Number System
Concise Mathematics Selina — Exercises 1(A) to 1(D), objective questions, statement questions and assertion–reason questions explained step by step.
Place Value and Number Formation
14 fully explained questions
Find the difference in the place values of the two sevens in the number 8,72,574.
In the number 8,72,574 :
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.
The face value of a digit is the digit itself, irrespective of its position in the number.
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
(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 |
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.
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
(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
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.
Hence, greatest = 7532 and smallest = 2357.
(ii) Digits : 6, 1, 4 and 9.
Hence, greatest = 9641 and smallest = 1469.
(iii) Digits : 7, 0, 4 and 2.
Hence, greatest = 7420 and smallest = 2047.
(iv) Digits : 1, 8, 5 and 3.
Hence, greatest = 8531 and smallest = 1358.
(v) Digits : 9, 6, 0 and 7.
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 tens place.
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.
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.
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.
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.
Hence, the largest number = 965320.
Write the smallest and the greatest 4-digit numbers, without repetition of any digit.
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 :
To form the greatest 4-digit number with different digits, use the greatest digits 9, 8, 7 and 6 in descending order :
Hence, smallest number = 1023 and greatest number = 9876.
Find the sum of the largest and the smallest four-digit numbers.
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 hundreds place and with all the different digits .
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.
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.
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?
(i) The four-digit numbers between 999 and 3000 start from 1000 and end at 2999.
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.
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.
(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 :
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 :
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 :
Hence, greatest number = 9475 and smallest number = 5479.
Operations and Word Problems
15 fully explained questions
Population of a city was 3,54,976 in the year 2014. In the year 2015, it increased by 68,438. What was the population of the city at the end of the year 2015?
Hence, the population of the city at the end of the year 2015 = 4,23,414.
A = 7,43,000 and B = 8,00,100. Which is greater A or B? And by how much?
Both A and B have an equal number of digits. Comparing them at the lakhs place, B has 8 and A has 7.
Since 8 > 7, B is greater than A.
Hence, B is greater than A by 57,100.
A notebook has 56 pages. How many pages will 5326 such notebooks have?
Hence, 5326 notebooks will have 2,98,256 pages.
A person has 75,000 sheets of paper. If each sheet makes 8 pages of a notebook, how many notebooks of 200 pages can be made using the above sheets?
Hence, 3000 notebooks of 200 pages can be made.
Add 1,76,209 ; 4,50,923 and 44,83,947.
Hence, the required sum = 51,11,079.
A cricket player has scored 7,849 runs so far in test matches. He wishes to complete 10,000 runs; how many more runs does he need?
Hence, the player needs 2151 more runs.
In an election two candidates A and B are the only contestants. If candidate A scored 9,32,567 votes and candidate B scored 9,00,235 votes, by how much margin did A win or lose the election?
Since 9,32,567 > 9,00,235, candidate A won the election.
Hence, A won the election by 32332 votes.
Find the difference between the largest and the smallest five digit numbers that can be formed using the digits 5, 1, 6, 3 and 2 without repeating any digit.
Hence, the required difference = 52965.
A man has ₹ 1,57,184 with him. He placed an order for purchasing 80 articles at ₹ 125 each. How much money will remain with him after the purchase?
Hence, ₹ 1,47,184 will remain with him after the purchase.
A student multiplied 8,035 by 87 instead of multiplying by 78. By how much was his answer greater than or less than the correct answer?
Since 87 > 78, the student's answer was greater than the correct answer.
Hence, his answer was greater than the correct answer by 72,315.
Mohani has 30 m cloth and she wants to make some shirts for her son. If each shirt requires 2 m 30 cm cloth, how many shirts, in all, can be made and how much length of the cloth will be left?
So, 13 shirts can be made and 10 cm of cloth is left as remainder.
Hence, 13 shirts can be made and 10 cm of cloth will be left.
The distance between two places A and B is 3 km 760 m. A boy travels from A to B and then B to A every day. How much distance does he travel in 8 days?
Hence, the boy travels 60.160 km in 8 days.
An oil-tin contains 6 litre 60 ml oil. How many identical bottles can this oil fill, if the capacity of each bottle is 30 ml?
Hence, 202 identical bottles can be filled.
The sale receipt of a company in a certain year was ₹ 83,73,540. In the following year, it decreased by ₹ 7,84,670.
(i) What was the sale receipt of the company during the following year?
(ii) What was the total sale receipt of the company during these two years?
Hence, the sale receipt during the following year = ₹ 75,88,870.
Hence, the total sale receipt during these two years = ₹ 1,59,62,410.
A number exceeds 8,59,470 by 3,00,999. What is the number?
Hence, the required number = 11,60,469.
Rounding Off Numbers
10 fully explained questions
Round off (approximate) each of the following to the nearest ten:
(i) 62
(ii) 265
(iii) 543
(iv) 6296
To round off to the nearest ten, look at the ones digit. If it is less than 5, replace it by 0; if it is 5 or more, increase the tens digit by 1 and replace the ones digit by 0.
Round off (approximate) each of the following to the nearest hundred:
(i) 748
(ii) 784
(iii) 6998
(iv) 59997
To round off to the nearest hundred, look at the tens digit. If it is less than 5, replace the tens and ones digits by 0; if it is 5 or more, increase the hundreds digit by 1 and replace the tens and ones digits by 0.
Round off (approximate) each of the following to the nearest thousand:
(i) 6475
(ii) 6732
(iii) 9948
(iv) 89994
To round off to the nearest thousand, look at the hundreds digit. If it is less than 5, replace the hundreds, tens and ones digits by 0; if it is 5 or more, increase the thousands digit by 1 and replace the digits on its right by 0.
Round off (approximate):
(i) 578 to the nearest ten.
(ii) 578 to the nearest hundred.
(iii) 4327 to the nearest thousand.
(iv) 32974 to the nearest ten thousand.
(v) 27487 to the nearest ten thousand.
(i) To round off 578 to the nearest ten, look at the ones digit. It is 8 (more than 5), so increase the tens digit by 1 and replace the ones digit by 0.
Hence, 578 to the nearest ten = 580.
(ii) To round off 578 to the nearest hundred, look at the tens digit. It is 7 (more than 5), so increase the hundreds digit by 1 and replace the tens and ones digits by 0.
Hence, 578 to the nearest hundred = 600.
(iii) To round off 4327 to the nearest thousand, look at the hundreds digit. It is 3 (less than 5), so replace the hundreds, tens and ones digits by 0 and keep the other digit as it is.
Hence, 4327 to the nearest thousand = 4000.
(iv) To round off 32974 to the nearest ten thousand, look at the thousands digit. It is 2 (less than 5), so replace the thousands, hundreds, tens and ones digits by 0 and keep the other digit as it is.
Hence, 32974 to the nearest ten thousand = 30000.
(v) To round off 27487 to the nearest ten thousand, look at the thousands digit. It is 7 (more than 5), so increase the ten thousands digit by 1 and replace the digits on its right by 0.
Hence, 27487 to the nearest ten thousand = 30000.
Round off (approximate) each of the following to the nearest ten, nearest hundred and nearest thousand.
(i) 864
(ii) 1249
(iii) 54,547
(iv) 68,076
(v) 56,293
(vi) 7,293
(vii) 89,24,379
For the nearest ten, look at the ones digit; for the nearest hundred, look at the tens digit; for the nearest thousand, look at the hundreds digit. If that digit is less than 5, replace it (and the digits on its right) by 0; if it is 5 or more, increase the digit on its left by 1 and replace the digits on its right by 0.
(i) 864 :
∴ 860, 900 and 1000.
(ii) 1249 :
∴ 1250, 1200 and 1000.
(iii) 54,547 :
∴ 54550, 54500 and 55000.
(iv) 68,076 :
∴ 68080, 68100 and 68000.
(v) 56,293 :
∴ 56290, 56300 and 56000.
(vi) 7,293 :
∴ 7290, 7300 and 7000.
(vii) 89,24,379 :
∴ 8924380, 8924400 and 8924000.
Approximate each of the following to the nearest ten:
(i) ₹ 562
(ii) 837 m
(iii) 545 cm
(iv) ₹ 27
To round off to the nearest ten, look at the ones digit. If it is less than 5, replace it by 0; if it is 5 or more, increase the tens digit by 1 and replace the ones digit by 0.
List all the natural numbers which can be rounded off to 30.
A number rounds off to 30 (to the nearest ten) when its ones digit decides whether it moves to 30 or away from it.
The numbers from 25 to 29 have ones digit 5 or more, so they round up to 30.
The numbers from 30 to 34 have ones digit less than 5, so they round down to 30.
Hence, the required numbers are 25, 26, 27, 28, 29, 30, 31, 32, 33 and 34.
List all the whole numbers which are approximate to 50.
A number rounds off to 50 (to the nearest ten) depending on its ones digit.
The numbers from 45 to 49 have ones digit 5 or more, so they round up to 50.
The numbers from 50 to 54 have ones digit less than 5, so they round down to 50.
Hence, the required numbers are 45, 46, 47, 48, 49, 50, 51, 52, 53 and 54.
Write the smallest and the largest numbers which are rounded off to 90.
A number rounds off to 90 (to the nearest ten) depending on its ones digit.
The numbers from 85 to 89 have ones digit 5 or more, so they round up to 90.
The numbers from 90 to 94 have ones digit less than 5, so they round down to 90.
So, the numbers that round off to 90 lie from 85 up to 94.
Hence, the smallest number is 85 and the largest number is 94.
Write the smallest and the largest numbers which are approximate to 130.
A number rounds off to 130 (to the nearest ten) depending on its ones digit.
The numbers from 125 to 129 have ones digit 5 or more, so they round up to 130.
The numbers from 130 to 134 have ones digit less than 5, so they round down to 130.
So, the numbers that round off to 130 lie from 125 up to 134.
Hence, the smallest number is 125 and the largest number is 134.
Estimation
14 fully explained questions
Estimate the sum after rounding off each pair of numbers to the nearest ten:
(i) 34 and 87
(ii) 78 and 18
(iii) 96 and 55
(iv) 76 and 62
(v) 457 and 175
(vi) 527 and 267
Estimate the sum after rounding off each pair of numbers to the nearest hundred:
(i) 336 and 782
(ii) 270 and 495
(iii) 4280 and 5295
(iv) 30047 and 39287
Estimate the sum after rounding off the following pairs of numbers to the nearest thousand:
(i) 53826 and 36455
(ii) 56802 and 22475
Estimate the following differences after rounding off each number to the nearest ten:
(i) 82 - 27
(ii) 96 - 36
(iii) 508 - 248
Estimate each difference after rounding off each given number to the nearest hundred:
(i) 769 - 314
(ii) 856 - 687
(iii) 6352 - 2086
Estimate each difference after rounding off each given number to the nearest thousand:
(i) 45974 - 38766
(ii) 76003 - 48399
Estimate each of the following products by rounding off each number to the nearest ten:
(i) 49 × 52
(ii) 63 × 38
(iii) 27 × 54
(iv) 53 × 85
(v) 74 × 67
(vi) 25 × 33
Estimate each of the following products by rounding off each number to the nearest hundred:
(i) 477 × 213
(ii) 624 × 236
(iii) 333 × 247
(iv) 537 × 283
(v) 382 × 127
(vi) 472 × 328
Estimate each of the following products by rounding off the first number correct to the nearest ten and the other number correct to the nearest hundred:
(i) 28 × 287
(ii) 432 × 128
(iii) 48 × 165
(iv) 72 × 258
(v) 83 × 664
(vi) 44 × 250
Estimate each of the following quotients by rounding off each number to the nearest ten:
(i) 87 ÷ 28
(ii) 84 ÷ 23
(iii) 77 ÷ 22
(iv) 198 ÷ 24
(v) 355 ÷ 26
(vi) 444 ÷ 42
(vii) 843 ÷ 33
There are 768 houses in a town and about 7 people live in each house. Round off these two numbers to the nearest ten and estimate the total number of people living in the town.
Hence, the estimated total number of people living in the town is 7,700.
There are 514 mango trees in an orchard. The owner plans to plant 178 more mango trees. Estimate how many mango trees will be there in the orchard after rounding off to the nearest hundred?
Hence, there will be approximately 700 mango trees in the orchard.
A machine manufactures 5,782 screws every day. If we round off the number of screws produced in a day to the nearest thousand, approximately how many screws will the machine manufacture in the month of April?
Hence, the machine will manufacture approximately 1,80,000 screws in the month of April.
The weight of a box is 4 kg 859 g. What will be the approximate weight of 150 such boxes if we round off the weight of each box to 5 kg?
Hence, the approximate weight of 150 boxes is 750 kg.
Multiple Choice Questions
10 fully explained questions
The sum of place value of two 3s in the number 73423 is:
In the number 73423 :
Hence, option 3 is the correct option.
53729 is greater than 53972:
Comparing 53729 and 53972, the first two digits (5 and 3) are the same.
Comparing the next digit, 7 < 9.
So, 53729 < 53972.
Therefore, the given statement is false.
Hence, option 2 is the correct option.
The difference between the largest and the smallest 3-digit numbers is:
Hence, option 2 is the correct option.
The number of two digit numbers from 40 to 70 is:
The two digit numbers from 40 to 70 (both included) are 40, 41, 42, ..., 70.
Hence, option 2 is the correct option.
If A = 83, B = 12 and C = 17, the value of A - B + C is:
A - B + C
Hence, option 3 is the correct option.
3,294 + 1,320 after rounding off to the nearest 10 is:
Since 4,610 is not among the given options,
Hence, option 4 is the correct option.
The difference between 652 and 4,038 after rounding off each given number to the nearest hundred is:
Hence, option 1 is the correct option.
2,228 rounded off to the nearest hundred minus 228 to the nearest ten is:
Hence, option 2 is the correct option.
Sum of digits 3 and 8 used in the numbers 5,03,489 is:
The digits referred to are 3 and 8.
Hence, option 3 is the correct option.
80 is multiplied with 23 instead of 32; the result of multiplication differs by:
Hence, option 2 is the correct option.
Statement I–II Questions
2 fully explained questions
Which of the following options is correct?
Hence, option 2 is the correct option.
25 - 42 + 63 = ?
25 - 42 + 63 = 30 - 40 + 60 = 50
So, Statement 1 is false and Statement 2 is true.
Hence, option 4 is the correct option.
Assertion–Reason Questions
2 fully explained questions
Number 89623
So, A is true and R is false.
Hence, option 1 is the correct option.
So, both A and R are true.
Hence, option 3 is the correct option.
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Frequently Asked Questions
The face value of a digit is the digit itself. Its place value depends on the digit’s position in the number.
First compare the number of digits. If both numbers have the same number of digits, compare their digits from left to right until the first different digit appears.
Arrange the digits in descending order. For the smallest number, arrange them in ascending order, but never place zero at the leftmost position.
Look at the ones digit. If it is 0 to 4, round down. If it is 5 to 9, round up.
Round each given number to the required place value first, and then perform the operation using the rounded numbers.
Yes. It covers Exercises 1(A), 1(B), 1(C), 1(D), multiple-choice questions, statement questions and assertion–reason questions from the supplied content.
