Knowing Our Numbers

Write the smallest natural number. Can you write the largest natural number?

Answer

The counting numbers 1, 2, 3, 4, ... are called natural numbers.

The smallest natural number is 1.

No, we cannot write the largest natural number, because there are infinitely many natural numbers. For every natural number, there exists a greater natural number obtained by adding 1 to it.

Hence, the smallest natural number is 1, and there is no largest natural number.

Insert commas suitably and write each of the following numbers in words in the Indian system and the International system of numeration:

(i) 506723

(ii) 180018018

Answer

(i) 506723

In the Indian system, the first comma comes after 3 digits from the right, and the next comma comes after every 2 digits.

506723 = 5,06,723

In words (Indian system): Five lakh six thousand seven hundred twenty three.

In the International system, commas come after every 3 digits from the right.

506723 = 506,723

In words (International system): Five hundred six thousand seven hundred twenty three.

(ii) 180018018

In the Indian system: 18,00,18,018

In words (Indian system): Eighteen crore eighteen thousand eighteen.

In the International system: 180,018,018

In words (International system): One hundred eighty million eighteen thousand eighteen.

Write the following numbers in expanded form:

(i) 750687

(ii) 5032109

Answer

(i) Expanding,

750687 = 7 × 1,00,000 + 5 × 10,000 + 0 × 1,000 + 6 × 100 + 8 × 10 + 7 × 1

= 7,00,000 + 50,000 + 600 + 80 + 7.

(ii) Expanding,

5032109 = 5 × 10,00,000 + 0 × 1,00,000 + 3 × 10,000 + 2 × 1,000 + 1 × 100 + 0 × 10 + 9 × 1

= 50,00,000 + 30,000 + 2,000 + 100 + 9.

Write the following numbers in figures:

(i) Seven lakh three thousand four hundred twenty

(ii) Eighty crore twenty three thousand ninety three

Also write the above numbers in the place value chart.

Answer

(i) Seven lakh three thousand four hundred twenty

The number consists of 7 lakhs, 0 ten thousands, 3 thousands, 4 hundreds, 2 tens and 0 ones.

In figures: 7,03,420

In the Indian place value chart:

(ii) Eighty crore twenty three thousand ninety three

The number consists of 8 ten crores, 0 crores, 0 ten lakhs, 0 lakhs, 2 ten thousands, 3 thousands, 0 hundreds, 9 tens and 3 ones.

In figures: 80,00,23,093

In the Indian place value chart:

Write each of the following numbers in numeral form and place commas correctly:

(i) Seventy three lakh seventy thousand four hundred seven.

(ii) Nine crore five lakh forty one.

(iii) Fifty eight million four hundred twenty three thousand two hundred two.

Answer

(i) Seventy three lakh seventy thousand four hundred seven.

The number consists of 73 lakhs, 70 thousands, 4 hundreds, 0 tens and 7 ones.

In figures (Indian system): 73,70,407

(ii) Nine crore five lakh forty one.

The number consists of 9 crores, 5 lakhs, 0 thousands, 0 hundreds, 4 tens and 1 in ones place.

In figures (Indian system): 9,05,00,041

(iii) Fifty eight million four hundred twenty three thousand two hundred two.

The number consists of 58 millions, 423 thousands and 202 ones.

In figures (International system): 58,423,202

Write the face value and place value of the digit 6 in the number 756032.

Answer

In the Indian system, the given number can be written as 7,56,032.

The face value of a digit in a number is the digit itself.

So, the face value of 6 = 6.

The place value of 6 = 6 × 1,000 = 6,000.

Hence, the face value = 6 and place value = 6,000.

Write the face value and the place value of the odd digits in the number 36510692.

Answer

In the Indian system, the given number can be written as 3,65,10,692.

The odd digits in the given number are 3, 5, 1 and 9.

For digit 3:

Face value = 3

Place value = 3 × 1,00,00,000 = 3,00,00,000

For digit 5:

Face value = 5

Place value = 5 × 1,00,000 = 5,00,000

For digit 1:

Face value = 1

Place value = 1 × 10,000 = 10,000

For digit 9:

Face value = 9

Place value = 9 × 10 = 90

Hence, the face values of the odd digits 3, 5, 1 and 9 are 3, 5, 1 and 9 respectively, and their place values are 3,00,00,000; 5,00,000; 10,000 and 90 respectively.

Find the difference between the place value and the face value of the digit 9 in the number 229301.

Answer

In the Indian system, the given number can be written as 2,29,301.

The face value of 9 = 9.

The place value of 9 = 9 × 1,000 = 9,000.

Difference = Place value - Face value

= 9,000 - 9

= 8,991

Hence, the difference between the place value and the face value of the digit 9 in 2,29,301 is 8,991.

Determine the product of place value and the face value of the digit 4 in the number 5437.

Answer

The face value of 4 = 4.

The place value of 4 = 4 × 100 = 400.

Product = Place value × Face value

= 400 × 4

= 1,600

Hence, the product of the place value and the face value of the digit 4 in 5437 is 1,600.

Find the difference between the number 895 and that obtained on reversing its digits.

Answer

The given number = 895.

On reversing the digits of 895, the new number = 598.

Difference = 895 - 598

= 297

Hence, the required difference = 297.

Determine the difference of the place value of two 7's in 37014472 and write it in words in International system.

Answer

In the International system, the given number can be written as 37,014,472.

The place value of 7 at the millions place = 7 × 1,000,000 = 7,000,000.

The place value of 7 at the tens place = 7 × 10 = 70.

Difference = 7,000,000 - 70

= 6,999,930

In words (International system): Six million nine hundred ninety nine thousand nine hundred thirty.

Hence, the required difference is 6,999,930, i.e., six million nine hundred ninety nine thousand nine hundred thirty.

Use the appropriate symbol < or > to fill in the blanks:

(i) 173 ... 189

(ii) 1058 ... 1074

(iii) 8315 ... 8037

Answer

(i) 173 ... 189

Both numbers have 3 digits.

Comparing from the leftmost digit:

The hundreds digit is 1 in both.

The tens digit is 7 in 173 and 8 in 189.

Since 7 is less than 8, 173 is smaller than 189.

Hence, 173 < 189.

(ii) 1058 ... 1074

Both numbers have 4 digits.

Comparing from the leftmost digit:

The thousands digit is 1 in both.

The hundreds digit is 0 in both.

The tens digit is 5 in 1058 and 7 in 1074.

Since 5 is less than 7, 1058 is smaller than 1074.

Hence, 1058 < 1074.

(iii) 8315 ... 8037

Both numbers have 4 digits.

Comparing from the leftmost digit:

The thousands digit is 8 in both.

The hundreds digit is 3 in 8315 and 0 in 8037.

Since 3 is greater than 0, 8315 is greater than 8037.

Hence, 8315 > 8037.

In each of the following pairs of numbers, state which number is smaller:

(i) 553, 503

(ii) 41338, 1139

(iii) 25431, 24531

Answer

(i) 553, 503

Both numbers have 3 digits.

Comparing from the leftmost digit, the hundreds digit is 5 in both, the tens digit is 5 in 553 and 0 in 503.

Since 0 is less than 5, 503 is smaller than 553.

Hence, 503 is the smaller number.

(ii) 41338, 1139

41338 has 5 digits and 1139 has 4 digits.

The number with fewer digits is smaller.

Hence, 1139 is the smaller number.

(iii) 25431, 24531

Both numbers have 5 digits.

Comparing from the leftmost digit, the ten thousands digit is 2 in both, the thousands digit is 5 in 25431 and 4 in 24531.

Since 4 is less than 5, 24531 is smaller than 25431.

Hence, 24531 is the smaller number.

Find the greatest and the smallest numbers in each row:

(i) 71834, 75284, 571, 2333, 594

(ii) 9853, 7691, 9999, 12002

Answer

(i) 71834, 75284, 571, 2333, 594

The number of digits in each number:

71834 → 5 digits

75284 → 5 digits

571 → 3 digits

2333 → 4 digits

594 → 3 digits

Comparing the 5-digit numbers 71834 and 75284:

The ten thousands digit is 7 in both, the thousands digit is 1 in 71834 and 5 in 75284. Since 5 > 1, 75284 > 71834.

So the greatest number is 75284.

Comparing the 3-digit numbers 571 and 594:

The hundreds digit is 5 in both, the tens digit is 7 in 571 and 9 in 594. Since 7 < 9, 571 < 594.

So the smallest number is 571.

Hence, the greatest number is 75284 and the smallest number is 571.

(ii) 9853, 7691, 9999, 12002

The number of digits in each number:

9853 → 4 digits

7691 → 4 digits

9999 → 4 digits

12002 → 5 digits

12002 has 5 digits while the others have 4 digits, so 12002 is the greatest.

Comparing the 4-digit numbers:

9853, 7691, 9999. The thousands digit is 9, 7, 9 respectively. So 7691 is the smallest among them.

Hence, the greatest number is 12002 and the smallest number is 7691.

Arrange the following numbers in ascending order:

304, 340, 34, 4503, 43, 430

Answer

Given, 304, 340, 34, 4503, 43, 430

The number of digits in each number:

304, 340, 430 → 3 digits

34, 43 → 2 digits

4503 → 4 digits

The 2-digit numbers are smaller than the 3-digit numbers, which are smaller than the 4-digit number.

Comparing the 2-digit numbers 34 and 43: 3 < 4, so 34 < 43.

Comparing the 3-digit numbers 304, 340 and 430:

The hundreds digit is 3, 3, 4 respectively. So 430 > 340 > 304 (after also comparing 304 and 340 at the tens digit: 0 < 4).

Hence, the ascending order is: 34 < 43 < 304 < 340 < 430 < 4503.

Arrange the following numbers in descending order:

53, 7333, 553, 7529, 377, 335

Answer

Given, 53, 7333, 553, 7529, 377, 335

The number of digits in each number:

53 → 2 digits

7333, 7529 → 4 digits

553, 377, 335 → 3 digits

The 4-digit numbers are largest.

Comparing 7333 and 7529: thousands digit 7 in both, hundreds digit 3 vs 5. Since 5 > 3, 7529 > 7333.

Comparing the 3-digit numbers 553, 377, 335: the hundreds digit is 5, 3, 3 respectively. 553 is the largest. Comparing 377 and 335: tens digit 7 vs 3, so 377 > 335.

Hence, the descending order is: 7529 > 7333 > 553 > 377 > 335 > 53.

Arrange the following numbers in ascending order (use place value chart):

67201, 743819, 748391, 67218, 85344

Answer

Writing the given numbers in the Indian place value chart:

Out of the given numbers, three numbers are 5-digit and two numbers are 6-digit.

5-digit numbers are smaller than 6-digit numbers.

For the 5-digit numbers 67201, 67218 and 85344:

Comparing 67201 and 67218: ten thousands 6 = 6, thousands 7 = 7, hundreds 2 = 2, tens 0 < 1. So 67201 < 67218.

85344 has the ten thousands digit 8, which is greater than 6. So 85344 is the largest 5-digit number.

For the 6-digit numbers 743819 and 748391:

Comparing them: lakhs 7 = 7, ten thousands 4 = 4, thousands 3 < 8. So 743819 < 748391.

Hence, the ascending order is: 67201 < 67218 < 85344 < 743819 < 748391.

Arrange the following numbers in descending order (use place value chart):

36951, 37140, 723891, 732142, 5718901, 37401

Answer

Writing the given numbers in the Indian place value chart:

Out of the given numbers, three are 5-digit, two are 6-digit and one is 7-digit.

7-digit number i.e. 5718901 is the largest.

For the 6-digit numbers 723891 and 732142:

Comparing them: lakhs 7 = 7, ten thousands 2 < 3. So 732142 > 723891.

For the 5-digit numbers 36951, 37140 and 37401:

Comparing 36951 and 37140: ten thousands 3 = 3, thousands 6 < 7. So 36951 < 37140.

Comparing 37140 and 37401: ten thousands 3 = 3, thousands 7 = 7, hundreds 1 < 4. So 37140 < 37401.

So among the 5-digit numbers, 37401 is largest, then 37140, then 36951.

Hence, the descending order is: 5718901 > 732142 > 723891 > 37401 > 37140 > 36951.

Write all possible 2-digit numbers that can be formed by using the digits 2, 3 and 4. Repetition of digits is not allowed. Also find their sum.

Answer

We are required to write 2-digit numbers using the digits 2, 3 and 4, where repetition of digits is not allowed.

Out of the given digits, the possible ways of choosing two digits are: 2, 3; 2, 4; 3, 4.

Using the digits 2 and 3, the numbers are 23 and 32.

Using the digits 2 and 4, the numbers are 24 and 42.

Using the digits 3 and 4, the numbers are 34 and 43.

All possible 2-digit numbers: 23, 32, 24, 42, 34, 43.

Sum = 23 + 32 + 24 + 42 + 34 + 43

= 198

Hence, all possible 2-digit numbers are 23, 32, 24, 42, 34, 43 and their sum is 198.

Write all possible 3-digit numbers using the digits 3, 1 and 5. Repetition of digits is not allowed.

Answer

We are required to write 3-digit numbers using the digits 3, 1 and 5, where repetition of digits is not allowed.

Keeping 1 at unit's place, 3-digit numbers are 351 and 531.

Keeping 3 at unit's place, 3-digit numbers are 153 and 513.

Keeping 5 at unit's place, 3-digit numbers are 135 and 315.

Hence, all possible 3-digit numbers are 135, 153, 315, 351, 513 and 531.

Write all possible 3-digit numbers using the digits 7, 0 and 6. Repetition of digits is not allowed. Also find their sum.

Answer

We are required to write 3-digit numbers using the digits 7, 0 and 6, where repetition of digits is not allowed.

The digit 0 cannot be put at hundred's place because that would make the number only 2-digit.

Keeping 0 at unit's place, 3-digit numbers are 760 and 670.

Keeping 0 at ten's place, 3-digit numbers are 706 and 607.

All possible 3-digit numbers: 706, 760, 607, 670.

Sum = 706 + 760 + 607 + 670

= 2,743

Hence, all possible 3-digit numbers are 706, 760, 607, 670 and their sum is 2,743.

Write all possible 2-digit numbers using the digits 4, 0 and 9. Repetition of digits is not allowed. Also find their sum.

Answer

We are required to write 2-digit numbers using the digits 4, 0 and 9, where repetition of digits is not allowed.

The digit 0 cannot be put at ten's place because that would make the number only 1-digit.

Keeping 4 at ten's place, 2-digit numbers are 40 and 49.

Keeping 9 at ten's place, 2-digit numbers are 90 and 94.

All possible 2-digit numbers: 40, 49, 90, 94.

Sum = 40 + 49 + 90 + 94

= 273

Hence, all possible 2-digit numbers are 40, 49, 90 and 94, and their sum is 273.

Write all possible 2-digit numbers that can be formed by using the digits 3, 7 and 9. Repetition of digits is allowed.

Answer

We are required to write 2-digit numbers using the digits 3, 7 and 9, where repetition of digits is allowed.

Keeping 3 at ten's place, 2-digit numbers are 33, 37 and 39.

Keeping 7 at ten's place, 2-digit numbers are 73, 77 and 79.

Keeping 9 at ten's place, 2-digit numbers are 93, 97 and 99.

Hence, all possible 2-digit numbers are 33, 37, 39, 73, 77, 79, 93, 97 and 99.

Write all possible numbers using the digits 3, 1 and 5. Repetition of digits is not allowed.

Answer

We are required to write all possible numbers using the digits 3, 1 and 5, where repetition of digits is not allowed.

1-digit numbers: 1, 3, 5.

2-digit numbers:

Keeping 1 at unit's place: 31, 51.

Keeping 3 at unit's place: 13, 53.

Keeping 5 at unit's place: 15, 35.

2-digit numbers: 13, 15, 31, 35, 51, 53.

3-digit numbers:

Keeping 1 at unit's place: 351, 531.

Keeping 3 at unit's place: 153, 513.

Keeping 5 at unit's place: 135, 315.

3-digit numbers: 135, 153, 315, 351, 513, 531.

Hence, all possible numbers are: 1, 3, 5, 13, 15, 31, 35, 51, 53, 135, 153, 315, 351, 513 and 531.

How many 6-digit numbers are there in all?

Answer

The greatest 6-digit number = 9,99,999.

The greatest 5-digit number = 99,999.

The total number of 6-digit numbers = 9,99,999 - 99,999

= 9,00,000

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

Write down the greatest number and the smallest number of 4 digits that can be formed by the digits 7, 5, 0 and 4 using each digit only once.

Answer

The given digits are 7, 5, 0 and 4, and the repetition of digits is not allowed.

For the greatest 4-digit number, the digit at the thousand's place has to be the greatest, which is 7; followed by the next greatest digit 5 at the hundred's place, then 4 at the ten's place and 0 at the unit's place.

The greatest 4-digit number = 7,540.

For the smallest 4-digit number, we have to follow just the reverse. The digit at the thousand's place has to be the smallest non-zero digit, which is 4 (since 0 cannot be at the thousand's place); followed by the next smallest digit 0 at the hundred's place, then 5 at the ten's place and 7 at the unit's place.

The smallest 4-digit number = 4,057.

Hence, the greatest 4-digit number is 7,540 and the smallest 4-digit number is 4,057.

Rearrange the digits of the number 5701024 to get the largest number and the smallest number of 7 digits.

Answer

The digits of the given number 5701024 are 5, 7, 0, 1, 0, 2 and 4.

For the largest 7-digit number, arrange the digits in descending order:

7, 5, 4, 2, 1, 0, 0.

The largest 7-digit number = 75,42,100.

For the smallest 7-digit number, the digit at the highest place (ten lakhs) cannot be 0. So, place the smallest non-zero digit (1) at the highest place, and arrange the remaining digits (0, 0, 2, 4, 5, 7) in ascending order after it.

The smallest 7-digit number = 10,02,457.

Hence, the largest 7-digit number is 75,42,100 and the smallest 7-digit number is 10,02,457.

Keeping the place value of digit 3 in the number 730265 same, rearrange the digits of the given number to get the largest number and smallest number of 6 digits.

Answer

In the number 7,30,265, the digit 3 is at the ten thousand's place.

The place value of 3 = 3 × 10,000 = 30,000.

To keep the place value of 3 same, the digit 3 must remain at the ten thousand's place.

The remaining digits are 7, 0, 2, 6 and 5.

For the largest 6-digit number, arrange the remaining digits in descending order, with 3 fixed at the ten thousand's place:

Lakhs place: 7 (largest)

Ten thousands place: 3 (fixed)

Thousands place: 6

Hundreds place: 5

Tens place: 2

Ones place: 0

The largest 6-digit number = 7,36,520.

For the smallest 6-digit number, the digit at the lakh's place cannot be 0. So, place the smallest non-zero digit (excluding 3) at the lakh's place, which is 2. Then arrange the remaining digits in ascending order:

Lakhs place: 2

Ten thousands place: 3 (fixed)

Thousands place: 0

Hundreds place: 5

Tens place: 6

Ones place: 7

The smallest 6-digit number = 2,30,567.

Hence, the largest 6-digit number is 7,36,520 and the smallest 6-digit number is 2,30,567.

Form the smallest and greatest 4-digit numbers by using any one digit twice from the digits:

(i) 5, 2, 3, 9

(ii) 6, 0, 1, 4

(iii) 4, 6, 1, 5, 8

Answer

(i) The given digits are 5, 2, 3 and 9.

For the greatest 4-digit number, use the largest digit twice. So, use 9 twice. The next largest digits are 5 and 3.

Arrange in descending order: 9, 9, 5, 3.

The greatest 4-digit number = 9,953.

For the smallest 4-digit number, use the smallest digit twice. So, use 2 twice. The other two smaller digits are 3 and 5.

Arrange in ascending order: 2, 2, 3, 5.

The smallest 4-digit number = 2,235.

Hence, the greatest 4-digit number is 9,953 and the smallest 4-digit number is 2,235.

(ii) The given digits are 6, 0, 1 and 4.

For the greatest 4-digit number, use the largest digit twice. So, use 6 twice. The next largest digits are 4 and 1.

Arrange in descending order: 6, 6, 4, 1.

The greatest 4-digit number = 6,641.

For the smallest 4-digit number, use the smallest digit twice. So, use 0 twice. The other two smaller digits are 1 and 4.

Since 0 cannot be at the thousand's place, the next smallest non-zero digit (1) is placed at the thousand's place, then 0, 0, 4 in order.

The smallest 4-digit number = 1,004.

Hence, the greatest 4-digit number is 6,641 and the smallest 4-digit number is 1,004.

(iii) The given digits are 4, 6, 1, 5 and 8.

For the greatest 4-digit number, use the largest digit twice. So, use 8 twice. The next largest digits from the remaining are 6 and 5.

Arrange in descending order: 8, 8, 6, 5.

The greatest 4-digit number = 8,865.

For the smallest 4-digit number, use the smallest digit twice. So, use 1 twice. The other two smaller digits from the remaining are 4 and 5.

Arrange in ascending order: 1, 1, 4, 5.

The smallest 4-digit number = 1,145.

Hence, the greatest 4-digit number is 8,865 and the smallest 4-digit number is 1,145.

Write (i) the greatest number of 6 digits (ii) the smallest number of 7 digits. Also find their difference.

Answer

(i) The greatest number of 6 digits is formed by placing the largest digit (9) in all six positions.

The greatest 6-digit number = 9,99,999.

(ii) The smallest number of 7 digits is formed by placing the smallest non-zero digit (1) at the highest place and 0s in all the remaining positions.

The smallest 7-digit number = 10,00,000.

Difference = 10,00,000 - 9,99,999 = 1.

Hence, the greatest 6-digit number is 9,99,999, the smallest 7-digit number is 10,00,000, and their difference is 1.

Write the greatest 4-digit number having distinct digits.

Answer

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

For the greatest 4-digit number with distinct digits, arrange the largest digits in descending order:

Thousands place: 9 (largest)

Hundreds place: 8 (next largest)

Tens place: 7 (next largest)

Ones place: 6 (next largest)

Hence, the greatest 4-digit number having distinct digits is 9,876.

Write the smallest 4-digit number having distinct digits.

Answer

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

For the smallest 4-digit number with distinct digits, the thousands place cannot be 0. So, the smallest non-zero digit (1) is placed at the thousand's place. Then arrange the remaining smallest digits in ascending order:

Thousands place: 1

Hundreds place: 0

Tens place: 2

Ones place: 3

Hence, the smallest 4-digit number having distinct digits is 1,023.

Write the greatest 6-digit number using three different digits.

Answer

For the greatest 6-digit number using exactly three different digits, the digits at the higher places should be as large as possible.

Use the digit 9 as many times as possible (without exceeding three different digits in total).

Using 9 four times makes only 3 different digits (9, 8, 7) when combined with 8 and 7. Let us check: in the number 9,99,987, the digits are 9, 9, 9, 9, 8, 7. The different digits are 9, 8 and 7, which gives exactly 3 different digits.

Arranging in descending order: 9, 9, 9, 9, 8, 7.

The greatest 6-digit number = 9,99,987.

Hence, the greatest 6-digit number using three different digits is 9,99,987.

Write the smallest 7-digit number using four different digits.

Answer

For the smallest 7-digit number using exactly four different digits, the digit at the highest place (ten lakhs) cannot be 0. So, the smallest non-zero digit (1) is placed at the highest place.

Then use 0 in as many remaining positions as possible. To make exactly four different digits in total, we need two more digits apart from 1 and 0. The smallest such digits are 2 and 3.

The digits to be used: 1, 0, 0, 0, 0, 2, 3.

Arranging the digits with 1 at the highest place and the rest in ascending order:

Ten lakhs place: 1

Lakhs place: 0

Ten thousands place: 0

Thousands place: 0

Hundreds place: 0

Tens place: 2

Ones place: 3

Hence, the smallest 7-digit number using four different digits is 10,00,023.

Write the greatest and the smallest 4-digit numbers using four different digits with the conditions as given:

(i) Digit 7 is always at units place.

(ii) Digit 4 is always at tens place.

(iii) Digit 9 is always at hundreds place.

(iv) Digit 2 is always at thousands place.

Answer

(i) Digit 7 is always at units place.

The number is of the form _ _ _ 7 (with all four digits different).

For the greatest 4-digit number, the remaining three digits should be as large as possible (different from 7 and from each other). The largest such digits are 9, 8 and 6.

Greatest 4-digit number = 9867.

For the smallest 4-digit number, the thousands place cannot be 0. The smallest non-zero digit (other than 7) is 1. Then the remaining digits are 0 and 2 (smallest available, different from 1 and 7).

Smallest 4-digit number = 1027.

Hence, the greatest 4-digit number is 9,867 and the smallest 4-digit number is 1,027.

(ii) Digit 4 is always at tens place.

The number is of the form _ _ 4 _ (with all four digits different).

For the greatest 4-digit number, fill the remaining places with the largest distinct digits other than 4. Use 9 at thousands, 8 at hundreds and 7 at units.

Greatest 4-digit number = 9847.

For the smallest 4-digit number, the thousands place cannot be 0. Place 1 at thousands, 0 at hundreds, 4 at tens and 2 at units.

Smallest 4-digit number = 1042.

Hence, the greatest 4-digit number is 9,847 and the smallest 4-digit number is 1,042.

(iii) Digit 9 is always at hundreds place.

The number is of the form _ 9 _ _ (with all four digits different).

For the greatest 4-digit number, fill the remaining places with the largest distinct digits other than 9. Use 8 at thousands, 7 at tens and 6 at units.

Greatest 4-digit number = 8976.

For the smallest 4-digit number, the thousands place cannot be 0. Place 1 at thousands, 9 at hundreds, 0 at tens and 2 at units.

Smallest 4-digit number = 1902.

Hence, the greatest 4-digit number is 8,976 and the smallest 4-digit number is 1,902.

(iv) Digit 2 is always at thousands place.

The number is of the form 2 _ _ _ (with all four digits different).

For the greatest 4-digit number, fill the remaining places with the largest distinct digits other than 2. Use 9 at hundreds, 8 at tens and 7 at units.

Greatest 4-digit number = 2987.

For the smallest 4-digit number, fill the remaining places with the smallest distinct digits other than 2 (since 2 is already at thousands). Use 0 at hundreds, 1 at tens and 3 at units.

Smallest 4-digit number = 2013.

Hence, the greatest 4-digit number is 2,987 and the smallest 4-digit number is 2,013.

In a particular year, a company manufactured 8570435 bicycles and next year it manufactured 8756430 bicycles. In which year more bicycles were manufactured and by how many?

Answer

Number of bicycles manufactured in the first year = 85,70,435.

Number of bicycles manufactured in the next year = 87,56,430.

Both numbers have 7 digits. Comparing from the leftmost digit:

The ten lakhs digit is 8 in both, the lakhs digit is 5 in 85,70,435 and 7 in 87,56,430.

Since 7 > 5, 87,56,430 > 85,70,435.

So, more bicycles were manufactured in the next year.

Difference = 87,56,430 - 85,70,435

= 1,85,995

Hence, more bicycles were manufactured in the next year, by 1,85,995.

What number must be subtracted from 1,02,59,756 to get 77,63,835?

Answer

Let the required number to be subtracted be x.

Then, 1,02,59,756 - x = 77,63,835

⇒ x = 1,02,59,756 - 77,63,835

⇒ x = 24,95,921

Hence, the required number to be subtracted is 24,95,921.

The sale receipt of a company during a year was ₹ 30587850. Next year it increased by ₹ 6375490. What was the total sale receipt of the company during these two years?

Answer

Sale receipt of the company in the first year = ₹3,05,87,850.

Increase in sale receipt in the next year = ₹63,75,490.

Sale receipt of the company in the next year = ₹3,05,87,850 + ₹63,75,490

= ₹3,69,63,340

Total sale receipts during these two years = Sale in first year + Sale in next year

= ₹3,05,87,850 + ₹3,69,63,340

= ₹6,75,51,190

Hence, the total sale receipt of the company during these two years was ₹6,75,51,190.

A machine manufactures 23875 screws per day. How many screws did it produce in the year 2016? Assume that the machine worked on all the days of the year.

Answer

Number of screws manufactured per day = 23,875.

The year 2016 is a leap year (since 2016 is divisible by 4 and is not a century year), so it has 366 days.

Total number of screws produced in the year 2016 = Number of screws per day × Number of days

= 23,875 × 366

= 87,38,250

Hence, the machine produced 87,38,250 screws in the year 2016.

A merchant had ₹78,592 with him. He placed an order for purchasing 30 bicycles at ₹2450 each. How much money will remain with him after the purchase?

Answer

Money the merchant had = ₹78,592.

Number of bicycles ordered = 30.

Cost of each bicycle = ₹2,450.

Total cost of 30 bicycles = 30 × ₹2,450

= ₹73,500

Money remaining with him after the purchase = ₹78,592 - ₹73,500

= ₹5,092

Hence, the money remaining with the merchant after the purchase is ₹5,092.

Amitabh is 1 m 82 cm tall and his wife is 35 cm shorter than him. What is his wife's height?

Answer

Amitabh's height = 1 m 82 cm

= (1 × 100 + 82) cm

= 182 cm

His wife is 35 cm shorter than him.

Wife's height = 182 cm - 35 cm

= 147 cm

= (100 + 47) cm

= 1 m 47 cm

Hence, his wife's height is 1 m 47 cm.

The mass of each gas cylinder is 21 kg 270 g. What is total mass of 28 such cylinders?

Answer

Mass of each gas cylinder = 21 kg 270 g

= (21 × 1,000 + 270) g

= 21,270 g

Total mass of 28 such cylinders = 28 × 21,270 g

= 5,95,560 g

= (595 × 1,000 + 560) g

= 595 kg 560 g

Hence, the total mass of 28 such cylinders is 595 kg 560 g.

In order to make a shirt, 2 m 25 cm cloth is needed. What length of cloth is required to make 18 such shirts?

Answer

Cloth needed for 1 shirt = 2 m 25 cm

= (2 × 100 + 25) cm

= 225 cm

Cloth needed for 18 such shirts = 18 × 225 cm

= 4,050 cm

= (40 × 100 + 50) cm

= 40 m 50 cm

Hence, the length of cloth required to make 18 such shirts is 40 m 50 cm.

The total mass of 12 packets of sweets, each of the same size, is 15 kg 600 g. What is the mass of each such packet?

Answer

Total mass of 12 packets = 15 kg 600 g

= (15 × 1,000 + 600) g

= 15,600 g

Mass of each packet = $\dfrac{\text{Total mass}}{\text{Number of packets}}$

= $\dfrac{15{,}600 \text{ g}}{12}$

= 1,300 g

= (1 × 1,000 + 300) g

= 1 kg 300 g

Hence, the mass of each packet is 1 kg 300 g.

A vessel has 4 litres 500 millilitres of orange juice. In how many glasses, each of 25 mL capacity, can it be filled?

Answer

Total amount of orange juice = 4 L 500 mL

= (4 × 1,000 + 500) mL

= 4,500 mL

Capacity of each glass = 25 mL.

Number of glasses = $\dfrac{\text{Total amount of juice}}{\text{Capacity of each glass}}$

= $\dfrac{4{,}500 \text{ mL}}{25 \text{ mL}}$

= 180

Hence, the orange juice can be filled in 180 glasses.

To stitch a trouser, 1 m 30 cm cloth is needed. Out of 25 m cloth, how many trousers can be stitched and how much cloth will remain?

Answer

Length of cloth available = 25 m

= (25 × 100) cm

= 2,500 cm

Length of cloth required to stitch a trouser = 1 m 30 cm

= (1 × 100 + 30) cm

= 130 cm

Dividing 2,500 by 130:

Quotient = 19, remainder = 30.

So, 19 trousers can be stitched and 30 cm of cloth will remain.

Hence, the number of trousers that can be stitched is 19 and the length of the remaining cloth is 30 cm.

Round off each of the following numbers to their nearest tens:

(i) 77

(ii) 903

(iii) 1205

(iv) 999

Answer

(i) 77

The digit at ones place is 7. Since 7 is greater than 5, we increase the tens digit by 1 and replace the ones digit by 0.

Hence, the rounded off number to the nearest tens = 80.

(ii) 903

The digit at ones place is 3. Since 3 is less than 5, we replace the ones digit by 0 and keep all other digits as they are.

Hence, the rounded off number to the nearest tens = 900.

(iii) 1205

The digit at ones place is 5. So, we increase the tens digit by 1 and replace the ones digit by 0.

Hence, the rounded off number to the nearest tens = 1,210.

(iv) 999

The digit at ones place is 9. Since 9 is greater than 5, we increase the tens digit by 1 and replace the ones digit by 0. As the tens digit is 9, increasing it by 1 carries over to the hundreds digit.

Hence, the rounded off number to the nearest tens = 1,000.

Estimate each of the following numbers to their nearest hundreds:

(i) 1246

(ii) 32057

(iii) 53961

(iv) 555555

Answer

(i) 1246

The digit at tens place is 4. Since 4 is less than 5, we replace each of the digits at tens place and ones place by 0.

Hence, the rounded off number to the nearest hundreds = 1,200.

(ii) 32057

The digit at tens place is 5. So, we increase the digit at hundreds place by 1 and replace each of the digits at tens place and ones place by 0.

Hence, the rounded off number to the nearest hundreds = 32,100.

(iii) 53961

The digit at tens place is 6. Since 6 is greater than 5, we increase the digit at hundreds place by 1 and replace each of the digits at tens place and ones place by 0. As the hundreds digit is 9, increasing it by 1 carries over.

Hence, the rounded off number to the nearest hundreds = 54,000.

(iv) 555555

The digit at tens place is 5. So, we increase the digit at hundreds place by 1 and replace each of the digits at tens place and ones place by 0.

Hence, the rounded off number to the nearest hundreds = 5,55,600.

Estimate each of the following numbers to their nearest thousands:

(i) 5706

(ii) 378

(iii) 47,599

(iv) 1,09,736

Answer

(i) 5706

The digit at hundreds place is 7. Since 7 is greater than 5, we increase the digit at thousands place by 1 and replace each of the digits at hundreds, tens and ones place by 0.

Hence, the rounded off number to the nearest thousands = 6,000.

(ii) 378

The digit at hundreds place is 3. Since 3 is less than 5, we replace each of the digits at hundreds, tens and ones place by 0.

Hence, the rounded off number to the nearest thousands = 0.

(iii) 47,599

The digit at hundreds place is 5. So, we increase the digit at thousands place by 1 and replace each of the digits at hundreds, tens and ones place by 0.

Hence, the rounded off number to the nearest thousands = 48,000.

(iv) 1,09,736

The digit at hundreds place is 7. Since 7 is greater than 5, we increase the digit at thousands place by 1 and replace each of the digits at hundreds, tens and ones place by 0.

Hence, the rounded off number to the nearest thousands = 1,10,000.

Give a rough estimate (by rounding off to nearest hundreds) and also a closer estimate (by rounding off to nearest tens):

(i) 439 + 334 + 4317

(ii) 8325 - 491

(iii) 1,08,734 - 47,599

(iv) 4,89,348 - 48,365

Answer

(i) 439 + 334 + 4317

Rough estimate (nearest hundreds):

439 → 400 (tens digit 3 < 5)

334 → 300 (tens digit 3 < 5)

4317 → 4300 (tens digit 1 < 5)

Estimated sum = 400 + 300 + 4300 = 5,000.

Closer estimate (nearest tens):

439 → 440 (ones digit 9 > 5)

334 → 330 (ones digit 4 < 5)

4317 → 4320 (ones digit 7 > 5)

Estimated sum = 440 + 330 + 4320 = 5,090.

Hence, the rough estimate is 5,000 and the closer estimate is 5,090.

(ii) 8325 - 491

Rough estimate (nearest hundreds):

8325 → 8300 (tens digit 2 < 5)

491 → 500 (tens digit 9 > 5)

Estimated difference = 8300 - 500 = 7,800.

Closer estimate (nearest tens):

8325 → 8330 (ones digit 5)

491 → 490 (ones digit 1 < 5)

Estimated difference = 8330 - 490 = 7,840.

Hence, the rough estimate is 7,800 and the closer estimate is 7,840.

(iii) 1,08,734 - 47,599

Rough estimate (nearest hundreds):

1,08,734 → 1,08,700 (tens digit 3 < 5)

47,599 → 47,600 (tens digit 9 > 5)

Estimated difference = 1,08,700 - 47,600 = 61,100.

Closer estimate (nearest tens):

1,08,734 → 1,08,730 (ones digit 4 < 5)

47,599 → 47,600 (ones digit 9 > 5)

Estimated difference = 1,08,730 - 47,600 = 61,130.

Hence, the rough estimate is 61,100 and the closer estimate is 61,130.

(iv) 4,89,348 - 48,365

Rough estimate (nearest hundreds):

4,89,348 → 4,89,300 (tens digit 4 < 5)

48,365 → 48,400 (tens digit 6 > 5)

Estimated difference = 4,89,300 - 48,400 = 4,40,900.

Closer estimate (nearest tens):

4,89,348 → 4,89,350 (ones digit 8 > 5)

48,365 → 48,370 (ones digit 5)

Estimated difference = 4,89,350 - 48,370 = 4,40,980.

Hence, the rough estimate is 4,40,900 and the closer estimate is 4,40,980.

Estimate each of the following by rounding off each number nearest to its greatest place:

(i) 730 + 998

(ii) 5,290 + 17,986

(iii) 796 - 314

(iv) 28,292 - 21,496

Answer

(i) 730 + 998

Rounding off 730 to its greatest place (hundreds): tens digit is 3 (< 5), so 730 → 700.

Rounding off 998 to its greatest place (hundreds): tens digit is 9 (≥ 5), so 998 → 1000.

Estimated sum = 700 + 1000 = 1,700.

Hence, the estimated sum = 1,700.

(ii) 5,290 + 17,986

Rounding off 5,290 to its greatest place (thousands): hundreds digit is 2 (< 5), so 5,290 → 5,000.

Rounding off 17,986 to its greatest place (ten thousands): thousands digit is 7 (≥ 5), so 17,986 → 20,000.

Estimated sum = 5,000 + 20,000 = 25,000.

Hence, the estimated sum = 25,000.

(iii) 796 - 314

Rounding off 796 to its greatest place (hundreds): tens digit is 9 (≥ 5), so 796 → 800.

Rounding off 314 to its greatest place (hundreds): tens digit is 1 (< 5), so 314 → 300.

Estimated difference = 800 - 300 = 500.

Hence, the estimated difference = 500.

(iv) 28,292 - 21,496

Rounding off 28,292 to its greatest place (ten thousands): thousands digit is 8 (≥ 5), so 28,292 → 30,000.

Rounding off 21,496 to its greatest place (ten thousands): thousands digit is 1 (< 5), so 21,496 → 20,000.

Estimated difference = 30,000 - 20,000 = 10,000.

Hence, the estimated difference = 10,000.

Estimate the following products by rounding off each of its factors nearest to its greatest place:

(i) 578 × 161

(ii) 9650 × 27

Answer

(i) 578 × 161

Rounding off 578 to its greatest place (hundreds): tens digit is 7 (≥ 5), so 578 → 600.

Rounding off 161 to its greatest place (hundreds): tens digit is 6 (≥ 5), so 161 → 200.

Estimated product = 600 × 200 = 1,20,000.

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

(ii) 9650 × 27

Rounding off 9650 to its greatest place (thousands): hundreds digit is 6 (≥ 5), so 9650 → 10,000.

Rounding off 27 to its greatest place (tens): ones digit is 7 (≥ 5), so 27 → 30.

Estimated product = 10,000 × 30 = 3,00,000.

Hence, the estimated product = 3,00,000.

Estimate the following products by rounding off each of its factors nearest to its hundreds:

(i) 5281 × 3491

(ii) 1387 × 888

Answer

(i) 5281 × 3491

Rounding off 5281 to its nearest hundreds: tens digit is 8 (≥ 5), so 5281 → 5,300.

Rounding off 3491 to its nearest hundreds: tens digit is 9 (≥ 5), so 3491 → 3,500.

Estimated product = 5,300 × 3,500 = 1,85,50,000.

Hence, the estimated product = 1,85,50,000.

(ii) 1387 × 888

Rounding off 1387 to its nearest hundreds: tens digit is 8 (≥ 5), so 1387 → 1,400.

Rounding off 888 to its nearest hundreds: tens digit is 8 (≥ 5), so 888 → 900.

Estimated product = 1,400 × 900 = 12,60,000.

Hence, the estimated product = 12,60,000.

Estimate the following quotients by rounding off each number to its nearest tens:

(i) 423 ÷ 29

(ii) 777 ÷ 27

Answer

(i) 423 ÷ 29

Rounding off 423 to its nearest tens: ones digit is 3 (< 5), so 423 → 420.

Rounding off 29 to its nearest tens: ones digit is 9 (≥ 5), so 29 → 30.

Estimated quotient = 420 ÷ 30 = 14.

Hence, the estimated quotient = 14.

(ii) 777 ÷ 27

Rounding off 777 to its nearest tens: ones digit is 7 (≥ 5), so 777 → 780.

Rounding off 27 to its nearest tens: ones digit is 7 (≥ 5), so 27 → 30.

Estimated quotient = 780 ÷ 30 = 26.

Hence, the estimated quotient = 26.

Estimate the following quotients by rounding off each number to its nearest hundreds:

(i) 2472 ÷ 493

(ii) 7459 ÷ 286

Answer

(i) 2472 ÷ 493

Rounding off 2472 to its nearest hundreds: tens digit is 7 (≥ 5), so 2472 → 2,500.

Rounding off 493 to its nearest hundreds: tens digit is 9 (≥ 5), so 493 → 500.

Estimated quotient = 2,500 ÷ 500 = 5.

Hence, the estimated quotient = 5.

(ii) 7459 ÷ 286

Rounding off 7459 to its nearest hundreds: tens digit is 5, so 7459 → 7,500.

Rounding off 286 to its nearest hundreds: tens digit is 8 (≥ 5), so 286 → 300.

Estimated quotient = 7,500 ÷ 300 = 25.

Hence, the estimated quotient = 25.

Fill in the blanks:

(i) The digit ... has the highest place value in the number 2309

(ii) The digit ... has the highest face value in the number 2039

(iii) The digit ... has the lowest place value in the number 2039

(iv) Both Indian and International systems of numeration have ... period in common.

(v) In the International system of numeration, commas are placed from ... after every ... digits.

(vi) The bigger number from the numbers 57,631 and 57,361 is ....

(vii) 1 crore = ... million

(viii) The smallest 4-digit number with 3 different digits is ...

(ix) The greatest 4-digit number with 3 different digits is ...

(x) 15 km 300 m = .... m

(xi) 7850 cm = ... m ... cm

(xii) The number 5079 when estimated to the nearest hundreds is ...

Answer

(i) In the number 2309, the digit 2 is at the thousands place with place value 2000, which is the highest.

Hence, the digit 2 has the highest place value in the number 2309.

(ii) In the number 2039, the digits are 2, 0, 3 and 9. The face value of a digit is the digit itself, and 9 is the largest.

Hence, the digit 9 has the highest face value in the number 2039.

(iii) In the number 2039, the digit 0 is at the hundreds place.

Place value of 0 = 0 x 100 = 0

Hence, the digit 0 has the lowest place value in the number 2039.

(iv) Both the Indian and the International systems of numeration have ones period in common.

Hence, both the Indian and International systems of numeration have ones period in common.

(v) In the International system of numeration, commas are placed from the right after every 3 digits.

Hence, commas are placed from the right after every 3 digits.

(vi) Comparing 57,631 and 57,361: 5 = 5, 7 = 7, 6 > 3. So 57,631 is bigger.

Hence, the bigger number is 57,631.

(vii) 1 crore = 10 million.

(viii) The smallest 4-digit number with 3 different digits is formed by placing the smallest non-zero digit (1) at the thousand's place, then 0, 0, 2 — using 3 different digits (1, 0, 2).

Hence, the smallest 4-digit number with 3 different digits is 1,002.

(ix) The greatest 4-digit number with 3 different digits is formed using 9, 9, 8, 7 — using 3 different digits (9, 8, 7).

Hence, the greatest 4-digit number with 3 different digits is 9,987.

(x) 15 km 300 m = (15 × 1,000 + 300) m = 15,300 m.

(xi) 7,850 cm = $\dfrac{7850}{100}$ m

= 78 m 50 cm.

(xii) In 5079, the tens digit is 7 (≥ 5). So we round up the hundreds digit (0 + 1 = 1) and replace the tens and ones digits by 0.

Hence, 5,079 rounded to the nearest hundreds is 5,100.

State whether the following statements are true (T) or false (F):

(i) The difference between the place value and the face of the digit 7 in the number 2701 is 693.

(ii) The smallest 4-digit number -1 = the greatest 3-digit number.

(iii) The place of a digit is independent of whether the number is written in the Indian system or International system of numeration.

(iv) In the International system, a number having less number of digits is always smaller than the number having more number of digits.

(v) The estimated value of 9999 to the nearest tens is 10000

Answer

(i) In 2701, the digit 7 is at the hundreds place.

Place value of 7 = 7 × 100 = 700.

Face value of 7 = 7.

Difference = 700 - 7 = 693.

Hence, the statement is True.

(ii) The smallest 4-digit number = 1,000.

The greatest 3-digit number = 999.

1,000 - 1 = 999.

Hence, the statement is True.

(iii) The position (place) of a digit in a number is the same in both the Indian and International systems; only the names of higher places differ between the two systems.

Hence, the statement is True.

(iv) When comparing two natural numbers, the number having more digits is always greater. Equivalently, the number having less digits is always smaller. This holds in both the Indian and International systems.

Hence, the statement is True.

(v) In 9999, the ones digit is 9 (≥ 5). So we increase the tens digit by 1 and replace the ones digit by 0. As the tens digit is 9, the increase carries over, eventually giving 10,000.

Hence, the statement is True.

The face value of the digit 5 in the number 36,503 is

1. 5

2. 503

3. 500

4. none of these

Answer

The face value of a digit in a number is the digit itself, regardless of the place it occupies.

So, the face value of 5 in 36,503 is 5.

Hence, option 1 is the correct option.

The difference between the place values of 6 and 3 in 76834 is

1. 3

2. 5700

3. 5930

4. 5970

Answer

In the number 76,834:

The digit 6 is at the thousands place. Place value of 6 = 6 × 1,000 = 6,000.

The digit 3 is at the tens place. Place value of 3 = 3 × 10 = 30.

Difference = 6,000 - 30 = 5,970.

Hence, option 4 is the correct option.

The sum of the place values of all the digits in 5003 is

1. 8

2. 53

3. 5003

4. 8000

Answer

In the number 5003:

Place value of 5 (thousands place) = 5 × 1,000 = 5,000.

Place value of 0 (hundreds place) = 0.

Place value of 0 (tens place) = 0.

Place value of 3 (ones place) = 3 × 1 = 3.

Sum = 5,000 + 0 + 0 + 3 = 5,003.

Hence, option 3 is the correct option.

The total number of 4-digit numbers is

1. 9000

2. 9999

3. 10000

4. none of these

Answer

The greatest 4-digit number = 9,999.

The smallest 4-digit number = 1,000.

Total number of 4-digit numbers = 9,999 - 1,000 + 1

= 9,000.

Hence, option 1 is the correct option.

The product of the place values of two-threes in 73532 is

1. 9000

2. 90000

3. 99000

4. 1000

Answer

In the number 73,532:

The first 3 (from the left) is at the thousands place. Place value = 3 × 1,000 = 3,000.

The second 3 is at the tens place. Place value = 3 × 10 = 30.

Product = 3,000 × 30 = 90,000.

Hence, option 2 is the correct option.

The smallest 4-digit number having distinct digits is

1. 1234

2. 1023

3. 1002

4. 3210

Answer

For the smallest 4-digit number with distinct digits, the thousand's place must be the smallest non-zero digit (1). Then the remaining digits in ascending order are 0, 2, 3.

So, the smallest 4-digit number with distinct digits = 1,023.

(Note: 1,002 has the digit 0 repeated, so its digits are not all distinct.)

Hence, option 2 is the correct option.

The largest 4-digit number having distinct digits is

1. 9999

2. 9867

3. 9786

4. 9876

Answer

For the largest 4-digit number with distinct digits, arrange the largest digits in descending order: 9, 8, 7, 6.

So, the largest 4-digit number with distinct digits = 9,876.

Hence, option 4 is the correct option.

The largest 4-digit number is

1. 9999

2. 9876

3. 9990

4. none of these

Answer

The largest 4-digit number is formed by placing the largest digit (9) in all four positions.

So, the largest 4-digit number = 9,999.

Hence, option 1 is the correct option.

The difference between the largest number of 3-digit and the largest number of 3-digit with distinct digits is

1. 0

2. 10

3. 12

4. 14

Answer

The largest 3-digit number = 999.

The largest 3-digit number with distinct digits = 987.

Difference = 999 - 987 = 12.

Hence, option 3 is the correct option.

If we write numbers from 1 to 100, the number of times the digit 5 has been written is

1. 11

2. 15

3. 19

4. 20

Answer

The digit 5 appears at the units place in: 5, 15, 25, 35, 45, 55, 65, 75, 85, 95 — that is, 10 times.

The digit 5 appears at the tens place in: 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 — that is, 10 times.

Total number of times the digit 5 is written = 10 + 10 = 20.

Hence, option 4 is the correct option.

The number 28,549 when rounded off to the nearest hundreds is

1. 28,000

2. 28,500

3. 28,600

4. 29,000

Answer

In 28,549, the digit at the tens place is 4. Since 4 is less than 5, we replace each of the digits at tens place and ones place by 0 and keep all other digits as they are.

Hence, option 2 is the correct option.

The smallest natural number which when rounded off to the nearest hundreds as 500 is

1. 499

2. 501

3. 450

4. 549

Answer

The natural numbers that round off to 500 when rounded to the nearest hundreds are those from 450 to 549.

The smallest such natural number is 450.

Hence, option 3 is the correct option.

The greatest natural number which when rounded off to the nearest hundreds as 500 is

1. 549

2. 599

3. 450

4. none of these

Answer

The natural numbers that round off to 500 when rounded to the nearest hundreds are those from 450 to 549.

The greatest such natural number is 549.

Hence, option 1 is the correct option.

The greatest 5-digit number formed by the digits 3, 0, 7 is

1. 33077

2. 77730

3. 77330

4. none of these

Answer

To form the greatest 5-digit number using the digits 3, 0 and 7 (with each digit used at least once and repetition allowed), use the largest digit (7) as many times as possible at the higher places, then the next largest digit, and so on.

Use 7 three times, then 3, then 0:

Greatest 5-digit number = 77,730.

Hence, option 2 is the correct option.

In the International place value system, we write 1 billion for

1. 10 lakh

2. 1 crore

3. 10 crore

4. 100 crore

Answer

In the International system, 1 billion = 1,000,000,000.

In the Indian system, 1 crore = 1,00,00,000 (8 digits) and 100 crore = 1,00,00,00,000 (10 digits).

So, 1 billion = 100 crore.

Hence, option 4 is the correct option.

Statement I: The place value of 6 in the numbers 126 and 621 is different.

Statement II: The place value of a non-zero digit depends upon the place it occupies in the given number.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Let us first consider Statement I.

In 126, the digit 6 is at the ones place. Place value of 6 = 6 × 1 = 6.

In 621, the digit 6 is at the hundreds place. Place value of 6 = 6 × 100 = 600.

Since 6 ≠ 600, the place values are different.

∴ Statement I is true.

Now let us consider Statement II.

The place value of a non-zero digit depends upon the place it occupies in the given number. This is a fundamental property of the place value system.

∴ Statement II is true.

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Statement I: Both the face value and the place value of 1 in 531 is 1.

Statement II: The face value of a digit in a number is the digit itself.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Let us first consider Statement I.

In 531, the digit 1 is at the ones place.

Face value of 1 = 1.

Place value of 1 = 1 × 1 = 1.

So both the face value and the place value of 1 are 1.

∴ Statement I is true.

Now let us consider Statement II.

The face value of a digit in a number is the digit itself, regardless of the place it occupies.

∴ Statement II is true.

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Statement I: 4560 < 1234

Statement II: Given two numbers, the number having more digits is lesser.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Let us first consider Statement I.

Both 4560 and 1234 have 4 digits. Comparing the leftmost digit: 4 > 1. So 4560 > 1234.

∴ Statement I is false.

Now let us consider Statement II.

Given two numbers, the number having more digits is greater (not lesser).

∴ Statement II is false.

Thus, both Statement I and Statement II are false.

Hence, option 4 is the correct option.

Statement I: The largest five-digit number is 99999.

Statement II: Ten million = one lakh

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Let us first consider Statement I.

The largest 5-digit number is formed by placing the largest digit (9) in all five positions.

So the largest 5-digit number = 99,999.

∴ Statement I is true.

Now let us consider Statement II.

Ten million = 10,000,000 in International system = 1,00,00,000 (which is 1 crore) in Indian system.

One lakh = 1,00,000.

So, ten million ≠ one lakh. (In fact, ten million = 1 crore.)

∴ Statement II is false.

Thus, Statement I is true but Statement II is false.

Hence, option 1 is the correct option.

Statement I: The estimated value of 2132 + 1234 by estimating the numbers to the nearest hundred = 3300.

Statement II: Sum of two natural numbers is always a natural number.

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Let us first consider Statement I.

Rounding off 2132 to the nearest hundreds: tens digit is 3 (< 5), so 2132 → 2,100.

Rounding off 1234 to the nearest hundreds: tens digit is 3 (< 5), so 1234 → 1,200.

Estimated sum = 2,100 + 1,200 = 3,300.

∴ Statement I is true.

Now let us consider Statement II.

The sum of two natural numbers is always a natural number (closure property of natural numbers under addition).

∴ Statement II is true.

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Write the numeral for each of the following numbers and insert commas correctly:

(i) Six crore nine lakh forty seven.

(ii) One hundred four million seven hundred twenty two thousand three hundred ninety four.

Answer

(i) Six crore nine lakh forty seven.

The number consists of 6 crores, 9 lakhs, 0 thousands, 0 hundreds, 4 tens and 7 ones.

In figures (Indian system): 6,09,00,047

(ii) One hundred four million seven hundred twenty two thousand three hundred ninety four.

The number consists of 104 millions, 722 thousands and 394 ones.

In figures (International system): 104,722,394

Insert commas suitably and write the number 30189301 in words in Indian and International system of numeration.

Answer

In the Indian system: 3,01,89,301

In words (Indian system): Three crore one lakh eighty nine thousand three hundred one.

In the International system: 30,189,301

In words (International system): Thirty million one hundred eighty nine thousand three hundred one.

Find the difference between the place value and the face value of the digit 6 in the number 72601

Answer

In the Indian system, the given number can be written as 72,601.

Counting from the right in 72,601:

1 is in the ones place.

0 is in the tens place.

6 is in the hundreds place.

2 is in the thousands place.

7 is in the ten thousands place.

The face value of 6 = 6.

The place value of 6 = 6 × 100 = 600.

Difference = Place value - Face value

= 600 - 6

= 594

Hence, the difference between the place value and the face value of the digit 6 in 72,601 is 594.

Write all possible two-digit numbers using the digits 4 and 0. Repetition of digits is allowed.

Answer

We are required to write 2-digit numbers using the digits 4 and 0, where repetition of digits is allowed.

The digit 0 cannot be at the ten's place because that would make the number only 1-digited.

Keeping 4 at ten's place, the 2-digit numbers are 40 and 44.

Hence, all possible 2-digit numbers are 40 and 44.

Write all possible natural numbers using the digits 7, 0, 6. Repetition of digits is not allowed.

Answer

We are required to write all possible natural numbers using the digits 7, 0 and 6, where repetition of digits is not allowed.

1-digit natural numbers: 6, 7. (0 is not a natural number.)

2-digit numbers: The digit 0 cannot be at the ten's place.

Keeping 6 at ten's place: 60, 67.

Keeping 7 at ten's place: 70, 76.

2-digit numbers: 60, 67, 70, 76.

3-digit numbers: The digit 0 cannot be at the hundred's place.

Keeping 6 at hundred's place: 607, 670.

Keeping 7 at hundred's place: 706, 760.

3-digit numbers: 607, 670, 706, 760.

Hence, all possible natural numbers are: 6, 7, 60, 67, 70, 76, 607, 670, 706 and 760.

Arrange the following numbers in ascending order:

3706, 58019, 3760, 59801, 560023

Answer

Given, 3706, 58019, 3760, 59801, 560023

The number of digits in each number:

3706, 3760 → 4 digits

58019, 59801 → 5 digits

560023 → 6 digits

The 4-digit numbers are smaller than the 5-digit numbers, which are smaller than the 6-digit number.

Comparing 3706 and 3760: 3 = 3, 7 = 7, 0 < 6. So 3706 < 3760.

Comparing 58019 and 59801: 5 = 5, 8 < 9. So 58019 < 59801.

Hence, the ascending order is: 3,706 < 3,760 < 58,019 < 59,801 < 5,60,023.

Write the greatest six-digit number using four different digits.

Answer

For the greatest 6-digit number using exactly four different digits, the digits at the higher places should be as large as possible.

Use the digit 9 as many times as possible. With 4 different digits in 6 positions, use 9 three times and the next three largest different digits 8, 7, 6 once each.

Total digits used: 9, 9, 9, 8, 7, 6 — four different digits (9, 8, 7, 6).

Arranging in descending order: 9, 9, 9, 8, 7, 6.

Hence, the greatest 6-digit number using four different digits is 9,99,876.

Write the smallest eight-digit number using four different digits.

Answer

For the smallest 8-digit number using exactly four different digits, the digit at the highest place (crores) cannot be 0. So, the smallest non-zero digit (1) is placed at the highest place.

Use 0 in as many remaining positions as possible. To have exactly four different digits in total, we need two more different digits besides 1 and 0. The smallest such digits are 2 and 3.

Total digits used: 1, 0, 0, 0, 0, 0, 2, 3 — four different digits (1, 0, 2, 3).

Arranging the digits with 1 at the highest place and the rest in ascending order:

1, 0, 0, 0, 0, 0, 2, 3.

Hence, the smallest 8-digit number using four different digits is 1,00,00,023.

Find the difference between the greatest and the smallest 4-digit numbers formed by the digits 0, 3, 6, 9

Answer

The given digits are 0, 3, 6 and 9, and each digit is used only once.

For the greatest 4-digit number, arrange the digits in descending order: 9, 6, 3, 0.

Greatest 4-digit number = 9,630.

For the smallest 4-digit number, the thousand's place cannot be 0. So, place the smallest non-zero digit (3) at the thousand's place, then arrange the remaining digits (0, 6, 9) in ascending order.

Smallest 4-digit number = 3,069.

Difference = 9,630 - 3,069

= 6,561

Hence, the difference between the greatest and the smallest 4-digit numbers is 6,561.

Find the sum of the greatest and the smallest 6-digit numbers formed by the digits 2, 0, 4, 7, 6, 5 using each digit only once.

Answer

The given digits are 2, 0, 4, 7, 6 and 5, and each digit is used only once.

For the greatest 6-digit number, arrange the digits in descending order: 7, 6, 5, 4, 2, 0.

Greatest 6-digit number = 7,65,420.

For the smallest 6-digit number, the lakh's place cannot be 0. So, place the smallest non-zero digit (2) at the lakh's place, then arrange the remaining digits (0, 4, 5, 6, 7) in ascending order.

Smallest 6-digit number = 2,04,567.

Sum = 7,65,420 + 2,04,567

= 9,69,987

Hence, the sum of the greatest and the smallest 6-digit numbers is 9,69,987.

Find the sum of the four-digit greatest number and the five-digit smallest number, each number having three different digits.

Answer

For the greatest 4-digit number with three different digits:

Use 9 as many times as possible. Use 9 twice, then the next two largest different digits are 8 and 7.

Greatest 4-digit number = 9,987.

For the smallest 5-digit number with three different digits:

The ten thousand's place cannot be 0. Place 1 (smallest non-zero) at the ten thousand's place. Then 0s in remaining places. To have exactly three different digits, use one more digit (the smallest available, which is 2).

Smallest 5-digit number = 10,002.

Sum = 9,987 + 10,002

= 19,989

Hence, the required sum is 19,989.

Write the greatest and the smallest four-digit numbers using four different digits with the conditions:

(i) Digit 3 always at hundred's place.

(ii) Digit 0 always at ten's place.

Answer

(i) Digit 3 always at hundred's place.

The number is of the form _ 3 _ _ (with all four digits different).

For the greatest 4-digit number, fill the remaining places with the largest distinct digits other than 3. Use 9 at thousands, 8 at tens and 7 at units.

Greatest 4-digit number = 9387.

For the smallest 4-digit number, the thousand's place cannot be 0. Place 1 (smallest non-zero, ≠ 3) at thousands, 3 at hundreds, 0 at tens and 2 at units.

Smallest 4-digit number = 1302.

Hence, the greatest 4-digit number is 9,387 and the smallest 4-digit number is 1,302.

(ii) Digit 0 always at ten's place.

The number is of the form _ _ 0 _ (with all four digits different).

For the greatest 4-digit number, fill the remaining places with the largest distinct digits other than 0. Use 9 at thousands, 8 at hundreds and 7 at units.

Greatest 4-digit number = 9807.

For the smallest 4-digit number, the thousand's place cannot be 0. Place 1 (smallest non-zero) at thousands, 2 at hundreds, 0 at tens and 3 at units.

Smallest 4-digit number = 1203.

Hence, the greatest 4-digit number is 9,807 and the smallest 4-digit number is 1,203.

Find the greatest and the smallest 5-digit numbers using different digits in which 5 appears at ten's place.

Answer

The number is a 5-digit number with all five digits different and 5 at the ten's place.

The number is of the form _ _ _ 5 _.

For the greatest 5-digit number, fill the remaining places with the largest distinct digits other than 5. Use 9 at ten thousands, 8 at thousands, 7 at hundreds and 6 at units.

Greatest 5-digit number = 98,756.

For the smallest 5-digit number, the ten thousand's place cannot be 0. Place 1 (smallest non-zero, ≠ 5) at ten thousands, 0 at thousands, 2 at hundreds, 5 at tens and 3 at units.

Smallest 5-digit number = 10,253.

Hence, the greatest 5-digit number is 98,756 and the smallest 5-digit number is 10,253.

A mobile number consists of ten digits. First four digits are 9, 9, 7 and 9. Make the smallest mobile number by using only one digit twice from the digits 8, 3, 5, 0, 6

Answer

The first four digits of the mobile number are 9, 9, 7 and 9.

So, the remaining six digits are to be filled using the digits 8, 3, 5, 0 and 6 (five digits), with one digit used twice (so total six digits).

For the smallest mobile number, use the smallest digit twice. So, use 0 twice. The other digits are 3, 5, 6 and 8.

Total six digits to be placed: 0, 0, 3, 5, 6, 8.

Arrange these six digits in ascending order: 0, 0, 3, 5, 6, 8.

Mobile number = 9 9 7 9 0 0 3 5 6 8

Hence, the smallest mobile number is 9979003568.

To stitch a uniform, 1 m 75 cm cloth is needed. Out of 153 m cloth, how many uniforms can be stitched and how much cloth will remain?

Answer

Length of cloth available = 153 m

= (153 × 100) cm

= 15,300 cm

Length of cloth required to stitch one uniform = 1 m 75 cm

= (1 × 100 + 75) cm

= 175 cm

Dividing 15,300 by 175:

Quotient = 87, remainder = 75.

So, 87 uniforms can be stitched and 75 cm of cloth will remain.

Hence, the number of uniforms that can be stitched is 87 and the length of the remaining cloth is 75 cm.

Medicine is packed in boxes, each weighing 4 kg 500 g. How many such boxes can be loaded in a van which cannot carry beyond 800 kg?

Answer

Mass of each box = 4 kg 500 g

= (4 × 1,000 + 500) g

= 4,500 g

Maximum mass the van can carry = 800 kg

= (800 × 1,000) g

= 8,00,000 g

Number of boxes that can be loaded = $\dfrac{\text{Maximum mass}}{\text{Mass of each box}}$

= $\dfrac{8{,}00{,}000}{4{,}500}$

Dividing 8,00,000 by 4,500:

Quotient = 177, remainder = 3,500.

So, the maximum number of boxes that can be loaded is 177 (since 178 boxes would weigh 178 × 4,500 = 8,01,000 g = 801 kg, which is more than 800 kg).

Hence, the number of boxes that can be loaded in the van is 177.

Estimate: 6554 - 677 by estimating the numbers to their nearest

(i) thousands

(ii) hundreds

(iii) greatest places

Also point out the most reasonable estimate.

Answer

The given numbers are 6554 and 677.

(i) Estimating the given numbers to their nearest thousands:

6554: hundreds digit is 5, so 6554 → 7,000.

677: hundreds digit is 6 (≥ 5), so 677 → 1,000.

Estimated difference = 7,000 - 1,000 = 6,000.

(ii) Estimating the given numbers to their nearest hundreds:

6554: tens digit is 5, so 6554 → 6,600.

677: tens digit is 7 (≥ 5), so 677 → 700.

Estimated difference = 6,600 - 700 = 5,900.

(iii) Estimating the given numbers to their greatest places:

The greatest place of 6554 is thousands. So 6554 → 7,000 (hundreds digit 5).

The greatest place of 677 is hundreds. So 677 → 700 (tens digit 7 ≥ 5).

Estimated difference = 7,000 - 700 = 6,300.

Actual difference = 6,554 - 677 = 5,877.

Comparing the estimates with the actual difference (5,877), the estimate to the nearest hundreds (5,900) is closest to the actual difference.

Hence, the estimate to the nearest hundreds (5,900) is the most reasonable estimate.

Write all 4-digit numbers that can be formed with the digits 2 and 5, using both digits equal number of time. Also find their sum.

Answer

We are required to write 4-digit numbers using only the digits 2 and 5, with each digit used the same number of times. So each digit is used twice.

The digits to be arranged are 2, 2, 5 and 5.

The possible 4-digit numbers are:

2255, 2525, 2552, 5225, 5252, 5522.

Sum = 2255 + 2525 + 2552 + 5225 + 5252 + 5522

= 23,331

Hence, the required 4-digit numbers are 2255, 2525, 2552, 5225, 5252 and 5522, and their sum is 23,331.

What is the difference between the smallest 6-digit number with five different digits and the greatest 5-digit number with four different digits?

Answer

For the smallest 6-digit number with five different digits:

The lakh's place cannot be 0. Place 1 (smallest non-zero) at the lakh's place. Use 0 in as many remaining positions as possible. To have exactly five different digits, we need three more digits besides 1 and 0. The smallest such digits are 2, 3 and 4.

Total six digits: 1, 0, 0, 2, 3, 4 — five different digits (1, 0, 2, 3, 4).

Arranging with 1 at the lakh's place and the rest in ascending order:

Smallest 6-digit number = 1,00,234.

For the greatest 5-digit number with four different digits:

Use 9 as many times as possible. Use 9 twice, then the next three largest different digits are 8, 7 and 6.

Total five digits: 9, 9, 8, 7, 6 — four different digits (9, 8, 7, 6).

Arranging in descending order:

Greatest 5-digit number = 99,876.

Difference = 1,00,234 - 99,876

= 358

Hence, the required difference is 358.

Write the smallest 7-digit number using all the even digits.

Answer

The even digits are 0, 2, 4, 6 and 8 (five different digits).

To form a 7-digit number using all five even digits (each at least once), we need two extra positions, which can be filled by repeating any of the even digits.

The digit at the highest place (ten lakhs) cannot be 0. So, place the smallest non-zero even digit (2) at the highest place.

To make the number smallest, fill as many of the remaining positions as possible with 0 (smallest digit).

Total seven digits: 2, 0, 0, 0, 4, 6, 8 — uses all five even digits (0, 2, 4, 6, 8).

Arranging with 2 at the highest place and the rest in ascending order:

Smallest 7-digit number = 20,00,468.

Hence, the smallest 7-digit number using all the even digits is 20,00,468.

How many times does the digit 3 occur at ten's place in natural numbers from 100 to 1000?

Answer

We need to count natural numbers from 100 to 1000 in which the digit at the ten's place is 3.

Such numbers are of the form 3 (a 3-digit number with 3 at the ten's place), as the number 1000 has 0 at its ten's place.

For the hundred's place, the digit can be any of 1, 2, 3, 4, 5, 6, 7, 8 or 9 (9 choices).

For the ten's place, the digit is fixed as 3 (1 choice).

For the unit's place, the digit can be any of 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 (10 choices).

Total such numbers = 9 × 1 × 10

= 90

Hence, the digit 3 occurs at the ten's place 90 times in natural numbers from 100 to 1000.