Number System

(i) Eight lakh five thousand twelve: 8,05,012

(ii) Thirteen lakh three thousand eight: 13,03,008

(iii) Three crore three lakh three thousand three: 3,03,03,003

(iv) Five crore twelve lakh eighteen: 5,12,00,018

(v) Nine crore nineteen lakh five thousand eight: 9,19,05,008

(vi) Six crore thirty five lakh nineteen thousand sixteen: 6,35,19,016

(vii) Eleven crore twenty two lakh thirty three thousand four hundred fifty: 11,22,33,450

(i) 8,08,080: Eight lakh eight thousand eighty.

(ii) 15,07,063: Fifteen lakh seven thousand sixty-three.

(iii) 87,08,109: Eighty-seven lakh eight thousand one hundred nine.

(iv) 2,14,05,063: Two crore fourteen lakh five thousand sixty-three.

(v) 3,03,03,103: Three crore three lakh three thousand one hundred three.

(vi) 10,06,05,368: Ten crore six lakh five thousand three hundred sixty-eight.

Counting from the right in 8,46,572 :

2 is in the ones place.

7 is in the tens place.

5 is in the hundreds place.

6 is in the thousands place.

4 is in the ten thousands place.

Hence, the place value of 4 in 8,46,572 is 40,000.

Counting from the right in 7,30,493 :

3 is in the ones place.

9 is in the tens place.

4 is in the hundreds place.

0 is in the thousands place.

3 is in the ten thousands place.

7 is in the lakhs place.

Hence, the place value of 7 in 7,30,493 is 7,00,000.

Counting from the right in 23,76,400 :

0 is in the ones place.

0 is in the tens place.

4 is in the hundreds place.

6 is in the thousands place.

7 is in the ten thousands place.

3 is in the lakhs place.

2 is in the ten lakhs place.

Hence, the place value of 6 in 23,76,400 is 6,000.

Counting from the right in 19,63,605 :

5 is in the ones place.

0 is in the tens place.

6 is in the hundreds place.

3 is in the thousands place.

6 is in the ten thousands place.

9 is in the lakhs place.

1 is in the ten lakhs place.

Hence, the place value of 9 in 19,63,605 is 9,00,000.

Counting from the right in 20,07,189 :

9 is in the ones place.

8 is in the tens place.

1 is in the hundreds place.

7 is in the thousands place.

0 is in the ten thousands place.

0 is in the lakhs place.

2 is in the ten lakhs place.

Hence, the place value of 8 in 20,07,189 is 80.

Counting from the right in 23,608 :

8 is in the ones place.

0 is in the tens place.

6 is in the hundreds place.

3 is in the thousands place.

2 is in the ten thousands place.

Hence, the place value of 3 in 23,608 is 3,000.

In 6,56,348, counting from the right:

8 is in the ones place.

4 is in the tens place.

3 is in the hundreds place.

6 is in the thousands place.

So, its place value is 6 × 1,000 = 6,000.

5 is in the ten thousands place.

6 is in the lakhs place.

So, its place value is 6 × 1,00,000 = 6,00,000.

Difference between the place-values of two sixes in 6,56,348 = 6,00,000 - 6,000 = 5,94,000

Hence, the difference between the place-values of two sixes in 6,56,348 is 5,94,000.

In the numeral 5,86,273, the digit '8' is in the ten thousands place.

So, its place value is 8 × 10,000 = 80,000.

The face value of a digit is the digit itself.

So, the face value of '8' is 8.

Difference = Place-value - Face-value

Difference = 80,000 − 8

Difference = 79,992

Hence, the difference between the place-value and the face-value of 8 in the numeral 5,86,273 = 79,992.

(i) Expanding,

5,16,287 = 5 × 1,00,000 + 1 × 10,000 + 6 × 1,000 + 2 × 100 + 8 × 10+ 7 × 1

= 5,00,000 + 10,000 + 6,000 + 200 + 80 + 7.

(ii) Expanding,

13,25,694 = 1 × 10,00,000 + 3 × 1,00,000 + 2 × 10,000 + 5 × 1,000 + 6 × 100 + 9 × 10 + 4 × 1

= 10,00,000 + 3,00,000 + 20,000 + 5,000 + 600 + 90 + 4.

(iii) Expanding,

8,08,808 = 8 × 1,00,000 + 0 × 10,000 + 8 × 1,000 + 8 × 100 + 0 × 10 + 8 × 1

= 8,00,000 + 8,000 + 800 + 8.

(iv) Expanding,

64,72,319 = 6 × 10,00,000 + 4 × 1,00,000 + 7 × 10,000 + 2 × 1,000 + 3 × 100 + 1 × 10 + 9 × 1

= 60,00,000 + 4,00,000 + 70,000 + 2,000 + 300 + 10 + 9.

(v) Expanding,

1,36,04,107 = 1 × 1,00,00,000 + 3 × 10,00,000 + 6 × 1,00,000 + 0 × 10,000 + 4 × 1,000 + 1 × 100 + 0 × 10 + 7 × 1

= 1,00,00,000 + 30,00,000 + 6,00,000 + 4,000 + 100 + 7.

(vi) Expanding,

9,36,50,519 = 9 × 1,00,00,000 + 3 × 10,00,000 + 6 × 1,00,000 + 5 × 10,000 + 0 × 1,000 + 5 × 100 + 1 × 10 + 9 × 1

= 9,00,00,000 + 30,00,000 + 6,00,000 + 50,000 + 500 + 10 + 9.

To find the number corresponding to the expanded form, we will calculate each term and then sum them up:

5 × 1,00,000 = 5,00,000

1 × 10,000 = 10,000

4 × 1,000 = 4,000

7 × 100 = 700

3 × 10 = 30

8 × 1 = 8

Now, add these values together:

5,00,000 + 10,000 + 4,000 + 700 + 30 + 8 = 5,14,738

Hence, the number corresponding to the given expanded form is 5,14,738.

To find the number corresponding to the expanded form, we will calculate each term and then sum them up:

6 × 1,00,000 = 6,00,000

6 × 1,000 = 6,000

3 × 10 = 30

6 × 1 = 6

Now, add these values together:

6,00,000 + 6,000 + 30 + 6 = 6,06,036

Hence, the number corresponding to the given expanded form is 6,06,036.

To find the number corresponding to the expanded form, we will calculate each term and then sum them up:

1 × 10,00,000 = 10,00,000

2 × 1,00,000 = 2,00,000

3 × 10,000 = 30,000

4 × 100 = 400

6 × 10 = 60

9 × 1 = 9

Now, add these values together:

10,00,000 + 2,00,000 + 30,000 + 400 + 60 + 9 = 12,30,469

Hence, the number corresponding to the given expanded form is 12,30,469.

To find the number corresponding to the expanded form, we will calculate each term and then sum them up:

2 × 10,00,000 = 20,00,000

3 × 1,00,000 = 3,00,000

7 × 1,000 = 7,000

9 × 100 = 900

4 × 10 = 40

5 × 1 = 5

Now, add these values together:

20,00,000 + 3,00,000 + 7,000 + 900 + 40 + 5 = 23,07,945.

Hence, the number corresponding to the given expanded form is 23,07,945.

To find the number corresponding to the expanded form, we will calculate each term and then sum them up:

9 × 10,00,000 = 90,00,000

8 × 1,000 = 8,000

8 × 100 = 800

8 × 1 = 8

Now, add these values together:

90,00,000 + 8,000 + 800 + 8 = 90,08,808.

Hence, the number corresponding to the given expanded form is 90,08,808.

(i) The successor of 6,001 is 6,001 + 1 = 6,002.

(ii) The successor of 1,099 is 1,099 + 1 = 1,100.

(iii) The successor of 12,749 is 12,749 + 1 = 12,750.

(iv) The successor of 2,19,708 is 2,19,708 + 1 = 2,19,709.

(v) The successor of 62,399 is 62,399 + 1 = 62,400.

(i) The predecessor of 99 is 99 - 1 = 98.

(ii) The predecessor of 1,305 is 1,305 - 1 = 1,304.

(iii) The predecessor of 32,000 is 32,000 - 1 = 31,999.

(iv) The predecessor of 1,65,000 is 1,65,000 - 1 = 1,64,999.

(i) The whole number whose successor is 100 is 100 − 1 = 99.

(ii) The whole number whose successor is 6,299 is 6,299 − 1 = 6,298.

(iii) The whole number whose successor is 71,650 is 71,650 − 1 = 71,649.

(iv) The whole number whose successor is 42,000 is 42,000 − 1 = 41,999.

(i) The whole number whose predecessor is 1,000 is 1,000 + 1 = 1,001.

(ii) The whole number whose predecessor is 3,189 is 3,189 + 1 = 3,190.

(iii) The whole number whose predecessor is 3,001 is 3,001 + 1 = 3,002.

(iv) The whole number whose predecessor is 8,000 is 8,000 + 1 = 8,001.

(v) The whole number whose predecessor is 9,999 is 9,999 + 1 = 10,000.

The three consecutive whole numbers succeeding 72,597 are:

1. 72,597 + 1 = 72,598

2. 72,598 + 1 = 72,599

3. 72,599 + 1 = 72,600

Hence, three consecutive whole numbers succeeding 72,597 are 72,598, 72,599 and 72,600.

To find the three consecutive whole numbers just preceding 5,10,001, we subtract 1 repeatedly:

1. 5,10,001 - 1 = 5,10,000

2. 5,10,000 - 1 = 5,09,999

3. 5,09,999 - 1 = 5,09,998

Hence, the three consecutive whole numbers just preceding 5,10,001 are 5,10,000, 5,09,999, and 5,09,998.

There are 6-digit numbers from 1,00,000 to 9,99,999, inclusive.

To find the total count, you can use the formula:

Total numbers = (Largest number) - (Smallest number) + 1

Largest 6-digit number = 9,99,999

Smallest 6-digit number = 1,00,000

Total 6-digit numbers = 9,99,999 − 1,00,000 + 1

Total 6-digit numbers = 8,99,999 + 1

Total 6-digit numbers = 9,00,000

Hence, there are 9,00,000 six-digit numbers in all.

The largest 8-digit number is formed by placing the largest digit (9) in all eight positions.

Hence, the largest 8-digit number is 9,99,99,999.

The smallest 8-digit number is formed by placing the second smallest digit (1) in first position and smallest digit (0) in all remaining positions.

Hence, the smallest 8-digit number is 1,00,00,000.

All possible 2-digit numbers formed by the digits 3, 7 and 9, when repetition of digits is not allowed are;

Starting with 3:

37

39

Starting with 7:

73

79

Starting with 9:

93

97

Hence, the possible 2-digit numbers are 37, 39, 73, 79, 93, 97.

Starting with 1:

137

173

Starting with 3:

317

371

Starting with 7:

713

731

Hence, the possible 3-digit numbers are 137, 173, 317, 371, 713 and 731.

When forming 3-digit numbers using the digits 9, 2, and 0, with each digit used only once, we must remember that a 3-digit number cannot start with 0.

If the hundreds digit is 9 : the remaining digits are 2 and 0.

Possible numbers: 920, 902

If the hundreds digit is 2 : the remaining digits are 9 and 0.

Possible numbers: 290, 209

Hence, the possible 3-digit numbers are 920, 902, 290, 209.

The first digit of a 4-digit number cannot be 0. So, the smallest available non-zero digit is 1.

Thousands place: 1

The remaining digits are 0, 3, and 6. To make the number as small as possible, arrange these remaining digits in ascending order: 0, 3, 6.

Hundreds place: 0

Tens place: 3

Units place: 6

Combining all the data, the smallest 4-digit number is 1036.

Hence, the smallest 4-digit number is 1036.

The given digits are: 0, 2, 7, 5.

Arranging them in descending order:

Largest digit: 7 (for the thousands place)

Next largest: 5 (for the hundreds place)

Next largest: 2 (for the tens place)

Smallest digit: 0 (for the units place)

Hence, the greatest 4-digit number that can be formed is 7520.

The digits available are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Thousands digit: The smallest digit is 0, but a number cannot start with 0 if it's meant to be a 4-digit number. So, the smallest non-zero digit, which is 1, must be in the thousands place.

Current digits used: {1}

Hundreds digit: Now that 1 is used, the smallest remaining digit available is 0.

Current digits used: {1, 0}

Tens digit: With 1 and 0 used, the next smallest digit available is 2.

Current digits used: {1, 0, 2}

Units digit: With 1, 0, and 2 used, the next smallest digit available is 3.

Current digits used: {1, 0, 2, 3}

Combining these digits in order, the smallest 4-digit number with four different digits is 1023.

Hence, the smallest 4-digit number with four different digits is 1,023.

The digits available are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Thousands digit: The largest digit is 9.

Current digits used: {9}

Hundreds digit: From the remaining digits, the next largest is 8.

Current digits used: {9, 8}

Tens digit: From the remaining digits, the next largest is 7.

Current digits used: {9, 8, 7}

Units digit: From the remaining digits, the next largest is 6.

Current digits used: {9, 8, 7, 6}

Combining these digits in order, the greatest 4-digit number with four different digits is 9876.

Hence, the greatest 4-digit number with four different digits is 9,876.

On comparing 98,760 and 1,02,345,

The number 98,760 has 5 digits.

The number 1,02,345 has 6 digits.

When comparing two positive whole numbers, the number with more digits is always the larger number.

Hence, 98,760 < 1,02,345.

On comparing 5,43,210 and 98,306,

The number 5,43,210 has 6 digits.

The number 98,306 has 5 digits.

When comparing two positive whole numbers, the number with more digits is always the larger number.

Hence, 5,43,210 > 98,306.

On comparing 2,13,576 and 67,531,

The number 2,13,576 has 6 digits.

The number 67,531 has 5 digits.

When comparing two positive whole numbers, the number with more digits is always the larger number.

Hence, 2,13,576 > 67,531.

On comparing 3,02,368 and 50,545,

The number 3,02,368 has 6 digits.

The number 50,545 has 5 digits.

When comparing two positive whole numbers, the number with more digits is always the larger number.

Hence, 3,02,368 > 50,545.

On comparing 76,134 and 76,079,

Both numbers have 5 digits.

When the number of digits is the same, we start comparing from the leftmost digit (the highest place value).

Compare the digits in each place value, moving from left to right:

Tens Thousands Place (Leftmost digit):

In 76,134, the digit is 7.

In 76,079, the digit is 7.

Result: These digits are the same. We need to move to the next place value.

Thousands Place:

In 76,134, the digit is 6.

In 76,079, the digit is 6.

Result: These digits are also the same. Move to the next place value.

Hundreds Place:

In 76,134, the digit is 1.

In 76,079, the digit is 0.

Result: Here, 1 is greater than 0.

Hence, 76,134 > 76,079.

On comparing 6,40,729 and 6,41,304,

Both numbers have 6 digits.

When the number of digits is the same, we start comparing from the leftmost digit (the highest place value).

Compare the digits in each place value, moving from left to right:

Lakhs Place (Leftmost digit):

In 6,40,729, the digit is 6.

In 6,41,304, the digit is 6.

Result: These digits are the same. We need to move to the next place value.

Tens Thousands Place :

In 6,40,729, the digit is 4.

In 6,41,304, the digit is 4.

Result: These digits are the same. We need to move to the next place value.

Thousands Place:

In 6,40,729, the digit is 0.

In 6,41,304, the digit is 1.

Result: Here, 1 is greater than 0.

Hence, 6,40,729 < 6,41,304.

On comparing 35,79,018 and 35,80,024,

Both numbers have 7 digits.

When the number of digits is the same, we start comparing from the leftmost digit (the highest place value).

Compare the digits in each place value, moving from left to right:

Ten Lakhs Place (Leftmost digit):

In 35,79,018, the digit is 3.

In 35,80,024, the digit is 3.

Result: These digits are the same. We need to move to the next place value.

Lakhs Place :

In 35,79,018, the digit is 5.

In 35,80,024, the digit is 5.

Result: These digits are the same. We need to move to the next place value.

Tens Thousands Place :

In 35,79,018, the digit is 7.

In 35,80,024, the digit is 8.

Result: Here, 8 is greater than 7.

Hence, 35,79,018 < 35,80,024.

On comparing 83,21,067 and 83,12,076,

Both numbers have 7 digits.

When the number of digits is the same, we start comparing from the leftmost digit (the highest place value).

Compare the digits in each place value, moving from left to right:

Ten Lakhs Place (Leftmost digit):

In 83,21,067, the digit is 8.

In 83,12,076, the digit is 8.

Result: These digits are the same. We need to move to the next place value.

Lakhs Place :

In 83,21,067, the digit is 3.

In 83,12,076, the digit is 3.

Result: These digits are the same. We need to move to the next place value.

Tens Thousands Place :

In 83,21,067, the digit is 2.

In 83,12,076, the digit is 1.

Result: Here, 2 is greater than 1.

Hence, 83,21,067 > 83,12,076.

On comparing 54,63,217 and 54,62,378,

Both numbers have 7 digits.

When the number of digits is the same, we start comparing from the leftmost digit (the highest place value).

Compare the digits in each place value, moving from left to right:

Ten Lakhs Place (Leftmost digit):

In 54,63,217, the digit is 5.

In 54,62,378, the digit is 5.

Result: These digits are the same. We need to move to the next place value.

Lakhs Place :

In 54,63,217, the digit is 4.

In 54,62,378, the digit is 4.

Result: These digits are the same. We need to move to the next place value.

Tens Thousands Place :

In 54,63,217, the digit is 6.

In 54,62,378, the digit is 6.

Result: These digits are the same. We need to move to the next place value.

Thousands Place :

In 54,63,217, the digit is 3.

In 54,62,378, the digit is 2.

Result: Here, 3 is greater than 2.

Hence, 54,63,217 > 54,62,378.

On comparing 35,780,624 and 35,876,002,

Both numbers have 8 digits.

When the number of digits is the same, we start comparing from the leftmost digit (the highest place value).

Compare the digits in each place value, moving from left to right:

Ten Millions Place (Leftmost digit):

In 35,780,624, the digit is 3.

In 35,876,002, the digit is 3.

Result: These digits are the same. We need to move to the next place value.

Millions Place :

In 35,780,624, the digit is 5.

In 35,876,002, the digit is 5.

Result: These digits are the same. We need to move to the next place value.

Hundreds Thousands Place :

In 35,780,624, the digit is 7.

In 35,876,002, the digit is 8.

Result: Here, 8 is greater than 7.

Hence, 35,780,624 < 35,876,002.

Given, 57,860,    60,375,    57,906,    61,435,    60,296, 61,380

To arrange the numbers in ascending order, we compare them digit by digit, starting from the leftmost digit.

Hence, 57,860 < 57,906 < 60,296 < 60,375 < 61,380 < 61,435.

Given, 7,15,840,    98,756,    8,94,105,    17,18,195,    98,678,    8,95,103

To arrange the numbers in ascending order, we compare them digit by digit, starting from the leftmost digit.

Hence, 98,678 < 98,756 < 7,15,840 < 8,94,105 < 8,95,103 < 17,18,195.

Given, 5,01,462,    5,01,076,    5,00,984,    5,02,000,    5,10,010,    4,56,780

To arrange the numbers in ascending order, we compare them digit by digit, starting from the leftmost digit.

Hence, 4,56,780 < 5,00,984 < 5,01,076 < 5,01,462 < 5,02,000 < 5,10,010.

Given, 16,13,241,    7,54,328,    7,52,987,    16,04,895,    7,53,024,    16,20,342

To arrange the numbers in descending order, we compare them digit by digit, starting from the leftmost digit.

Hence, 16,20,342 > 16,13,241 > 16,04,895 > 7,54,328 > 7,53,024 > 7,52,987.

Given, 93,68,516,    1,05,40,603,    9,10,32,401,    93,67,839,    1,05,41,201

To arrange the numbers in descending order, we compare them digit by digit, starting from the leftmost digit.

Hence, 9,10,32,401 > 1,05,41,201 > 1,05,40,603 > 93,68,516 > 93,67,839.

Given, 1,48,65,710,    2,05,07,106,    3,00,08,215,    27,86,789,    28,76,879

To arrange the numbers in descending order, we compare them digit by digit, starting from the leftmost digit.

Hence, 3,00,08,215 > 2,05,07,106 > 1,48,65,710 > 28,76,879 > 27,86,789.

Given, Population of Arunachal Pradesh = 13,82,611

Population of Meghalaya = 29,64,007

Population of Tripura = 36,71,032

Total population = 13,82,611 + 29,64,007 + 36,71,032

= 80,17,650.

Hence, the total population of these three states = 80,17,650.

Given, Population of Andhra Pradesh = 6,30,20,208

Population of Telangana = 3,52,86,757

Difference between population = 6,30,20,208 - 3,52,86,757

= 2,77,33,451.

Hence, the population of Andhra Pradesh exceeds that of Telangana by 2,77,33,451.

Given, Initial stock of wheat in the godown: 85,90,865 quintals

Wheat sent to Punjab: 17,89,564 quintals

Wheat sent to Delhi: 23,17,986 quintals

Total sent = Wheat sent to Punjab + Wheat sent to Delhi

= 17,89,564 + 23,17,986

= 41,07,550 quintals

Balance stock = Initial stock - Total sent

= 85,90,865 - 41,07,550

= 44,83,315 quintals

Hence, the balance stock of wheat is 44,83,315 quintals.

Given, the initial number = 47,56,908

The amount by which it is exceeded = 16,34,096

The number = 47,56,908 + 16,34,096

= 63,91,004

Hence, the number is 63,91,004.

Given, the larger number = 20,00,000

The smaller number = 15,69,748

Difference = 20,00,000 - 15,69,748

= 4,30,252

Hence, 15,69,748 is smaller than 20,00,000 by 4,30,252.

Given, the larger number = 1,43,56,803

The smaller number = 86,78,215

Difference = 1,43,56,803 - 86,78,215

= 56,78,588

Hence, 1,43,56,803 is larger than 86,78,215 by 56,78,588.

Let required number be x.

Thus,

23,56,714 - x = 8,69,873

x = 23,56,714 - 8,69,873

x = 14,86,841

Hence, the number that must be subtracted from 23,56,714 to get 8,69,873 is 14,86,841.

Given, the larger number = 63,00,000

The smaller number = 46,97,859

Difference = 63,00,000 - 46,97,859

= 16,02,141

Hence, 16,02,141 must be added to 46,97,859 to get 63,00,000.

Given, sum of two numbers = 60,10,203

One of the numbers = 48,21,325

Other number = 60,10,203 - 48,21,325

= 11,88,878

Hence, the other number is 11,88,878.

Given, total amount = ₹1,35,00,000

Amount given to wife = ₹56,32,560

Amount given to son = ₹37,84,890

Amount given to daughter = Total amount - (Amount given to wife and son combined)

= ₹1,35,00,000 - (₹56,32,560 + ₹37,84,890)

= ₹1,35,00,000 - ₹94,17,450

= ₹40,82,550.

Hence, the daughter gets ₹40,82,550.

Given, Swati's monthly savings = ₹6,750

Time period = 12 years = 12 × 12 months

= 144 months

Total savings = Monthly savings × Total months

= ₹6,750 × 144

= ₹9,72,000

Hence, Swati will save ₹9,72,000 in 12 years.

Given, cost of a chair = ₹1,586

Number of chairs = 245

Total cost = Cost of a chair × Number of chairs

= ₹1,586 × 245

= ₹3,88,570

Hence, cost of 245 chairs = ₹3,88,570.

Given, number of screws produced in a day = 9,675

Number of days = 1 year = 365 days

Total number of screws = Number of screws produced in a day × Number of days

= 9,675 × 365

= 35,31,375

Hence, total screws produced in a year = 35,31,375.

Given,

Speed of aeroplane = 1,685 km per hour

Total hours = 58 hours

By formula, distance = speed × time

= 1,685 × 58

= 97,730 km

Hence, distance covered by the aeroplane = 97,730 km.

Given, cost of a scooter = ₹46,850

Number of scooters = 234

Total cost = Cost of a scooter × Number of scooters

= ₹46,850 × 234

= ₹1,09,62,900

Hence, the total cost of 234 scooters = ₹1,09,62,900.

Given, contribution of each student = ₹740

Number of students = 1,625

Total contribution = Number of students × Contribution of each student

= 1,625 × ₹740

= ₹12,02,500

Hence, total contribution = ₹12,02,500.

Given, total cost of refrigerators = ₹48,46,400

Number of refrigerators = 65

Cost of each refrigerator = $\frac{\text{total cost of refrigerators}}{\text{number of refrigerators}}$

= $\frac{₹48,46,400}{65}$

= ₹74,560

Hence, the cost of each refrigerator = ₹74,560.

Given, total cost of flats = ₹1,24,94,300

Number of flats = 35

Cost of each flat = $\frac{\text{total cost of flats}}{\text{number of flats}}$

= $\frac{₹1,24,94,300}{35}$

= ₹3,56,980

Hence, the cost of each flat = ₹3,56,980.

Given, product of two numbers = 17,20,740.

One of the numbers = 1,785

The other number = $\frac{\text{product of two numbers}}{\text{one of the numbers}}$

= $\frac{17,20,740}{1,785}$

= 964

Hence, the other number = 964.

Given, total distance = 4,320 km

Total time = 45 hours

By formula, Speed = $\frac{\text{Distance}}{\text{Time}}$

= $\frac{4,320}{45}$

= 96 km/hr.

Hence, the speed of the car = 96 km/hr.

Given, total amount of milk = 13 kg 650 g = 13 × 1,000 + 650 g = 13,650 g

Total number of students = 21

Milk each student gets = $\frac{\text{Total amount of milk}}{\text{Total number of students}}$

= $\frac{13,650 g}{21}$

= 650 g

Hence, each student gets 650 g of milk.

Given, total distance = 36 km = 36 × 1,000 = 36,000 m

Total time = 1 hour = 1 × 60 × 60 sec = 3600 sec

By formula,

Speed = $\frac{\text{Total distance}}{\text{Total time}}$

= $\frac{36,000}{3600}$

= 10 m/s.

Hence, speed = 10 m/s.

(i) 54

The units digit is 4. Since 4 is less than 5, we round down.

Hence, number rounded to the nearest ten = 50.

(ii) 327

The units digit is 7. Since 7 is 5 or greater, we round up the tens digit.

The tens digit (2) becomes 3, and the units digit becomes 0.

Hence, number rounded to the nearest ten = 330.

(iii) 2,793

The units digit is 3. Since 3 is less than 5, we round down.

Hence, number rounded to the nearest ten = 2,790.

(iv) 8,049

The units digit is 9. Since 9 is 5 or greater, we round up the tens digit.

The tens digit (4) becomes 5, and the units digit becomes 0.

Hence, number rounded to the nearest ten = 8,050.

(v) 12,345

The units digit is 5. Since 5 is 5 or greater, we round up the tens digit.

The tens digit (4) becomes 5, and the units digit becomes 0.

Hence, number rounded to the nearest ten = 12,350.

(i) 925

The tens digit is 2. Since 2 is less than 5, we round down.

The hundreds digit (9) remains the same, and the tens and units digits become 00.

Hence, number rounded to the nearest hundred = 900.

(ii) 6,854

The tens digit is 5. Since 5 is 5 or greater, we round up the hundreds digit.

The hundreds digit (8) becomes 9, and the tens and units digits become 00.

Hence, number rounded to the nearest hundred = 6,900.

(iii) 41,263

The tens digit is 6. Since 6 is 5 or greater, we round up the hundreds digit.

The hundreds digit (2) becomes 3, and the tens and units digits become 00.

Hence, number rounded to the nearest hundred = 41,300.

(iv) 27,861

The tens digit is 6. Since 6 is 5 or greater, we round up the hundreds digit.

The hundreds digit (8) becomes 9, and the tens and units digits become 00.

Hence, number rounded to the nearest hundred = 27,900.

(v) 5,549

The tens digit is 4. Since 4 is less than 5, we round down.

The hundreds digit (5) remains the same, and the tens and units digits become 00.

Hence, number rounded to the nearest hundred = 5,500.

(i) 7,386

The hundreds digit is 3. Since 3 is less than 5, we round down.

The thousands digit (7) remains the same, and the hundreds, tens, and units digits become 000.

Hence, number rounded to the nearest thousand = 7,000.

(ii) 34,276

The hundreds digit is 2. Since 2 is less than 5, we round down.

The thousands digit (4) remains the same, and the hundreds, tens, and units digits become 000.

Hence, number rounded to the nearest thousand = 34,000.

(iii) 23,804

The hundreds digit is 8. Since 8 is 5 or greater, we round up the thousands digit.

The thousands digit (3) becomes 4, and the hundreds, tens, and units digits become 000.

Hence, number rounded to the nearest thousand = 24,000.

(iv) 76,540

The hundreds digit is 5. Since 5 is 5 or greater, we round up the thousands digit.

The thousands digit (6) becomes 7, and the hundreds, tens, and units digits become 000.

Hence, number rounded to the nearest thousand = 77,000.

(i) (72 + 37)

Round 72 to the nearest ten: The units digit is 2, so it rounds down to 70.

Round 37 to the nearest ten: The units digit is 7, so it rounds up to 40.

Approximate sum: 70 + 40 = 110

Hence, the approximate sum = 110.

(ii) (264 + 348)

Round 264 to the nearest ten: The units digit is 4, so it rounds down to 260.

Round 348 to the nearest ten: The units digit is 8, so it rounds up to 350.

Approximate sum: 260 + 350 = 610

Hence, the approximate sum = 610.

(iii) (2,538 + 6,274)

Round 2,538 to the nearest ten: The units digit is 8, so it rounds up to 2,540.

Round 6,274 to the nearest ten: The units digit is 4, so it rounds down to 6,270.

Approximate sum: 2,540 + 6,270 = 8,810

Hence, the approximate sum = 8,810.

(iv) (4,782 + 2,345)

Round 4,782 to the nearest ten: The units digit is 2, so it rounds down to 4,780.

Round 2,345 to the nearest ten: The units digit is 5, so it rounds up to 2,350.

Approximate sum: 4,780 + 2,350 = 7,130

Hence, the approximate sum = 7,130.

(i) (347 + 476)

Round 347 to the nearest hundred: The tens digit is 4, so it rounds down to 300.

Round 476 to the nearest hundred: The tens digit is 7, so it rounds up to 500.

Approximate sum: 300 + 500 = 800

Hence, the approximate sum = 800.

(ii) (654 + 247)

Round 654 to the nearest hundred: The tens digit is 5, so it rounds up to 700.

Round 247 to the nearest hundred: The tens digit is 4, so it rounds down to 200.

Approximate sum: 700 + 200 = 900

Hence, the approximate sum = 900.

(iii) (5,240 + 3,421)

Round 5,240 to the nearest hundred: The tens digit is 4, so it rounds down to 5,200.

Round 3,421 to the nearest hundred: The tens digit is 2, so it rounds down to 3,400.

Approximate sum: 5,200 + 3,400 = 8,600

Hence, the approximate sum = 8,600.

(iv) (9,483 + 6,572)

Round 9,483 to the nearest hundred: The tens digit is 8, so it rounds up to 9,500.

Round 6,572 to the nearest hundred: The tens digit is 7, so it rounds up to 6,600.

Approximate sum: 9,500 + 6,600 = 16,100

Hence, the approximate sum = 16,100.

(i) (43,728 + 36,275)

Round 43,728 to the nearest thousand: The hundreds digit is 7, so it rounds up to 44,000.

Round 36,275 to the nearest thousand: The hundreds digit is 2, so it rounds down to 36,000.

Approximate sum: 44,000 + 36,000 = 80,000

Hence, the approximate sum = 80,000.

(ii) (37,804 + 22,475)

Round 37,804 to the nearest thousand: The hundreds digit is 8, so it rounds up to 38,000.

Round 22,475 to the nearest thousand: The hundreds digit is 4, so it rounds down to 22,000.

Approximate sum: 38,000 + 22,000 = 60,000

Hence, the approximate sum = 60,000.

(iii) (6,785 + 2,476)

Round 6,785 to the nearest thousand: The hundreds digit is 7, so it rounds up to 7,000.

Round 2,476 to the nearest thousand: The hundreds digit is 4, so it rounds down to 2,000.

Approximate sum: 7,000 + 2,000 = 9,000

Hence, the approximate sum = 9,000.

(i) Round 64 to the nearest ten: The units digit is 4, so it rounds down to 60.

Round 29 to the nearest ten: The units digit is 9, so it rounds up to 30.

Approximate difference: 60 − 30 = 30

Hence, the approximate difference = 30.

(ii) (186 − 49)

Round 186 to the nearest ten: The units digit is 6, so it rounds up to 190.

Round 49 to the nearest ten: The units digit is 9, so it rounds up to 50.

Approximate difference: 190 − 50 = 140

Hence, the approximate difference = 140.

(iii) (429 − 206)

Round 429 to the nearest ten: The units digit is 9, so it rounds up to 430.

Round 206 to the nearest ten: The units digit is 6, so it rounds up to 210.

Approximate difference: 430 − 210 = 220

Hence, the approximate difference = 220.

(i) (769 - 435)

Round 769 to the nearest hundred: The tens digit is 6, so it rounds up to 800.

Round 435 to the nearest hundred: The tens digit is 3, so it rounds down to 400.

Approximate difference: 800 − 400 = 400

Hence, the approximate difference = 400.

(ii) (859 - 675)

Round 859 to the nearest hundred: The tens digit is 5, so it rounds up to 900.

Round 675 to the nearest hundred: The tens digit is 7, so it rounds up to 700.

Approximate difference: 900 − 700 = 200

Hence, the approximate difference = 200.

(iii) (8,359 - 4,317)

Round 8,359 to the nearest hundred: The tens digit is 5, so it rounds up to 8,400.

Round 4,317 to the nearest hundred: The tens digit is 1, so it rounds down to 4,300.

Approximate difference: 8,400 − 4,300 = 4,100

Hence, the approximate difference = 4,100.

(i) (45,783 - 38,695)

Round 45,783 to the nearest thousand: The hundreds digit is 7, so it rounds up to 46,000.

Round 38,695 to the nearest thousand: The hundreds digit is 6, so it rounds up to 39,000.

Approximate difference: 46,000 − 39,000 = 7,000

Hence, the approximate difference = 7,000.

(ii) (38,005 - 29,375)

Round 38,005 to the nearest thousand: The hundreds digit is 0, so it rounds down to 38,000.

Round 29,375 to the nearest thousand: The hundreds digit is 3, so it rounds down to 29,000.

Approximate difference: 38,000 − 29,000 = 9,000

Hence, the approximate difference = 9,000.

(iii) (7,654 - 4,368)

Round 7,654 to the nearest thousand: The hundreds digit is 6, so it rounds up to 8,000.

Round 4,368 to the nearest thousand: The hundreds digit is 3, so it rounds down to 4,000.

Approximate difference: 8,000 − 4,000 = 4,000

Hence, the approximate difference = 4,000.

(i) (49 × 72)

Round 49 to the nearest ten: The units digit is 9, so it rounds up to 50.

Round 72 to the nearest ten: The units digit is 2, so it rounds down to 70.

Approximate product: 50 × 70 = 3,500

Hence, the approximate product = 3,500.

(ii) (39 × 62)

Round 39 to the nearest ten: The units digit is 9, so it rounds up to 40.

Round 62 to the nearest ten: The units digit is 2, so it rounds down to 60.

Approximate product: 40 × 60 = 2,400

Hence, the approximate product = 2,400.

(iii) (63 × 57)

Round 63 to the nearest ten: The units digit is 3, so it rounds down to 60.

Round 57 to the nearest ten: The units digit is 7, so it rounds up to 60.

Approximate product: 60 × 60 = 3,600

Hence, the approximate product = 3,600.

(iv) (35 × 43)

Round 35 to the nearest ten: The units digit is 5, so it rounds up to 40.

Round 43 to the nearest ten: The units digit is 3, so it rounds down to 40.

Approximate product: 40 × 40 = 1,600

Hence, the approximate product = 1,600.

(i) 265 × 334

Round 265 to the nearest hundred: The tens digit is 6, so it rounds up to 300.

Round 334 to the nearest hundred: The tens digit is 3, so it rounds down to 300.

Approximate product: 300 × 300 = 90,000

Hence, the approximate product = 90,000.

(ii) 457 × 872

Round 457 to the nearest hundred: The tens digit is 5, so it rounds up to 500.

Round 872 to the nearest hundred: The tens digit is 7, so it rounds up to 900.

Approximate product: 500 × 900 = 4,50,000

Hence, the approximate product = 4,50,000.

(iii) 381 × 316

Round 381 to the nearest hundred: The tens digit is 8, so it rounds up to 400.

Round 316 to the nearest hundred: The tens digit is 1, so it rounds down to 300.

Approximate product: 400 × 300 = 1,20,000

Hence, the approximate product = 1,20,000.

(iv) 455 × 138

Round 455 to the nearest hundred: The tens digit is 5, so it rounds up to 500.

Round 138 to the nearest hundred: The tens digit is 3, so it rounds down to 100.

Approximate product: 500 × 100 = 50,000

Hence, the approximate product = 50,000.

(i) 86 ÷ 27

Round 86 to the nearest ten: 90

Round 27 to the nearest ten: 30

Approximate quotient: 90 ÷ 30 = 3

Hence, the approximate quotient = 3.

(ii) 83 ÷ 19

Round 83 to the nearest ten: 80

Round 19 to the nearest ten: 20

Approximate quotient: 80 ÷ 20 = 4

Hence, the approximate quotient = 4.

(iii) 286 ÷ 25

Round 286 to the nearest hundred: 300

Round 25 to the nearest ten: 30

Approximate quotient: 300 ÷ 30 = 10

Hence, the approximate quotient = 10.

(iv) 865 ÷ 38

Round 865 to the nearest hundred: 900

Round 38 to the nearest ten: 40

Approximate quotient: 900 ÷ 40 = 90 ÷ 4 = 22.5.

Since we are looking for an approximate quotient, we can round to the nearest whole number.

Approximate quotient: 23

Hence, the approximate quotient = 23.

(i) 1,64,839

Commas according to International number system: 164,839

164,839: One hundred sixty-four thousand eight hundred thirty-nine.

(ii) 35,84,267

Commas according to International number system: 3,584,267

3,584,267: Three million five hundred eighty-four thousand two hundred sixty-seven.

(iii) 57,93,64,021

Commas according to International number system: 579,364,021

579,364,021: Five hundred seventy-nine million three hundred sixty-four thousand twenty-one.

(iv) 90,09,09,098

Commas according to International number system: 900,909,098

900,909,098: Nine hundred million nine hundred nine thousand ninety-eight.

(i) 56,307,840

Fifty-six million three hundred seven thousand eight hundred forty.

(ii) 519,250,086

Five hundred nineteen million two hundred fifty thousand eighty-six.

(iii) 5,003,030

Five million three thousand thirty.

(iv) 67,010,206

Sixty-seven million ten thousand two hundred six.

(v) 101,011,110

One hundred one million eleven thousand one hundred ten.

(vi) 600,606,006

Six hundred million six hundred six thousand six.

(i) Forty million four hundred four thousand sixty four = 40,404,064.

(ii) Sixty three million six hundred five thousand two = 63,605,002.

(iii) Eight million eight thousand eight = 8,008,008.

(iv) Nine hundred thirty four million eight hundred seventy three thousand five hundred six = 934,873,506.

(v) Five hundred four million eight hundred seven thousand three hundred nineteen = 504,807,319.

(vi) Seven hundred thirteen million nine hundred ten thousand eighteen = 713,910,018.

(vii) Eighty million seventy thousand ten = 80,070,010.

(viii) One hundred million one hundred thousand one hundred = 100,100,100.

The numeral is 695803704.

(i) International System

With Commas: 695,803,704

695,803,704: Six hundred ninety-five million eight hundred three thousand seven hundred four.

(ii) Hindu-Arabic (Indian) System

With Commas: 69,58,03,704

69,58,03,704: Sixty-nine crore fifty-eight lakh three thousand seven hundred four.

The numeral is 25,79,206.

Let's look at the place values from right to left (Indian system):

6 is in the Ones place

0 is in the Tens place

2 is in the Hundreds place

9 is in the Thousands place

7 is in the Ten Thousands place

5 is in the Lakhs place

2 is in the Ten Lakhs place

The digit 7 is in the Ten Thousands place.

Therefore, its place value is 7 × 10,000 = 70,000.

Hence, option 3 is the correct option.

The numeral is 36,43,908.

The face value of a digit is the digit itself, regardless of its position in the number.

In the numeral 36,43,908, the digit 4 is present.

The face value of 4 is simply 4.

Hence, option 2 is the correct option.

The numeral is 32,53,619.

Place Value of 6:

In the numeral 32,53,619, the digit 6 is in the Hundreds place.

So, its place value is 6 × 100 = 600.

Face Value of 6:

The face value of a digit is the digit itself.

So, the face value of 6 is 6.

Difference between Place Value and Face Value:

Difference = Place Value - Face Value

Difference = 600 − 6 = 594

Hence, option 2 is the correct option.

Counting numbers (also known as natural numbers) are the numbers we use for counting objects. They start from 1.

0 is a whole number, but not a counting number in the mathematical sense.

Hence, option 2 is the correct option.

The successor of a number is obtained by adding 1 to that number.

Let the whole number be 'x'.

Its successor is given as 53,100.

So, we have the equation:

⇒ x + 1 = 53,100

To find x, subtract 1 from 53,100:

⇒ x = 53,100 - 1

⇒ x = 53,099

The whole number whose successor is 53,100 is 53,099.

Hence, option 2 is the correct option.

To make the largest number, arrange the digits in descending order: 8, 3, 0.

The largest number = 830.

To make the smallest 3-digit number, the hundreds digit cannot be 0. So, place the next smallest digit (3) in the hundreds place, followed by the remaining digits in ascending order (0, 8).

The smallest number = 308.

Difference = Largest number - Smallest number

= 830 - 308

= 522.

Hence, option 3 is the correct option.

The largest 7-digit number = 9,999,999.

The smallest 7-digit number = 1,000,000.

Total Count = (Largest Number - Smallest Number) + 1

Total 7-digit numbers = 9,999,999 - 1,000,000 + 1

= 8,999,999 + 1

= 9,000,000

Hence, option 1 is the correct option.

The number that comes just before 10,00,000

= 10,00,000 - 1

= 9,99,999.

Hence, option 3 is the correct option.

The largest number of 4 digits is 9,999.

To find the largest 4-digit number exactly divisible by 25, we divide 9,999 by 25:

9999 ÷ 25 = 25 × 399 + 24

The remainder is 24.

To get a number exactly divisible by 25, we subtract this remainder from 9,999:

9999 − 24 = 9975

So, the largest 4-digit number exactly divisible by 25 is 9,975.

Hence, option 4 is the correct option.

By formula,

Dividend = Divisor × Quotient + Remainder

= 23 × 17 + 19

= 391 + 19

= 410.

Hence, option 4 is the correct option.

Let the number be N.

The successor of the number is N + 1.

The predecessor of the number is N − 1.

According to the problem, the sum of the successor and the predecessor is 1326:

⇒ (N + 1) + (N − 1) = 1326

⇒ N + 1 + N − 1 = 1326

⇒ 2N = 1326

⇒ N = $\frac{1326}{2}$

⇒ N = 663

Hence, option 1 is the correct option.

(i) 4,80,798 ............... 4,81,612

Both numbers have 6 digits.

Comparing from the left:

The lakhs digit is 4 in both.

The ten thousands digit is 8 in both.

The thousands digit is 0 in 4,80,798 and 1 in 4,81,612.

Since 0 is less than 1, 4,80,798 is smaller than 4,81,612.

Hence, 4,80,798 < 4,81,612.

(ii) 5,43,611 ............... 5,43,584

Both numbers have 6 digits.

Comparing from the left:

The lakhs digit is 5 in both.

The ten thousands digit is 4 in both.

The thousands digit is 3 in both.

The hundreds digit is 6 in 5,43,611 and 5 in 5,43,584.

Since 6 is greater than 5, 5,43,611 is larger than 5,43,584.

Hence, 5,43,611 > 5,43,584.

(iii) 2,37,928 ............... 68,946

2,37,928 has 6 digits.

68,946 has 5 digits.

A number with more digits (when positive) is always greater than a number with fewer digits.

Hence, 2,37,928 > 68,946.

(i) The tens digit is 3, which is less than 5, so we round down.

Hence, 839 rounded to the nearest hundred is 800.

(ii) The hundreds digit is 8, which is 5 or greater, so we round up the thousands digit.

Hence, 83,867 rounded to the nearest thousand is 84,000.

(iii) The sum: 368 + 143 = 511.

Then, round 511 to the nearest ten. The units digit is 1, which is less than 5, so we round down.

Hence, 368 + 143 to the nearest ten is 510.

False.

Reason

Counting numbers (also known as natural numbers) are the numbers we use for counting objects. They start from 1.

0 is a whole number, but not a counting number in the mathematical sense.

False

Reason

In the Indian place value chart, starting from the right (ones place):

1. Ones

2. Tens

3. Hundreds

4. Thousands

5. Ten Thousands

6. Lakhs

7. Ten Lakhs

8. Crores

So, the 8th place is called Crores, and the 7th place is called Ten Lakhs.

True

Reason

In the Indian Numbering System: 10 Lakhs = 10,00,000

In the International Numbering System: 1 Million = 1,000,000

Both represent the same value.

True

Reason

The smallest number that has 7 digits = 1,000,000 (one million or ten lakh).

The predecessor of a number is the number that comes immediately before it.

Predecessor of 1,000,000 = 1,000,000 - 1 = 999,999.

The largest number that has 6 digits = 999,999.

The predecessor of the smallest 7-digit number = 999,999.

The largest 6-digit number is 999,999.

True

Reason

International System: 10 million = 10,000,000

Indian System: 1 crore = 1,00,00,000

Since 10,000,000 (10 million) is equal to 1,00,00,000 (1 crore).

True

Reason

Here's the breakdown of 13,564,030 in the International System:

13, (Millions period) - Thirteen million

564, (Thousands period) - Five hundred sixty-four thousand

030 (Ones period) - Thirty

So, 13,564,030 in International system is written as thirteen million five hundred sixty four thousand thirty.

The largest number of seven digits = 9,999,999.

The smallest number of eight digits = 10,000,000.

The difference = 10,000,000 − 9,999,999 = 1.

∴ Assertion (A) is true.

The largest number of seven digits = 9,999,999.

The successor of 9,999,999 = 9,999,999 + 1 = 10,000,000.

The smallest number of eight digits = 10,000,000.

∴ Reason (R) is true.

∴ Reason (R) is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.

Ten million in the International System is written as 10,000,000.

One crore in the Indian System is written as 1,00,00,000.

Both numbers represent the same value.

∴ Assertion (A) is true.

One crore (1,00,00,000) has 7 zeroes.

∴ Reason (R) is false.

Hence, option 3 is the correct option.