Find the difference in the place values of the two sevens in the number 8,72,574.
Answer
In the number 8,72,574 :
One 7 occurs at the ten-thousands place, so its place value = 7 × 10000 = 70000.
The other 7 occurs at the tens place, so its place value = 7 × 10 = 70.
Required difference = 70000 - 70 = 69930.
Hence, the difference in the place values of the two sevens = 69930.
Find the difference between the face value of 9 and the face value of 4 in the number 324798.
Answer
The face value of a digit is the digit itself, irrespective of its position in the number.
Face value of 9 = 9.
Face value of 4 = 4.
Required difference = 9 - 4 = 5.
Hence, the difference between the face values of 9 and 4 = 5.
By making a suitable chart, compare:
(i) 540276 and 369998
(ii) 6983245 and 6893254
Answer
(i) Writing the numbers in a place value chart :
| Number | L | TTh | Th | H | T | O |
|---|---|---|---|---|---|---|
| 540276 | 5 | 4 | 0 | 2 | 7 | 6 |
| 369998 | 3 | 6 | 9 | 9 | 9 | 8 |
Both numbers have an equal number of digits. Comparing the digits at the leftmost (lakhs) place, the first number is 5 and the second number is 3.
Since 5 > 3,
Hence, 540276 > 369998.
(ii) Writing the numbers in a place value chart :
| Number | TL | L | TTh | Th | H | T | O |
|---|---|---|---|---|---|---|---|
| 6983245 | 6 | 9 | 8 | 3 | 2 | 4 | 5 |
| 6893254 | 6 | 8 | 9 | 3 | 2 | 5 | 4 |
Both numbers have an equal number of digits. At the leftmost (ten-lakhs) place, both have the same digit 6. At the next (lakhs) place, the first number is 9 and the second number is 8.
Since 9 > 8,
Hence, 6983245 > 6893254.
Compare the numbers written in the following table by writing them in ascending order:
| C | TL | L | TTh | Th | H | T | O |
|---|---|---|---|---|---|---|---|
| 5 | 4 | 3 | 2 | 9 | 7 | 2 | |
| 2 | 3 | 1 | 0 | 6 | 2 | 9 | 3 |
| 5 | 2 | 2 | 3 | 7 | 9 | 1 | |
| 2 | 3 | 1 | 8 | 2 | 6 | 3 | 4 |
| 5 | 4 | 3 | 4 | 4 | 7 | 8 | 2 |
Answer
The numbers represented by the rows of the chart are :
5432972, 23106293, 5223791, 23182634 and 54344782.
The numbers 5432972 and 5223791 have 7 digits each; comparing them, at the lakhs place 5223791 has 2 and 5432972 has 4, so 5223791 < 5432972.
The numbers 23106293 and 23182634 have 8 digits each; comparing them, the first differing digit (ten-thousands place) is 0 in 23106293 and 8 in 23182634, so 23106293 < 23182634.
The number 54344782 has the most digits (8 digits) with the greatest leftmost digit, so it is the greatest.
Arranging in ascending order :
Hence, 5223791 < 5432972 < 23106293 < 23182634 < 54344782.
Use a place value chart to compare the given numbers and write them in descending order: 543287 ; 5482900 ; 2732940 ; 43877 ; 78396 and 4999.
Answer
Writing the numbers in a place value chart :
| Number | TL | L | TTh | Th | H | T | O |
|---|---|---|---|---|---|---|---|
| 543287 | 5 | 4 | 3 | 2 | 8 | 7 | |
| 5482900 | 5 | 4 | 8 | 2 | 9 | 0 | 0 |
| 2732940 | 2 | 7 | 3 | 2 | 9 | 4 | 0 |
| 43877 | 4 | 3 | 8 | 7 | 7 | ||
| 78396 | 7 | 8 | 3 | 9 | 6 | ||
| 4999 | 4 | 9 | 9 | 9 |
The numbers 5482900 and 2732940 have 7 digits each (the most digits); comparing them at the ten-lakhs place, 5 > 2, so 5482900 is the greatest and 2732940 is the next greatest.
The number 543287 has 6 digits, so it is next.
The numbers 78396 and 43877 have 5 digits each; comparing them at the ten-thousands place, 7 > 4, so 78396 > 43877.
The number 4999 has the least number of digits (4 digits), so it is the smallest.
Arranging in descending order :
Hence, 54,82,900 > 27,32,940 > 5,43,287 > 78,396 > 43,877 > 4,999.
Find the smallest and the greatest numbers in each of the cases given below:
(i) 983, 5754, 84 and 5942
(ii) 32849, 53628, 5499 and 54909
Answer
(i) Among 983, 5754, 84 and 5942 :
84 has the least number of digits (2 digits), so it is the smallest.
5754 and 5942 have 4 digits each; comparing them at the hundreds place, 9 > 7, so 5942 is the greatest.
Hence, smallest number = 84 and greatest number = 5942.
(ii) Among 32849, 53628, 5499 and 54909 :
5499 has the least number of digits (4 digits), so it is the smallest.
The numbers 32849, 53628 and 54909 have 5 digits each. Comparing 53628 and 54909 at the thousands place, 4 > 3, so 54909 is the greatest.
Hence, smallest number = 5499 and greatest number = 54909.
Form the greatest and the smallest 4-digit numbers using the given digits, without repetition:
(i) 3, 7, 2 and 5
(ii) 6, 1, 4 and 9
(iii) 7, 0, 4 and 2
(iv) 1, 8, 5 and 3
(v) 9, 6, 0 and 7
Answer
To form the greatest number, write the digits in descending order. To form the smallest number, write the digits in ascending order (without placing 0 at the leftmost place).
(i) Digits : 3, 7, 2 and 5.
Greatest number = 7532 and smallest number = 2357.
Hence, greatest = 7532 and smallest = 2357.
(ii) Digits : 6, 1, 4 and 9.
Greatest number = 9641 and smallest number = 1469.
Hence, greatest = 9641 and smallest = 1469.
(iii) Digits : 7, 0, 4 and 2.
Greatest number = 7420 and smallest number = 2047 (0 is not placed at the leftmost place).
Hence, greatest = 7420 and smallest = 2047.
(iv) Digits : 1, 8, 5 and 3.
Greatest number = 8531 and smallest number = 1358.
Hence, greatest = 8531 and smallest = 1358.
(v) Digits : 9, 6, 0 and 7.
Greatest number = 9760 and smallest number = 6079 (0 is not placed at the leftmost place).
Hence, greatest = 9760 and smallest = 6079.
Form the greatest and the smallest 4-digit numbers using any four different digits, with the condition that digit 5 is always at the ten's place.
Answer
The digit 5 is fixed at the tens place, so the number is of the form _ _ 5 _.
To form the greatest number, fill the remaining places (from the left) with the greatest possible different digits : 9, 8 and 7.
Greatest number = 9 8 5 7 = 9857.
To form the smallest number, fill the remaining places (from the left) with the smallest possible different digits, keeping 0 out of the leftmost place : 1, 0 and 2.
Smallest number = 1 0 5 2 = 1052.
Hence, greatest number = 9857 and smallest number = 1052.
Form the largest number with the digits 2, 3, 5, 9, 6 and 0, without repetition of any digit.
Answer
To form the largest number, write the given digits in descending order of their values.
Arranging 2, 3, 5, 9, 6 and 0 in descending order : 9, 6, 5, 3, 2, 0.
Largest number = 965320.
Hence, the largest number = 965320.
Write the smallest and the greatest 4-digit numbers, without repetition of any digit.
Answer
To form the smallest 4-digit number with different digits, use the smallest digits 1, 0, 2 and 3, keeping 0 out of the leftmost place :
Smallest number = 1023.
To form the greatest 4-digit number with different digits, use the greatest digits 9, 8, 7 and 6 in descending order :
Greatest number = 9876.
Hence, smallest number = 1023 and greatest number = 9876.
Find the sum of the largest and the smallest four-digit numbers.
Answer
The largest four-digit number = 9999.
The smallest four-digit number = 1000.
Required sum = 9999 + 1000 = 10999.
Hence, the sum of the largest and the smallest four-digit numbers = 10999.
Form the greatest and the smallest five-digit numbers with 8 in the hundred's place and with all the different digits .
Answer
The digit 8 is fixed at the hundreds place, so the number is of the form _ _ 8 _ _.
To form the greatest number, fill the remaining places (from the left) with the greatest different digits : 9, 7, 6 and 5.
Greatest number = 9 7 8 6 5.
To form the smallest number, fill the remaining places (from the left) with the smallest different digits, keeping 0 out of the leftmost place : 1, 0, 2 and 3.
Smallest number = 1 0 8 2 3.
Hence, greatest number = 97865 and smallest number = 10823.
(i) How many four-digit numbers are there between 999 and 3000?
(ii) How many four-digit numbers are there between 99 and 3000?
Answer
(i) The four-digit numbers between 999 and 3000 start from 1000 and end at 2999.
Number of such numbers = 2999 - 1000 + 1 = 2000.
Hence, there are 2000 four-digit numbers between 999 and 3000.
(ii) Every four-digit number is greater than 99, so the four-digit numbers between 99 and 3000 are the same as those from 1000 to 2999.
Number of such numbers = 2999 - 1000 + 1 = 2000.
Hence, there are 2000 four-digit numbers between 99 and 3000.
Form the greatest and the smallest 4-digit numbers using the digits 5, 4, 7 and 9 (without repeating the digits) and with the condition that:
(i) 7 is at the ones place.
(ii) 9 is at the tens place.
(iii) 4 is at the hundreds place.
Answer
(i) The digit 7 is fixed at the ones place, so the number is of the form _ _ _ 7.
Using the remaining digits 5, 4 and 9 :
Greatest number = 9 5 4 7 = 9547 (remaining digits in descending order).
Smallest number = 4 5 9 7 = 4597 (remaining digits in ascending order).
Hence, greatest number = 9547 and smallest number = 4597.
(ii) The digit 9 is fixed at the tens place, so the number is of the form _ _ 9 _.
Using the remaining digits 5, 4 and 7 :
Greatest number = 7 5 9 4 = 7594.
Smallest number = 4 5 9 7 = 4597.
Hence, greatest number = 7594 and smallest number = 4597.
(iii) The digit 4 is fixed at the hundreds place, so the number is of the form _ 4 _ _.
Using the remaining digits 5, 7 and 9 :
Greatest number = 9 4 7 5 = 9475.
Smallest number = 5 4 7 9 = 5479.
Hence, greatest number = 9475 and smallest number = 5479.