1,087,654,321 (one billion eighty-seven million...). For (c): Indian system groups for traditional naming (thousands, lakhs, crores); American groups uniformly in threes for easier conversion. Personal preference explanation required.

Assertion-Reason Questions for Large Numbers Around Us

Q1. Assertion (A):

One lakh in the Indian place value system is written as 1,00,000.

Reason (R):

In the Indian place value system, we use commas to separate lakhs and thousands places, so the smallest 6-digit number is 1,00,000.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why one lakh is written as 1,00,000 in the Indian system. The comma placement between the lakh and thousands places is a defining feature of Indian notation.

Q2. Assertion (A):

If a person ate one new variety of rice each day, they could taste approximately 36,500 varieties in 100 years.

Reason (R):

There are 365 days in a year, so 365 × 100 = 36,500 days total, meaning 36,500 varieties can be tasted.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true: 365 days/year × 100 years = 36,500 days = 36,500 varieties at one per day. The reason is also true (365 × 100 = 36,500), but it doesn't fully explain the assertion because it doesn't explicitly state the 1-to-1 relationship between days and varieties consumed daily. Both statements are correct, but the reason could be more complete.

Q3. Assertion (A):

The largest 5-digit number is 99,999 and the smallest 6-digit number is 1,00,000.

Reason (R):

Because the difference between them is 1, as shown by 1,00,000 - 99,999 = 1.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason directly explains and verifies the assertion. The consecutive nature of these boundary numbers (differing by exactly 1) demonstrates they are adjacent in the place value system.

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Case-Based Questions (CBQ) for Large Numbers Around Us

Case Study:

A seed bank in Karnataka decided to document the rice seed varieties they have collected. In their first year, they collected 25,000 seed packets. By the end of the second year, they had collected 56,000 packets total. By the third year, they reached 1,03,500 packets. The manager wants to compare these quantities with one lakh (1,00,000) to understand how close they are to their goal of preserving one lakh varieties.

Q1. How many seed packets short was the bank after the first year compared to one lakh?

  • A. 75,000 ✓
  • B. 25,000
  • C. 50,000
  • D. 100,000

Answer: A

Q2. How many more packets did they collect between year 1 and year 2?

  • A. 25,000
  • B. 31,000 ✓
  • C. 56,000
  • D. 30,000

Answer: B

Q3. By the third year, how much more than one lakh did they exceed?

  • A. 3,500 ✓
  • B. 1,000
  • C. 5,000
  • D. 2,500

Answer: A

Q4. What was the total increase in seed packets from year 1 to year 3?

  • A. 1,03,500
  • B. 78,500 ✓
  • C. 25,000
  • D. 56,000

Answer: B

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