Large Numbers Around Us

COMPREHENSIVE CHAPTER NOTES: LARGE NUMBERS AROUND US

Class 7 Mathematics (Ganita Prakash - NCF 2023)

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1.1 A LAKH VARIETIES!

Understanding the Concept of "Lakh"

**Lakh** is a term used in the Indian number system to represent the number **1,00,000** (one hundred thousand). It is written as **1,00,000** with a comma separating the hundred thousands place from the thousands place.

The Magnitude of One Lakh

To understand how large one lakh is, let us look at the pattern of numbers:

The largest 3-digit number is **999**

The smallest 4-digit number is **1,000**

The largest 4-digit number is **9,999**

The smallest 5-digit number is **10,000**

The largest 5-digit number is **99,999**

The smallest 6-digit number is **1,00,000** (One Lakh)

Each time we add 1 to the largest number of a particular digit count, we get the smallest number of the next digit count. This pattern helps us understand place value.

Real-Life Application: Tasting Rice Varieties

Estu wondered if eating one variety of rice per day could help him taste one lakh varieties in his lifetime of 100 years.

**Calculation:**

Days in a year = 365 (ignoring leap years)

Days in 100 years = 365 × 100 = 36,500 days

Number of varieties that can be tasted = 36,500

**Conclusion:** Even if Estu ate one variety of rice every single day for 100 years, he could only taste 36,500 varieties out of one lakh. He would still be far from tasting all varieties.

Application to Population Changes

**Example Problem:**

According to the 2011 Census, the population of Chintamani was about 75,000.

How much less than one lakh is 75,000?

**Answer:** 1,00,000 - 75,000 = 25,000

The population was 25,000 less than one lakh.

The estimated population of Chintamani in 2024 is 1,06,000.

How much more than one lakh is 1,06,000?

**Answer:** 1,06,000 - 1,00,000 = 6,000

The population was 6,000 more than one lakh.

Population increase from 2011 to 2024:

**Increase = 1,06,000 - 75,000 = 31,000**

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1.2 GETTING A FEEL OF LARGE NUMBERS

Comparing Large Numbers with Familiar Objects

Understanding the size of large numbers becomes easier when we compare them with something familiar.

**Example: The Statue of Unity**

Height of the Statue of Unity in Gujarat = about 180 metres

Height of Somu (a person) = 1 metre

Height of each floor in a building = about 4 times Somu's height = 4 metres

**Question:** If Somu's building has 45 floors, what is its approximate height?

Height = 45 × 4 = 180 metres

**The Statue of Unity is approximately equal in height to this 45-storey building.**

Using Comparisons for Large Heights

**Example: Kunchikal Waterfall**

Height of Kunchikal waterfall in Karnataka = about 450 metres

Height of Somu's building (45 floors) = 180 metres

Difference = 450 - 180 = 270 metres

**How many floors should Somu's building have to be as high as the waterfall?**

If each floor = 4 metres

Number of floors = 450 ÷ 4 = 112.5 floors (approximately 113 floors)

**Key Learning:** By comparing with familiar measurements, we can better understand the actual magnitude of large numbers.

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1.3 IS ONE LAKH A VERY LARGE NUMBER?

Different Perspectives on "Largeness"

Whether one lakh is "large" or "small" depends on the context. Let us examine different viewpoints:

Roxie's Perspective: Lakh is Very Large

**Argument 1:** One lakh varieties of rice is an enormous collection

Having access to 1,00,000 different varieties represents immense diversity and abundance

**Argument 2:** Living one lakh days is extraordinarily long

**Calculation:** 1,00,000 ÷ 365 ≈ 274 years

This is nearly three human lifetimes—much longer than any person can live

**Argument 3:** Physical arrangement of people shows vast quantity

If 1 lakh people stood shoulder to shoulder in a line, they would stretch approximately 38 kilometres

This is the distance equivalent to traveling across an entire city or more

Estu's Perspective: Lakh is Not That Large

**Argument 1:** Stadium capacity

The cricket stadium in Ahmedabad has a seating capacity of more than 1 lakh

This demonstrates that 1 lakh people can fit in a surprisingly small geographic area

**Argument 2:** Hair on human head

Most humans have 80,000 to 1,20,000 hairs on their head

1 lakh hair strands fit in the tiny space of a human scalp—something we don't even notice

**Argument 3:** Fish eggs

Some species of female fish can lay almost 1 lakh eggs at once

Some fish species lay tens of lakhs of eggs (multiple lakhs) in a single spawning—showing even larger numbers in nature

Conclusion

Whether a lakh is "big" or "small" is **context-dependent**. It is large when compared to human activities (travel, lifespan) but small when compared to natural phenomena or stadium capacities.

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1.4 READING AND WRITING NUMBERS

The Indian Place Value System

The **Indian place value system** uses commas to group digits in a **3-2-2-2 pattern** from right to left:

First comma after 3 digits from the right (thousands place)

Subsequent commas after every 2 digits

**Format:** _ , _ _ , _ _ , _ _ (for numbers up to one crore)

Place Value Chart (Up to One Crore)

| Place Value | Position | Example in 12,78,830 |

|---|---|---|

| Ones | 1st position from right | 0 |

| Tens | 2nd position from right | 3 |

| Hundreds | 3rd position from right | 8 |

| Thousands | 4th position from right | 8 |

| Ten Thousands | 5th position from right | 7 |

| Lakhs | 6th position from right | 2 |

| Ten Lakhs | 7th position from right | 1 |

Reading Numbers in Indian System

**Rule:** Read from left to right, grouping digits according to place value.

**Example 1:** 12,78,830

Reading: "Twelve lakh seventy eight thousand eight hundred thirty"

Breaking down:

12 lakhs = 12,00,000

78 thousands = 78,000

830 = 800 + 30

**Example 2:** 15,75,000

Reading: "Fifteen lakh seventy five thousand"

Breaking down:

15 lakhs = 15,00,000

75 thousands = 75,000

0 ones place (so we don't mention it)

Writing Numbers in Words: Practice Problems

**Problem:** Write 3,00,600 in words

**Answer:** Three lakh six hundred

Breaking down: 3 lakhs + 0 ten thousands + 0 thousands + 6 hundreds + 0 tens + 0 ones

**Problem:** Write 5,04,085 in words

**Answer:** Five lakh four thousand eighty five

Breaking down: 5 lakhs + 0 ten thousands + 4 thousands + 0 hundreds + 8 tens + 5 ones

**Problem:** Write 27,30,000 in words

**Answer:** Twenty seven lakh thirty thousand

Breaking down: 27 lakhs + 30 thousands

**Problem:** Write 70,53,138 in words

**Answer:** Seventy lakh fifty three thousand one hundred thirty eight

Breaking down: 70 lakhs + 53 thousands + 138

Converting Words to Numbers

**Problem:** Write "One lakh twenty three thousand four hundred and fifty six" as a numeral

1 lakh = 1,00,000

23 thousand = 23,000

456 = 456

**Answer:** 1,23,456

**Problem:** Write "Four lakh seven thousand seven hundred and four" as a numeral

4 lakhs = 4,00,000

7 thousand = 7,000

704 = 704

**Answer:** 4,07,704

**Problem:** Write "Fifty lakhs five thousand and fifty" as a numeral

50 lakhs = 50,00,000

5 thousand = 5,000

50 = 50

**Answer:** 50,05,050

**Problem:** Write "Ten lakhs two hundred and thirty five" as a numeral

10 lakhs = 10,00,000

235 = 235

**Answer:** 10,00,235

Key Points to Remember

In the Indian system, a **lakh** is the 6th digit from the right

Always use commas to separate: ones, thousands (after 3 digits), and then every 2 digits

Zero is important—if a place value is empty, write 0 in that position

Read from left to right, starting with the largest place value

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1.5 LAND OF TENS

The Concept of Place Value Through Special Calculators

The "Land of Tens" introduces the concept of place value through imaginary calculators that only have one button each. This helps us understand how numbers can be constructed using different powers of ten.

The Thoughtful Thousands Calculator (+1000 button)

This calculator only has a **+1000 button**. It starts at 0.

**Understanding:** Each button press adds 1000 to the previous number.

**Problem 1:** How many times should the +1000 button be pressed to show three thousand?

3 × 1000 = 3000

**Answer:** 3 times

**Problem 2:** How many times to show 10,000?

10,000 ÷ 1000 = 10

**Answer:** 10 times

**Problem 3:** How many times to show fifty three thousand?

53 × 1000 = 53,000

**Answer:** 53 times

**Problem 4:** How many times to show 90,000?

90,000 ÷ 1000 = 90

**Answer:** 90 times

**Problem 5:** How many times to show one lakh?

1,00,000 ÷ 1000 = 100

**Answer:** 100 times

**Problem 6:** If pressed 153 times, what number is shown?

153 × 1000 = 153,000

**Answer:** 1,53,000

**Problem 7:** How many thousands are required to make one lakh?

1,00,000 ÷ 1000 = 100

**Answer:** 100 thousands make one lakh

**Relationship:** 100 thousands = 1 lakh

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The Tedious Tens Calculator (+10 button)

This calculator only has a **+10 button**. Each press adds 10.

**Problem 1:** How many times to show five hundred?

500 ÷ 10 = 50

**Answer:** 50 times

**Problem 2:** How many times to show 780?

780 ÷ 10 = 78

**Answer:** 78 times

**Problem 3:** How many times to show 1000?

1000 ÷ 10 = 100

**Answer:** 100 times

**Problem 4:** How many times to show 3700?

3700 ÷ 10 = 370

**Answer:** 370 times

**Problem 5:** How many times to show 10,000?

10,000 ÷ 10 = 1000

**Answer:** 1000 times

**Problem 6:** How many times to show one lakh?

1,00,000 ÷ 10 = 10,000

**Answer:** 10,000 times

**Problem 7:** If pressed 435 times, what number is shown?

435 × 10 = 4,350

**Answer:** 4,350

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The Handy Hundreds Calculator (+100 button)

This calculator only has a **+100 button**. Each press adds 100.

**Problem 1:** How many times to show four hundred?

400 ÷ 100 = 4

**Answer:** 4 times

**Problem 2:** How many times to show 3,700?

3,700 ÷ 100 = 37

**Answer:** 37 times

**Problem 3:** How many times to show 10,000?

10,000 ÷ 100 = 100

**Answer:** 100 times

**Problem 4:** How many times to show fifty three thousand?

53,000 ÷ 100 = 530

**Answer:** 530 times

**Problem 5:** How many times to show 90,000?

90,000 ÷ 100 = 900

**Answer:** 900 times

**Problem 6:** How many times to show 97,600?

97,600 ÷ 100 = 976

**Answer:** 976 times

**Problem 7:** How many times to show 1,00,000?

1,00,000 ÷ 100 = 1000

**Answer:** 1000 times

**Problem 8:** If pressed 582 times, what number is shown?

582 × 100 = 58,200

**Answer:** 58,200

**Problem 9:** How many hundreds are required to make ten thousand?

10,000 ÷ 100 = 100

**Answer:** 100 hundreds make 10,000

**Problem 10:** How many hundreds are required to make one lakh?

1,00,000 ÷ 100 = 1000

**Answer:** 1000 hundreds make one lakh

Important Observation About Handy Hundreds

Handy Hundreds can show some numbers that the other two calculators cannot show easily. For example:

Tedious Tens (only +10) cannot easily show 250 without using 25 presses

The Thoughtful Thousands (only +1000) cannot show numbers less than 1000

But Handy Hundreds can show 250 with just 2.5 presses (conceptually)

**Note:** Some numbers can be shown by multiple calculators, but each has unique advantages.

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Creative Chitti: The Multi-Button Calculator

**Creative Chitti** has buttons for: +1, +10, +100, +1000, +10000, +100000, and +1000000

The calculator can reach the same number in **multiple different ways** by pressing different combinations of buttons.

#### Example 1: Getting 321

**Method 1:**

Press +10 button 32 times = 320

Press +1 button 1 time = 1

Total: 320 + 1 = 321

**Method 2:**

Press +100 button 3 times = 300

Press +10 button 2 times = 20

Press +1 button 1 time = 1

Total: 300 + 20 + 1 = 321

**Expression for Method 1:** (32 × 10) + (1 × 1) = 321

**Expression for Method 2:** (3 × 100) + (2 × 10) + (1 × 1) = 321

#### Example 2: Getting 5072

**Method 1:**

Press +100 button 50 times = 5000

Press +10 button 7 times = 70

Press +1 button 2 times = 2

Total: 5000 + 70 + 2 = 5072

**Expression:** (50 × 100) + (7 × 10) + (2 × 1) = 5072

**Method 2:**

Press +1000 button 3 times = 3000

Press +100 button 20 times = 2000

Press +1 button 72 times = 72

Total: 3000 + 2000 + 72 = 5072

**Expression:** (3 × 1000) + (20 × 100) + (72 × 1) = 5072

Key Concept: Flexibility in Place Value Representation

The same number can be represented in multiple ways using different place values. This shows that:

5072 = 5000 + 70 + 2

5072 = 3000 + 2000 + 72

5072 = 50 × 100 + 7 × 10 + 2 × 1

5072 = 3 × 1000 + 20 × 100 + 72 × 1

All these representations equal the same number but use different combinations of place values.

#### Practice Problems: Finding Different Ways

**Problem 1:** Find at least two different ways to get 8300

**Solution Method 1:**

8300 = 8000 + 300

8300 = 8 × 1000 + 3 × 100

Expression: **(8 × 1000) + (3 × 100) = 8300**

**Solution Method 2:**

8300 = 80 × 100 + 0

Expression: **(80 × 100) = 8300**

**Solution Method 3:**

8300 = 83 × 100

Expression: **(83 × 100) = 8300**

**Problem 2:** Find at least two different ways to get 40,629

**Solution Method 1:**

40,629 = 40,000 + 600 + 29

Expression: **(40 × 1000) + (6 × 100) + (2 × 10) + (9 × 1) = 40,629**

**Solution Method 2:**

40,629 = 4 × 10,000 + 62 × 10 + 9

Expression: **(4 × 10000) + (62 × 10) + (9 × 1) = 40,629**

**Problem 3:** Find at least two different ways to get 56,354

**Solution Method 1:**

56,354 = 50,000 + 6,000 + 300 + 50 + 4

Expression: **(5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1) = 56,354**

**Solution Method 2:**

56,354 = 56 × 1000 + 354

Expression: **(56 × 1000) + (3 × 100) + (5 × 10) + (4 × 1) = 56,354**

**Problem 4:** Find at least two different ways to get 66,666

**Solution Method 1:**

66,666 = 60,000 + 6,000 + 600 + 60 + 6

Expression: **(6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1) = 66,666**

**Solution Method 2:**

66,666 = 666 × 100 + 66

Expression: **(666 × 100) + (6 × 10) + (6 × 1) = 66,666**

**Problem 5:** Find at least two different ways to get 367,813

**Solution Method 1:**

367,813 = 300,000 + 60,000 + 7,000 + 800 + 13

Expression: **(3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1) = 367,813**

**Solution Method 2:**

367,813 = 36 × 10,000 + 781 × 10 + 3

Expression: **(36 × 10000) + (78 × 100) + (1 × 10) + (3 × 1) = 367,813**

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Creative Chitti's Challenges

#### Challenge 1: Maximum and Minimum 3-digit Numbers with Exactly 30 Button Presses

**Question:** Using exactly 30 button presses, what is the largest 3-digit number you can make? What is the smallest 3-digit number?

**For the Largest Number:**

To maximize the 3-digit number, we should use larger place values (like +100).

Press +100 button 9 times = 900

Press +10 button 9 times = 90

Press +1 button 12 times = 12

Total presses: 9 + 9 + 12 = 30

Total number: 900 + 90 + 12 = **912** ✗ (This is only 20 presses)

Let's recalculate:

Press +100 button 9 times = 900

Press +10 button 9 times = 90

Press +1 button 12 times = 12

Need: 9 + 9 + 12 = 30 ✓

**Answer: 999** (Press +100 nine times, +10 nine times, +1 twelve times)

**For the Smallest Number:**

To minimize but stay in 3-digit range (must be ≥ 100):

Press +100 button 1 time = 100

Press +10 button 0 times = 0

Press +1 button 29 times = 29

Total presses: 1 + 0 + 29 = 30

Total number: 100 + 29 = **129**

#### Challenge 2: Making 997 with Different Numbers of Clicks

**Given:** 997 can be made using 25 button presses

**Standard method:**

(9 × 100) + (9 × 10) + (7 × 1) = 900 + 90 + 7 = 997

Presses: 9 + 9 + 7 = 25 ✓

**Alternative methods with different presses:**

**Method 1 (More presses):**

(99 × 10) + (7 × 1) = 990 + 7 = 997

Presses: 99 + 7 = 106 presses (much more)

**Method 2 (Fewer presses):**

Not easily possible with the standard buttons, but conceptually we could express it differently.

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Systematic Sippy: The Efficient Calculator

**Systematic Sippy** has buttons: +1, +10, +100, +1000, +10000, +100000

**Goal:** Reach any number using the **minimum number of button presses**

**Key Principle:** To use the fewest button presses, we should use the largest possible place values first.

#### Example: Getting 5072 with Minimum Presses

**Method (Minimal Presses):**

Press +1000 button 5 times = 5000

Press +100 button 0 times = 0

Press +10 button 7 times = 70

Press +1 button 2 times = 2

Total presses: 5 + 0 + 7 + 2 = **14 presses**

**Expression:** (5 × 1000) + (0 × 100) + (7 × 10) + (2 × 1) = 5072

This matches the **Indian place value notation** of 5,072:

5 in the thousands place

0 in the hundreds place

7 in the tens place

2 in the ones place

**Note:** The earlier method using (3 × 1000) + (20 × 100) + (72 × 1) required 3 + 20 + 72 = 95 presses, which is much more!

#### Example: Getting 8300 with Minimum Presses

**Method (Minimal Presses):**

Press +1000 button 8 times = 8000

Press +100 button 3 times = 300

Press +10 button 0 times = 0

Press +1 button 0 times = 0

Total presses: 8 + 3 + 0 + 0 = **11 presses**

**Expression:** (8 × 1000) + (3 × 100) + (0 × 10) + (0 × 1) = 8300

Connection Between Place Value and Minimum Presses

**Key Discovery:** The minimum number of button presses for any number equals the **sum of the digits** in the number when written in place value form!

For example:

5072: digits are 5, 0, 7, 2 → minimum presses = 5 + 0 + 7 + 2 = 14

8300: digits are 8, 3, 0, 0 → minimum presses = 8 + 3 + 0 + 0 = 11

**Why?** Because the most efficient way to build a number is:

1. Use the largest place value (highest digit position)

2. Then the next largest place value

3. Continue until all digits are accounted for

This naturally follows the **Indian place value system** where each digit is multiplied by its corresponding place value.

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1.6 OF CRORES AND CRORES!

The Crore: The Next Large Number

If the +10,00,000 button is pressed 10 times:

10 × 10,00,000 = 1,00,00,000

This number is called **one crore**

It is written as: **1,00,00,000** (1 followed by 7 zeroes)

**Relationship:**

100 lakhs = 1 crore

1 crore = 1,00,00,000

Indian and American Number Systems Comparison

Both systems group digits using commas, but in different patterns:

#### Indian System

**Pattern:** 3-2-2-2 grouping from right to left

| Number | Writing | Reading |

|---|---|---|

| 1,000 | 1,000 | One thousand |

| 10,000 | 10,000 | Ten thousand |

| 1,00,000 | 1,00,000 | One lakh |

| 10,00,000 | 10,00,000 | Ten lakhs |

| 1,00,00,000 | 1,00,00,000 | One crore |

| 10,00,00,000 | 10,00,00,000 | Ten crores |

| 1,00,00,00,000 | 1,00,00,00,000 | One arab (One hundred crores) |

**Terminology:**

**Arab** = 100 crores = 1,00,00,00,000

Also called **one hundred crores**

Written with **11 zeroes** after 1

#### American/International System

**Pattern:** 3-3-3-3 grouping from right to left

| Number | Writing | Reading |

|---|---|---|

| 1,000 | 1,000 | One thousand |

| 10,000 | 10,000 | Ten thousand |

| 100,000 | 100,000 | Hundred thousand |

| 1,000,000 | 1,000,000 | One million |

| 10,000,000 | 10,000,000 | Ten million |

| 100,000,000 | 100,000,000 | Hundred million |

| 1,000,000,000 | 1,000,000,000 | One billion |

Origins of Indian Number Terms

**Lakh** comes from the Sanskrit word **लक्ष (lakṣha)**

**Crore** comes from the Sanskrit word **कोटि (koṭi)**

These terms are used across South Asian countries:

India

Bhutan

Nepal

Sri Lanka

Pakistan

Bangladesh

Maldives

Afghanistan

Myanmar

Key Observations About Zeroes

**Lakh:**

1 lakh = 1,00,000

Contains **5 zeroes**

**Crore:**

1 crore = 1,00,00,000

Contains **7 zeroes**

**Arab (Hundred Crores):**

1 arab = 1,00,00,00,000

Contains **11 zeroes**

**Pattern Recognition:**

1 thousand = 1,000 (3 zeroes)

1 lakh = 100 × 1 thousand = 1,00,000 (5 zeroes)

1 crore = 100 × 1 lakh = 1,00,00,000 (7 zeroes)

1 arab = 100 × 1 crore = 1,00,00,00,000 (11 zeroes)

**Relationship between consecutive units:**

1 crore = 100 lakhs

1 arab = 100 crores

1 lakh = 100 thousands

Practice: How Many Zeroes?

**Question 1:** How many zeroes does a thousand lakh have?

A thousand lakh = 1000 × 1,00,000

= 1,00,00,00,000

**Answer:** 9 zeroes

**Question 2:** How many zeroes does a hundred thousand have?

A hundred thousand = 100,000

**Answer:** 5 zeroes

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Reading Large Numbers in Both Systems

#### Example: 9876501234

**In Indian System (with commas: 9,87,65,01,234):**

Reading from right to left, grouping in 3-2-2-2 pattern:

234 → two hundred thirty four

01 (thousand) → one thousand

65 (lakh) → sixty five lakh

87 (crore) → eighty seven crore

9 (arab) → nine arab

**Number name:** Nine arab eighty seven crore sixty five lakh one thousand two hundred thirty four

**Alternative reading (using crores):** Nine hundred eighty seven crore sixty five lakh one thousand two hundred thirty four

**In American System (with commas: 9,876,501,234):**

Reading from right to left, grouping in 3-3-3-3 pattern:

234 → two hundred thirty four

501 (thousand) → five hundred one thousand

876 (million) → eight hundred seventy six million

9 (billion) → nine billion

**Number name:** Nine billion eight hundred seventy six million five hundred one thousand two hundred thirty four

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Practice Problems: Reading Large Numbers

**Problem 1:** Read 4050678 in Indian system

Insert commas: 40,50,678

Reading: Forty lakh fifty thousand six hundred seventy eight

**Problem 2:** Read 48121620 in Indian system

Insert commas: 48,12,1620

Wait, this is wrong. Let me recalculate: 4,81,21,620

Reading: Four crore eighty one lakh twenty one thousand six hundred twenty

**Problem 3:** Read 20022002 in Indian system

Insert commas: 2,00,22,002

Reading: Two crore twenty two thousand and two

**Problem 4:** Read 246813579 in Indian system

Insert commas: 2,46,81,3579

Wait, recalculate: 24,68,13,579

Reading: Twenty four crore sixty eight lakh thirteen thousand five hundred seventy nine

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Converting Words to Numbers in Both Systems

**Problem 1:** Write "One crore one lakh one