Chapter - 1 | Number System | Class 6 Mathematics | Selina Concise Mathematics - Middle School.
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📂 Chapter 1 - Number System.
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# Let's Discuss :
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There are ten digits in our numbers system (0, 1, 2, 3, ....., 9) and every number is made by using one or more digits with or without repetition.
E.g. 8, 18, 361, 4863, etc.
# Place Value of a Digit in a Number :
The place value (local value) of a digit depends upon the position it occupies in the number.
Example :
Consider a two digit number i.e. 69.
69 in Expanded form can be written as (60 + 9)
Or
[ (6 x 10) + (9 x 1) ]
Here,
- Place value of 6 is Tens.
- Place value of 9 is Ones.
📌NOTE : The place value of '0' is always 0 regardless of the place it occupies in the given number.
Similarly,
In a Six digit Number i.e. 123456.
Expanded form = 1,00,000 + 20,000 + 3000 + 400 + 50 + 6
Or
[ (1 x 1,00,000) + (2 x 10,000) + (3 x 1000) + (4 x 100) + (5 x 10) + (6 x 1) ]
Here,
- Place value of 1 is Lakhs.
- Place value of 2 is Ten Thousands.
- Place value of 3 is Thousands.
- Place value of 4 is Hundreds.
- Place value of 5 is Tens.
- Place value of 6 is Ones.
-- Place Value Table --
| Lakhs (L) | Ten Thousands (TTh) | Thousands (Th) | Hundreds (H) | Tens (T) | Ones (o) |
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 |
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# Face Value of a Digit in a Number :
The face value of a digit does not depend on the position of the digit in a number and is equal to the value of the digit itself.
Example :
Consider a two digit number i.e. 69.
Here,
- The face value of 6 is 6 itself.
- The face value of 9 is 9 itself.
Similarly,
In a six digit number i.e. 123456.
Here,
- The face value of 1 is 1 itself.
- The face value of 2 is 2 itself.
- The face value of 3 is 3 itself.
- The face value of 4 is 4 itself.
- The face value of 5 is 5 itself.
- The face value of 6 is 6 itself.
| Digits | Place Value (Local Value) | Face Value |
|---|---|---|
| 1 | 1 x 1,00,000 = 1,00,000 | 1 |
| 2 | 2 x 10,000 = 20,000 | 2 |
| 3 | 3 x 1000 = 3000 | 3 |
| 4 | 4 x 100 = 400 | 4 |
| 5 | 5 x 10 = 50 | 5 |
| 6 | 6 x 1 = 6 | 6 |
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# Comparing Numbers :
In order to compare two numbers, we follow the rules given below :
Rule 1 : If the two numbers have different numbers of digits then the number with more number of digits is greater.
E.g. In numbers 54932 and 942362, The number 942362 is greater as it has more number of digits (i.e. 6 digits).
Rule 2 : If two numbers have equal number of digits then we compare the place value of digits starting from leftmost place in both the numbers.
E.g. In numbers 53421 and 54321, The number of digits are same (i.e. 5 digits). Here, 54321 is greater as the digit on the thousands place is greater (i.e. 4000).
Rule 3 : If two numbers have equal number of digits then we compare the place value of digits starting from leftmost place in both the numbers using place value chart (table).
| Lakhs (L) | Ten Thousands (TTh) | Thousands (Th) | Hundreds (H) | Tens (T) | Ones (o) |
|---|---|---|---|---|---|
| 5 | 4 | 0 | 2 | 7 | 6 |
| 3 | 6 | 9 | 9 | 9 | 8 |
Here, The number 540276 is greater as it digit on the Lakhs place is greater.
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# Forming Smallest and Greatest number using Digits :
We can form different numbers from a given set of digits by arranging the given digits in various places.
(i) To form Greatest number : To form the greatest number arrange the digits in Descending order.
(ii) To form Smallest number : To form the smallest number arrange the digits in Ascending order.
📌Note : Descending order means greater to smaller and Ascending order means smaller to greater.
(a) When the given digits do not include 0.
We can form different numbers from a given set of digits by arranging the given digits in various places.
E.g.
Given Digits : 5, 4, 6, 3, 9
(i) To form Greatest number : Arrange the digits in Descending order like 96543.
(ii) To form Smallest number : Arrange the digits in Ascending order like 34569.
(b) When the given digits include 0.
We can form different numbers from a given set of digits by arranging the given digits in various places.
E.g.
Given Digits : 5, 4, 0, 3, 9
(i) To form Greatest number : Arrange the digits in Descending order like 95430.
(ii) To form Smallest number : Arrange the digits in Ascending order like 30459.
📌Note : To form the smallest number, The digit '0' should not be placed at extreme left, but it can be placed at second place from the left side.
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# Units of Length, Mass and Capacity :
Unit means single thing e.g. a litre, a day, a minute, a kilometre, etc.
The standard unit of length, weight (mass) and capacity (volume) are metre (m), gram (g) and litre (L) respectively.
| Units of | Length | Units of | Mass | Units of | Capacity | ||
|---|---|---|---|---|---|---|---|
| Millimetre (mm) | 1/1000 metre | Milligram (mg) | 1/1000 gram | Millilitre (mL) | 1/1000 litre | ||
| Centimetre (cm) | 1/100 metre | Centigram (cg) | 1/100 gram | Centilitre (cl) | 1/100 litre | ||
| Decimetre (dm) | 1/10 metre | Decigram (dg) | 1/10 gram | Decilitre (dl) | 1/10 litre | ||
| A metre (m) | Standard unit | A gram (g) | Standard unit | A Litre (L) | Standard unit | ||
| Decametre (dam) | 10 metre | Decagram (dag) | 10 gram | Decalitre (dal) | 10 litre | ||
| Hectometre (hm) | 100 metre | Hectogram (hg) | 100 gram | Hectolitre (hl) | 100 litre | ||
| Kilometre (km) | 1000 metre | kilogram (kg) | 1000 gram | Kilolitre (kl) | 1000 litre |
📌 Note : Numbers which do not refer to any particular unit such as 8, 12, 15, 23, etc. are called abstract numbers. whereas, the numbers which refers to a particular unit such as 8L, 12m, 15 days, 23 hours, etc. are called concrete numbers.
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# Estimation :
Estimation means to make a rough idea about quantities, cost, size, population, etc.
E.g.
(i) Cost of construction of a building is estimated to be 5 crores.
(ii) Population of a town in year 2025 is estimated around 25 lakhs.
# Approximation :
Approximation means rounding off the given quantity (number) to the nearest Ten, Hundred, Thousand, etc.
E.g.
(i) If Rohan cycled 2.8 km it is approximated to 3 km.
(ii) Price of a mobile phone is ₹24,750 which is approximated to ₹25,000.
📌Note : Approximation means nearest to actual amount or value.
# Rounding Off Numbers :
(a) Rounding off to the nearest 10 :
Rule 1 : If the digit at ones place is less than 5 (i.e. 1, 2, 3 and 4) then replace ones digit by 0, and keep the other digits as they are.
E.g.
541 to the nearest ten is 540.932 to the nearest ten is 930.753 to the nearest ten is 750.864 to the nearest ten is 860.
Rule 2 : If the digit at ones place is 5 or more than 5 (i.e. 5, 6, 7, 8 and 9) then replace ones digit by 0, and increase the tens digit by 1 and keep the other digits as they are.
E.g.
545 to the nearest ten is 550.936 to the nearest ten is 940.757 to the nearest ten is 760.868 to the nearest ten is 870.279 to the nearest ten is 280.
(b) Rounding off to the nearest 100 :
Rule 1 : If the digit at tens place is less than 5 (i.e. 1, 2, 3 and 4) then replace ones and tens digit by 0, and keep the other digits as they are.
E.g.
514 to the nearest hundred is 500.923 to the nearest hundred is 900.734 to the nearest hundred is 700.845 to the nearest hundred is 800.
Rule 2 : If the digit at tens place is 5 or more than 5 (i.e. 5, 6, 7, 8 and 9) then replace ones and tens digit by 0, and increase the hundreds digit by 1 and keep the other digits as they are.
E.g.
4352 to the nearest hundred is 4400.3961 to the nearest hundred is 4000.2376 to the nearest hundred is 2400.1685 to the nearest hundred is 1700.5291 to the nearest hundred is 5300.
(c) Rounding off to the nearest 1000 :
Rule 1 : If the digit at hundreds place is less than 5 (i.e. 1, 2, 3 and 4) then replace ones, tens and hundreds digit by 0, and keep the other digits as they are.
E.g.
5146 to the nearest thousand is 5000.9232 to the nearest thousand is 9000.7341 to the nearest thousand is 7000.8453 to the nearest thousand is 8000.
Rule 2 : If the digit at hundreds place is 5 or more than 5 (i.e. 5, 6, 7, 8 and 9) then replace ones and tens and hundreds digit by 0, and increase the thousands digit by 1 and keep the other digits as they are.
E.g.
4523 to the nearest thousand is 5000.3612 to the nearest thousand is 4000.2764 to the nearest thousand is 3000.1857 to the nearest thousand is 2000.5918 to the nearest thousand is 6000.
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# Exercise Solutions :
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-- EXERCISE - 1 (A) --
Q1. Fill in the blanks :
(i) In 24673, the place value of 6 is 600 (6 Hundreds)
(ii) In 8039, the place value of 8 is 8000 (8 Thousands)
(iii) In 325648, the local value of 2 is 20,000 (2 Ten Thousands)
(iv) In 6439, the local value of 3 is 30 ( 3 Tens)
Q2. Find the difference between the place values of 3 and 5 in the number 3945.
Solution :
Place value of 3 is thousands (3 x 1000 = 3000)
Place value of 5 is ones (5 x 1 = 5)
Difference = 3000 - 5 = 2995
Ans. 2995
Q3. In the number 40562,
(i) the local value of 5 = 500 (5 Hundreds)
(ii) the face value of 6 = 60 ( 6 Tens)
(iii) the sum of the place value of 5 and the face value of 6 = 560
Q4. Find the difference in the place values of the two sevens in the number 8,72,574.
Solution :
Place value of first 7 from right side is Tens (7 x 10 = 70)
Place value of second 7 from right side is Ten Thousands (7 x 10,000 = 70,000)
Difference = 70,000 - 70 = 69,930
Ans. 69,930
Q5. Find the difference between the face value of 9 and the face value of 4 in the number 324798.
Solution :
Face value of 9 is nine (09)
Face value of 4 is four (04)
Difference = 09 - 04 = 05
Ans. 05
-- EXERCISE - 1 (B) --
Q1. Which is greater ?
(i) 537 or 98
Solution :
The first number is 537 i.e. 500 + 30 + 7
The second number is 98 i.e. 90 + 8
Here,
The first number 537 has high place value i.e. 500
Therefore,
537 is the greater number.
Ans. 537
(ii) 2428 or 529
Solution :
The first number is 2428 i.e. 2000 + 400 + 20 + 8
The second number is 529 i.e. 500 + 20 + 9
Here,
The first number 2428 has high place value i.e. 2000
Therefore,
2428 is the greater number.
Ans. 2428
(iii) 259467 or 1035729
Solution :
The first number is 259467 i.e. 2,00,000 + 50,000 + 9,000 + 400 + 60 + 7
The second number is 1035729 i.e. 10,00,000 + 30,000 + 5000 + 700 + 20 + 9
Here,
The second number 1035729 has high place value i.e. 10,00,000
Therefore,
1035729 is the greater number.
Ans. 1036729
(iv) 63520 or 63250
Solution :
The first number is 63520 i.e. 60,000 + 3000 + 500 + 20
The second number is 63250 i.e. 60,000 + 3000 + 200 + 50
Here,
The first number 63520 has higher Hundreds place value i.e. 500
Therefore,
63520 is the greater number.
Ans. 63520
Q2. Which is smaller ?
(i) 428 or 437
Solution :
The first number is 428 i.e. 400 + 20 + 8
The second number is 437 i.e. 400 + 30 + 7
Here,
The first number 428 has smaller Tens place value i.e. 20
Therefore,
428 is the smaller number.
Ans. 428
(ii) 2497 or 2597
Solution :
The first number is 2497 i.e. 2000 + 400 + 90 + 7
The second number is 2597 i.e. 2000 + 500 + 90 + 7
Here,
The first number 2497 has smaller Hundreds place value i.e. 400
Therefore,
2497 is the smaller number.
Ans. 2497
(iii) 3297 or 3596
Solution :
The first number is 3297 i.e. 3000 + 200 + 90 + 7
The second number is 3596 i.e. 3000 + 500 + 90 + 6
Here,
The first number 3297 has smaller Hundreds place value i.e. 200
Therefore,
3297 is the smaller number.
Ans. 3297
(iv) 380362 or 381007
Solution :
The first number is 380362 i.e. 3,00,000 + 80,000 + 300 + 60 + 2
The second number is 381007 i.e. 3,00,000 + 81,000 + 7
Here,
The first number 380362 has smaller Thousands place value i.e. 80,000
Therefore,
380362 is the smaller number.
Ans. 380362
Q3. By making a suitable chart, compare :
(i) 540276 and 369998
Solution :
| Lakhs (L) | Ten Thousands (TTh) | Thousands (Th) | Hundreds (H) | Tens (T) | Ones (o) |
|---|---|---|---|---|---|
| 5 | 4 | 0 | 2 | 7 | 6 |
| 3 | 6 | 9 | 9 | 9 | 8 |
The first number is 540276 i.e. 5,00,000 + 40,000 + 200 + 70 + 6
The second number is 369998 i.e. 3,00,000 + 69,000 + 900 + 90 + 8
Here,
The second number 369998 has smaller Lakhs place value i.e. 3,00,000
Therefore,
369998 is the smaller number.
Ans. 369998
(ii) 6983245 and 6893254
Solution :
| Ten Lakhs (TL) | Lakhs (L) | Ten Thousands (TTh) | Thousands (Th) | Hundreds (H) | Tens (T) | Ones (o) |
|---|---|---|---|---|---|---|
| 6 | 9 | 8 | 3 | 2 | 4 | 5 |
| 6 | 8 | 9 | 3 | 2 | 5 | 4 |
The first number is 6983245 i.e. 60,00,000 + 9,00,000 + 83,000 + 200 + 40 + 5
The second number is 6893245 i.e. 60,00,000 + 8,00,000 + 93,000 + 200 + 50 + 4
Here,
The second number 6893245 has smaller Lakhs place value i.e. 8,00,000
Therefore,
6893254 is the smaller number.
Ans. 6893254
Q4. Compare the numbers written in the following table writing them in ascending order :
| Crores (C) | Ten Lakhs (TL) | Lakhs (L) | Ten Thousands (TTh) | Thousands (Th) | Hundreds (H) | Tens (T) | Ones (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 |
Solution :
Ascending Order - 52,23,791 < 54,32,972 < 2,31,06,293 < 2,31,82,634 < 5,43,44,782.
Q5. Use a place value chart to compare the given numbers and write them in descending order : 543287 ; 5482900 ; 2732940 ; 43877 ; 78396 and 4999.
Solution :
| Ten Lakhs (TL) | Lakhs (L) | Ten Thousands (TTh) | Thousands (Th) | Hundreds (H) | Tens (T) | Ones (o) |
|---|---|---|---|---|---|---|
| 5 | 4 | 3 | 2 | 8 | 7 | |
| 5 | 4 | 8 | 2 | 9 | 0 | 0 |
| 2 | 7 | 3 | 2 | 9 | 4 | 0 |
| 4 | 3 | 8 | 7 | 7 | ||
| 7 | 8 | 3 | 9 | 6 | ||
| 4 | 9 | 9 | 9 |
Descending order - 54,82,900 > 27,32,940 > 5,43,287 > 78,396 > 43,877 > 4,999
Q6. Find the smallest and the greatest numbers in each of the cases given below :
(i) 983, 5754, 84 and 5942
Solution :
The first number is 983 i.e. 900 + 80 + 3
The second number is 5754 i.e. 5000 + 700 + 50 + 4
The third number is 84 i.e. 80 + 4
The fourth number is 5942 i.e. 5000 + 900 + 40 + 2
Here,
The third number 84 has smallest Hundreds place value i.e. 000
The fourth number 5942 has greatest Thousands and Hundreds place value i.e. 5000 + 900
Therefore,
84 is the smallest number and 5942 is the greatest number.
Ans. Smallest = 84 and Greatest = 5942
(ii) 32849, 53628, 5499 and 54909.
Solution :
The first number is 32849 i.e. 30,000 + 2000 + 800 + 40 + 9
The second number is 53628 i.e. 50,000 + 3000 + 600 + 20 + 8
The third number is 5499 i.e. 5000 + 4000 + 90 + 9
The fourth number is 54909 i.e. 50,000 + 4000 + 900 + 9
Here,
The third number 5499 has smallest Ten Thousands place value i.e. 00,000
The fourth number 54909 has greatest Ten Thousands and Thousands place value i.e. 50,000 + 4000
Therefore,
5499 is the smallest number and 54909 is the greatest number.
Ans. Smallest = 5499 and Greatest = 54909
Q7. Form the greatest and the smallest 4-digit numbers using the given digits, without repetition :
📌 Note : To form the greatest number, we need to arrange these digits in descending order of their face value and to form the smallest number, we need to arrange these digits in ascending order of their face value.
(i) 3, 7, 2 and 5
Solution :
Smallest number : 2357
Greatest number : 7532
(ii) 6, 1, 4 and 9
Solution :
Smallest number : 1469
Greatest number : 9641
(iii) 7, 0, 4 and 2
Solution :
Smallest number : 2047
Greatest number : 7420
(iv) 1, 8, 5 and 3
Solution :
Smallest number : 1358
Greatest number : 8531
(v) 9, 6, 0 and 7
Solution :
Smallest number : 6079
Greatest number : 9760
(vi) 9, 5, 8 and 1
Solution :
Smallest number : 1589
Greatest number : 9851
Q8. 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.
Solution :
To form greatest number :
Let the 4 digits be 9, 8, 7, 5
Greatest number with 5 at the ten's place = 9857
To form smallest number :
Let the 4 digits be 0, 1, 2, 5
Smallest number with 5 at the ten's place = 1052
Q9. Form the largest number with the digits 2, 3, 5, 9, 6 and 0, without repetition of any digit.
Solution :
Largest number = 965320
Q10. Write the smallest and the greatest 4-digit numbers, without repetition of any digit.
Solution :
To form greatest number :
Let the 4 digits be 9, 8, 7, 6
Greatest number without repetition of digits = 9876
To form smallest number :
Let the 4 digits be 0, 1, 2, 3
Smallest number without repetition of digits = 1023
Q11. Find the sum of the largest and the smallest four-digit numbers.
Solution :
Largest 4 digit number = 9999
Smallest 4 digit number = 1000
Sum of numbers = 9999 + 1000 = 10,999
Ans. 10,999
Q12. Form the greatest and the smallest five-digit numbers with 8 in the hundred's place and with all the digits different.
Solution :
To form greatest number :
Let the 5 digits be 9, 8, 7, 6, 5
Greatest number with 8 at the hundred's place = 97865
To form smallest number :
Let the 5 digits be 0, 1, 2, 3, 8
Smallest number with 8 at the hundred's place = 10823
Q13. Find the difference between the smallest and the greatest six-digit numbers.
Solution :
Greatest 6 digit number = 9,99,999
Smallest 6 digit number = 1,00,000
Difference = 999999 - 100000 = 8,99,999
Ans. 8,99,999
Q14.
(i) How many four-digit numbers are there between 999 and 3000 ?
Solution :
Smallest 4 digit number between 999 and 3000 = 1000
Largest 4 digit number between 999 and 3000 = 3000
Difference = 3000 - 1000 = 2000.
Ans. 2000
(ii) How many four-digit numbers are there between 99 and 3000 ?
Solution :
Smallest 4 digit number between 99 and 3000 = 1000
Largest 4 digit number between 999 and 3000 = 3000
Difference = 3000 - 1000 = 2000.
Ans. 2000
Q15. How many four-digit numbers are there between 1001 and 4000 ?
Solution :
Smallest 4 digit number between 1001 and 4000 = 1002
Largest 4 digit number between 1001 and 4000 = 4000
Difference = 4000 - 1002 = 2998
Ans. 2998
Q16. Write all the possible three-digit numbers using the digits 3, 6 and 8, if repetition of any digit is not allowed.
Solution :
All possible 3 digit numbers without repetition = 368, 386, 638, 683, 836, 863.
Q17. 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.
Solution :
Smallest 4 digit number without repetition = 4597
Greatest 4 digit number without repetition = 9547
(ii) 9 is at the tens place.
Solution :
Smallest 4 digit number without repetition = 4597
Greatest 4 digit number without repetition = 7594
(iii) 4 is at the hundreds place
Solution :
Smallest 4 digit number without repetition = 5479
Greatest 4 digit number without repetition = 9475
-- EXERCISE 1 (C) --
Q1. 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 ?
Solution :
Population of city in the year 2014 = 3,54,976
Population of city increased by in the year 2015 = 68,438
Population of city at the end of the year 2015 = 3,54,976 + 68, 438 = 4,23,414.
Ans. Population of city at the end of the year 2015 is 4,23,414.
Q2. A = 7,43,000 and B = 8,00,100. Which is greater A or B ? And by how much ?
Solution :
The first number is 7,43,000 i.e. 7,00,000 + 40,000 + 3000
The second number is 8,00,100 i.e. 8,00,000 + 100
Here,
The second number 8,00,100 has high place value i.e. 8,00,000
Therefore, 8,00,100 is the greater number.
Difference between the two numbers = 8,00,100 - 7,43,000 = 57,100.
Ans. A is greater by 57,100.
Q3. A notebook has 56 pages. How many pages will 5326 such notebooks have ?
Solution :
Number of pages in one notebook = 56
Number of Notebooks = 5326
Total number of pages in 5326 notebooks = 5326 x 56 = 2,98,256.
Ans. 2,98,256.
Q4. 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 ?
Solution :
Number of sheets of paper a person has = 75,000
Number of pages each sheet can make = 8
Total number of pages obtained = 8 x 75,000 = 6,00,000
Number of pages in each notebook = 200
Total number of notebooks made = 6,00,000 ÷ 200 = 3,000
Ans. Total number of notebooks made out of 6,00,000 pages are 3,000.
Q5. Add 1,76,209 ; 4,50,923 and 44,83,947.
Solution :
Sum of 1,76,209 + 4,50,923 + 44,83,947
1,76,2094,50,923+ 44,83,94751,11,079
Ans. 51,11,079
Q6. 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 ?
Solution :
Runs scored by a cricket player = 7,849
Runs wishes to complete by the cricket player = 10,000
Runs need to be scored by the cricket player = 10,000 - 7849 = 2151
Ans. Runs need to be scored by the cricket player are 2151.
Q7. 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 loose the election ?
Solution :
Votes scored by candidate A = 9,32,567
Votes scored by candidate B = 9,00,235
Difference = 9,32,567 - 9,00,235 = 32,332
Ans. A wins by 32,332 votes.
Q8. 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.
Solution :
Smallest number = 12356
Greatest number = 65321
Difference = 65321 - 12356 = 52,965
Ans. 52,965
Q9. 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 ?
Solution :
Number of screws manufactured in a day = 5782
Rounding off to the nearest thousand = 6000
Number of days in the month of April = 30
Number of screws manufactured in the month of April = 6000 x 30 = 1,80,000
Ans. Number of screws manufactured in the month of April are 1,80,000
Q10. 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 ?
Solution :
Amount of money a man has = ₹1,57,184
Cost of one article purchase by the man = ₹125
Number of articles purchase by the man = 80
Money spent on purchase of articles = 80 x 125 = 10,000
Money remain with the man after purchase = 1,57,184 - 10,000 = ₹1,47,184
Ans. Money remain with the man after purchase of articles is ₹1,47,184
Q11. 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 ?
Solution :
Number to be multiplied = 8,035
Number by which it multiplied = 87
Number by which it should be multiplied = 78
Difference between the two numbers = 87-78 = 9
Difference between the answer = 9 x 8,035 = 72,315
Ans. The answer is greater by 72,315.
Alternate Method :
Number to be multiplied = 8,035
Number by which it multiplied = 87
Multiplication = 8,035 x 87 = 6,99,045
Number by which it should be multiplied = 78
Multiplication = 8,035 x 78 = 6,26,730
Difference between the answer = 6,99,045 - 6,26,730 = 72,315
Ans. The answer is greater by 72,315.
Q12. 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 ?
Solution :
Length of cloth Mohani has = 30m = 3000cm
Length of cloth required to make a shirt = 2m 30cm = 230cm
Number of shirts Mohani can made = 3000 ÷ 230 = 13
Cloth remained = 10 cm
Ans. Mohani can make 13 shirts and will left with 10 cm of cloth.
Q13. 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 ?
Solution :
Weight of one box = 4 kg 859g
Rounding off the weight of one box = 5 kg
Number of boxes = 150
Total weight of all boxes = 150 x 5 = 750kg
Ans. The approximate weight of 150 boxes is 750kg.
Q14. 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 ?
Solution :
Distance between the two points = 3km 760m = 3760m
Distance travelled in one day = 3760 x 2 = 7520m
Number of days he travelled = 8
Total distance travelled by the boy = 7520 x 8 = 60,160m = 60km 160m
Ans. Distance travelled by the boy in days is 60km 160m.
Q15. An oil-tin contains 6 litre and 60 ml oil. How many identical bottles can this oil fill, if the capacity of each bottle is 30 ml ?
Solution :
Quantity of oil a tin contains = 6L 60ml = 6060ml
Capacity of each bottle = 30ml
Total number of bottle that can be filled = 6060 ÷ 30 = 202
Ans. Number of bottles that can be filled are 202.
Q16. 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 ?
Solution :
Sale receipt of a company in first year = ₹83,73,540
Decrease in sale receipt of the company in following year = ₹7,84,670
Sale receipt of the company in the following year = 83,73,540 - 7,84,670 = ₹75,88,870
Ans. Sale receipt of the company in the following year is ₹75,88,870
(ii) What was the total sale receipt of the company during these two years ?
Solution :
Sale receipt of a company in first year = ₹83,73,540
Sale receipt of the company in the following year = ₹75,88,870
Total sale receipt of the company during these two years = 83,73,540 + 75,88,870 = 1,59,62,410.
Ans. Total sale receipt of the company during these two years is 1,59,62,410.
Q17. A number exceeds 8,59,470 by 3,00,999. What is the number ?
Solution :
First number = 8,59,470
Difference in two numbers = 3,00,999
Second number = 8,59,470 + 3,00,999 = 11,60,469
Ans. The second number is 11,60,469.
-- EXERCISE - 1 (D) --
Q1. Round off (approximate) each of the following to the nearest ten :
(i) 62
Ans. 60
(ii) 265
Ans. 270
(iii) 543
Ans. 540
(iv) 8261
Ans. 8260
(v) 6296
Ans. 6300
(vi) 3998
Ans. 4000
(vii) 72396
Ans. 72400
Q2. Round off (approximate) each of the following to the nearest hundred :
(i) 748
Ans. 700
(ii) 784
Ans. 800
(iii) 2667
Ans. 2700
(iv) 5432
Ans. 5400
(v) 6998
Ans. 7000
(vi) 59997
Ans. 60000
Q3. Round off (approximate) each of the following to the nearest thousand :
(i) 6475
Ans. 6000
(ii) 6732
Ans. 7000
(iii) 25352
Ans. 25000
(iv) 32568
Ans. 33000
(v) 9948
Ans. 10000
(vi) 89994
Ans. 90000
Q4. Round off (approximate) :
(i) 578 to the nearest ten.
Ans. 580
(ii) 578 to the nearest hundred.
Ans. 600
(iii) 4327 to the nearest thousand.
Ans. 4000
(iv) 32974 to the nearest ten thousand.
Ans. 30,000
(v) 27487 to the nearest ten thousand.
Ans. 30,000
Q5. Round off (approximate) each of the following to the nearest ten, nearest hundred and nearest thousand.
(i) 864
Solution :
Nearest Ten = 860
Nearest Hundred = 900
Nearest Thousand = 1000
(ii) 1249
Solution :
Nearest Ten = 1250
Nearest Hundred = 1200
Nearest Thousand = 1000
(iii) 54,547
Solution :
Nearest Ten = 54,550
Nearest Hundred = 54,500
Nearest Thousand = 55,000
(iv) 68,076
Solution :
Nearest Ten = 68,080
Nearest Hundred = 68,100
Nearest Thousand = 68,000
(v) 56,293
Solution :
Nearest Ten = 56,290
Nearest Hundred = 56,300
Nearest Thousand = 56,000
(vi) 7,293
Solution :
Nearest Ten = 7,290
Nearest Hundred = 7,300
Nearest Thousand = 7,000
(vii) 89,24,379
Solution :
Nearest Ten = 89,24,380
Nearest Hundred = 89,24,400
Nearest Thousand = 89,24,000
Q6. Approximate each of the following to the nearest ten :
(i) ₹562
Ans. ₹560
(ii) 837 m
Ans. 840 m
(iii) 545 cm
Ans. 550 cm
(iv) ₹27
Ans. ₹30
Q7. List all the natural numbers which can be rounded off to 30.
Ans. Natural numbers which can be rounded off to 30 are 25, 26, 27, 28, 29, 31, 32, 33, 34.
Q8. List all the whole numbers which are approximate to 50.
Ans. Whole numbers which are approximate to 50 are 45, 46, 47, 48, 49, 51, 52, 53, 54.
Q9. Write the smallest and the largest numbers which are rounded off to 90.
Ans. Smallest number which can be rounded off to 90 is 85 and Largest number which can be rounded off to 90 is 94.
Q10. Write the smallest and the largest numbers which are approximate to 130.
Ans. Smallest number which can be approximate to 130 is 125 and Largest number which can be approximate to 130 is 134.
-- EXERCISE - 1 (E) --
Q1. Estimate the sum of each pair of numbers to the nearest ten :
(i) 34 and 87
Solution :
34 round off to nearest ten = 30
87 round off to nearest ten = 90
Sum of two numbers = 30 + 90 = 120
Ans. Estimated Sum is 120
(ii) 78 and 18
Solution :
78 round off to nearest ten = 80
18 round off to nearest ten = 20
Sum of two numbers = 80 + 20 = 100
Ans. Estimated Sum is 100
(iii) 96 and 55
Solution :
96 round off to nearest ten = 100
55 round off to nearest ten = 60
Sum of two numbers = 100 + 60 = 160
Ans. Estimated Sum is 160
(iv) 76 and 62
Solution :
76 round off to nearest ten = 80
62 round off to nearest ten = 60
Sum of two numbers = 80 + 60 = 140
Ans. Estimated Sum is 140
(v) 457 and 175
Solution :
457 round off to nearest ten = 460
175 round off to nearest ten = 180
Sum of two numbers = 460 + 180 = 640
Ans. Estimated Sum is 640
(vi) 527 and 267
Solution :
527 round off to nearest ten = 530
267 round off to nearest ten = 270
Sum of two numbers = 530 + 270 = 800
Ans. Estimated Sum is 800
Q2. Estimate the sum of each pair of numbers to the nearest hundred :
(i) 336 and 782
Solution :
336 round off to nearest hundred = 300
782 round off to nearest hundred = 800
Sum of two numbers = 300 + 800 = 1100
Ans. Estimated Sum is 1100.
(ii) 270 and 495
Solution :
270 round off to nearest hundred = 300
495 round off to nearest hundred = 500
Sum of two numbers = 300 + 500 = 800
Ans. Estimated Sum is 800.
(iii) 4280 and 5295
Solution :
4280 round off to nearest hundred = 4300
5295 round off to nearest hundred = 5300
Sum of two numbers = 4300 + 5300 = 9600
Ans. Estimated Sum is 9600.
(iv) 30047 and 39287
Solution :
30047 round off to nearest hundred = 30,000
39287 round off to nearest hundred = 39,300
Sum of two numbers = 30,000 + 39,300 = 69,300
Ans. Estimated Sum is 69,300.
Q3. Estimate the sum of the following pairs of numbers to the nearest thousand :
(i) 53826 and 36455
Solution :
53826 round off to nearest thousand = 54,000
36455 round off to nearest thousand = 36,000
Sum of two numbers = 54,000 + 36,000 = 90,000
Ans. Estimated Sum is 90,000.
(ii) 56802 and 22475
Solution :
56802 round off to nearest thousand = 57,000
22475 round off to nearest thousand = 22,000
Sum of two numbers = 57,000 + 22,000 = 79,000
Ans. Estimated Sum is 79,000.
Q4. Estimate the following differences correct to the nearest ten :
(i) 82 - 27
Solution :
82 round off to nearest ten = 80
27 round off to nearest ten = 30
Difference of two numbers = 80 - 30 = 50
Ans. Estimated Difference is 50.
(ii) 96 - 36
Solution :
96 round off to nearest ten = 100
36 round off to nearest ten = 40
Difference of two numbers = 100 - 40 = 60
Ans.Estimated Difference is 60.
(iii) 508 - 248
Solution :
508 round off to nearest ten = 510
248 round off to nearest ten = 250
Difference of two numbers = 510 - 250 = 260
Ans. Estimated Difference is 260.
Q5. Estimate each difference to the nearest hundred :
(i) 769 - 314
Solution :
769 round off to nearest hundred = 800
314 round off to nearest hundred = 300
Difference of two numbers = 800 - 300 = 500
Ans. Estimated Difference is 500.
(ii) 856 - 687
Solution :
856 round off to nearest hundred = 900
687 round off to nearest hundred = 700
Difference of two numbers = 900 - 700 = 200
Ans. Estimated Difference is 200.
(iii) 6352 - 2086
Solution :
6352 round off to nearest hundred = 6400
2086 round off to nearest hundred = 2100
Difference of two numbers = 6400 - 2100 = 4300
Ans. Estimated Difference is 4300.
Q6. Estimate each difference to the nearest thousand :
(i) 45974 - 38766
Solution :
45974 round off to nearest thousand = 46,000
38766 round off to nearest thousand = 39,000
Difference of two numbers = 46,000 - 39,000 = 7,000
Ans. Estimated Difference is 7,000.
(ii) 76003 - 48399
Solution :
76003 round off to nearest thousand = 76,000
48399 round off to nearest thousand = 48,000
Difference of two numbers = 76,000 - 48,000 = 28,000
Ans. Estimated Difference is 28,000.
Q7. Estimate each of the following products by rounding off each number to the nearest ten :
(i) 49 × 52
Solution :
49 round off to nearest ten = 50
52 round off to nearest ten = 50
Product of two numbers = 50 x 50 = 2500
Ans. Estimated Product is 2500.
(ii) 63 x 38
Solution :
63 round off to nearest ten = 60
38 round off to nearest ten = 40
Product of two numbers = 60 x 40 = 2400
Ans. Estimated Product is 2400.
(iii) 27 x 54
Solution :
27 round off to nearest ten = 30
54 round off to nearest ten = 50
Product of two numbers = 20 x 50 = 1500
Ans. Estimated Product is 1500.
(iv) 53 x 85
Solution :
53 round off to nearest ten = 50
85 round off to nearest ten = 90
Product of two numbers = 50 x 90 = 4500
Ans. Estimated Product is 4500.
(v) 74 ×67
Solution :
74 round off to nearest ten = 70
67 round off to nearest ten = 70
Product of two numbers = 70 x 70 = 4900
Ans. Estimated Product is 4900.
(vi) 25 × 33
Solution :
25 round off to nearest ten = 30
33 round off to nearest ten = 30
Product of two numbers = 30 x 30 = 900
Ans. Estimated Product is 900.
Q8. Estimate each of the following products by rounding off each number to the nearest hundred :
(i) 477 ×213
Solution :
477 round off to nearest hundred = 500
213 round off to nearest hundred = 200
Product of two numbers = 500 x 200 = 1,00,000
Ans. Estimated Product is 1,00,000.
(ii) 624 × 236
Solution :
624 round off to nearest hundred = 600
236 round off to nearest hundred = 200
Product of two numbers = 600 x 200 = 1,20,000
Ans. Estimated Product is 1,20,000.
(iii) 333 x 247
Solution :
333 round off to nearest hundred = 300
247 round off to nearest hundred = 200
Product of two numbers = 300 x 200 = 60,000
Ans. Estimated Product is 60,000.
(iv) 537 x 283
Solution :
537 round off to nearest hundred = 500
283 round off to nearest hundred = 300
Product of two numbers = 500 x 300 = 1,50,000
Ans. Estimated Product is 1,50,000.
(v) 382 × 127
Solution :
382 round off to nearest hundred = 400
127 round off to nearest hundred = 100
Product of two numbers = 400 x 100 = 40,000
Ans. Estimated Product is 40,000.
(vi) 472 × 328
Solution :
472 round off to nearest hundred = 500
328 round off to nearest hundred = 300
Product of two numbers = 500 x 300 = 1,50,000
Ans. Estimated Product is 1,50,000.
Q9. 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 x 287
Solution :
28 round off to nearest ten = 30
287 round off to nearest hundred = 300
Product of two numbers = 30 x 300 = 9,000
Ans. Estimated Product is 9,000.
(ii) 432 x 128
Solution :
432 round off to nearest ten = 430
128 round off to nearest hundred = 100
Product of two numbers = 430 x 100 = 43,000
Ans. Estimated Product is 43,000.
(iii) 48 x 165
Solution :
48 round off to nearest ten = 50
165 round off to nearest hundred = 200
Product of two numbers = 50 x 200 = 10,000
Ans. Estimated Product is 10,000.
(iv) 72 x 258
Solution :
72 round off to nearest ten = 70
258 round off to nearest hundred = 300
Product of two numbers = 70 x 300 = 21,000
Ans. Estimated Product is 21,000.
(v) 83 x 664
Solution :
83 round off to nearest ten = 80
664 round off to nearest hundred = 700
Product of two numbers = 80 x 700 = 56,000
Ans. Estimated Product is 56,000.
(vi) 44 × 250
Solution :
44 round off to nearest ten = 40
250 round off to nearest hundred = 300
Product of two numbers = 40 x 300 = 12,000
Ans. Estimated Product is 12,000.
Q10. Estimate each of the following quotients by rounding each number to the nearest ten :
(i) 87 ÷ 28
Solution :
87 round off to nearest ten = 90
28 round off to nearest ten = 30
Division of two numbers = 90 ÷ 30 = 3
Ans. Estimated quotient is 3.
(ii) 84 ÷ 23
Solution :
84 round off to nearest ten = 80
23 round off to nearest ten = 20
Division of two numbers = 80 ÷ 20 = 4
Ans. Estimated quotient is 4.
(iii) 77 ÷ 22
Solution :
77 round off to nearest ten = 80
22 round off to nearest ten = 20
Division of two numbers = 80 ÷ 20 = 4
Ans. Estimated quotient is 4.
(iv) 198 ÷ 24
Solution :
198 round off to nearest ten = 200
24 round off to nearest ten = 20
Division of two numbers = 200 ÷ 20 = 10
Ans. Estimated quotient is 10.
(v) 355 ÷ 26
Solution :
355 round off to nearest ten = 360
26 round off to nearest ten = 30
Division of two numbers = 360 ÷ 30 = 12
Ans. Estimated quotient is 12.
(vi) 444 ÷ 42
Solution :
444 round off to nearest ten = 440
42 round off to nearest ten = 40
Division of two numbers = 440 ÷ 40 = 11
Ans. Estimated quotient is 11.
(vii) 843 ÷ 33
Solution :
843 round off to nearest ten = 840
33 round off to nearest ten = 30
Division of two numbers = 840 ÷ 30 = 28
Ans. Estimated quotient is 28.
Q11. 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.
Solution :
Number of houses in the town = 768
Round off to nearest ten = 770
Number of people in each house = 7
Round off to nearest ten = 10
Product of two numbers = 770 x 10 = 7700
Ans. Estimated number of people living in the town are 7700.
Q12. 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, correct to the nearest hundred ?
Solution :
Number of mango trees in an orchard = 514
Round off to nearest hundred = 500
Number of trees plan to be planted = 178
Round off to nearest hundred = 200
Total number of mango trees in the orchard = 500 + 200 = 700
Ans. Estimated number of mango trees in the orchard are 700.
-- Multiple Choice Type --
# Choose the correct answer from the options given below.
Q1. The sum of place value of two 3s in the number 73423 is :
(i) 76423
(ii) 73426
(iii) 3003 ✅
(iv) none of these
Q2. 53729 is greater than 53972 :
(i) true
(ii) false ✅
(iii) none of these
Q3. The difference between the largest and the smallest 3-digit numbers is :
(i) 10,000
(ii) 899 ✅
(iii) 1,099
(iv) none of these
Q4. The number of two digit numbers from 40 to 70 is :
(i) 30
(ii) 31 ✅
(iii) 29
(iv) none of these
Q5. If A = 83, B = 12 and C = 17, the value of A - B + C is :
(i) 54
(ii) 112
(iii) 88 ✅
(iv) none of these
Q6. 3,294 + 1,320 to the nearest 10 is :
(i) 4,430
(ii) 4,420
(iii) 4,426
(iv) none of these ✅
Q7. The difference between 652 and 4,038 to the nearest hundred is :
(i) 3,300 ✅
(ii) 3,400
(iii) 3,390
(iv) 3,350
Q8. 2,228 rounded off to the nearest hundred minus 228 to the nearest ten is :
(i) 1,790
(ii) 1,970 ✅
(iii) 2,170
(iv) 2,220
Q9. Sum of digits 3 and 8 used in the numbers 5,03,489 is :
(i) 3,800
(ii) 3,030
(iii) 11 ✅
(iv) none of these
Q10. 80 is multiplied with 23 instead of 32; the result of multiplication differs by :
(i) 80 x (23 - 32)
(ii) 720 ✅
(iii) 80 x 32-23
(iv) none of these
Q11. Statement 1 : The smallest number of 6-digit using only first three digits is 100022.
Statement 2 : To get the largest number, using the given digits, the largest digit is written at the right most and then the remaining digits are written in descending order of their values to the right.
Which of the following options is correct ?
(i) Both the statements are true.
(ii) Both the statements are false.
(iii) Statement 1 is true, and statement 2 is false.
(iv) Statement 1 is false, and statement 2 is true
Explanation :
For statement 1 : The first three digits are 0, 1 and 2 and so the smallest 6-digit number formed is 100002.
[The given statement is false].
For statement 2 : The given statement is false.
The largest digit is placed at the left most and then the other digits are written in descending order of their values to the right. [The given statement is false]
Ans. (ii) Both the statements are false.
Q12. The numbers 25, 42 and 63, also 25 - 42 + 63.
Statement 1 : 25 - 42 + 63 before rounding off the given numbers = 88 - 42 = 46 = 50
Statement 2 : After rounding off the given numbers; 25 - 42 + 63 = 30 - 40 + 60 = 50
(i) Both the statements are true.
(ii) Both the statements are false.
(iii) Statement 1 is true, and statement 2 is false.
(iv) Statement 1 is false, and statement 2 is true. ✅
Q13. Number 89623
Assertion (A) : The sum of place values of digits 6 and 2 is 620.
Reason (R) : The required sum = 600 + 2 = 602.
(1) A is true, R is false.
(2) A is false, R is true.
(3) Both A and R are true.
(4) Both A and R are false.
(i) 1 ✅
(ii) 2
(iii) 3
(iv) 4
Q14. Assertion (A) : In the number 709 , the sum of place value and face value of 0 is 0.
Reason (R) : The place value of a digit depends upon the position it occupies in the number, however the face value of a digit is always the digit itself regardless of the place it occupies in the given number.
(1) A is true, R is false.
(2) A is false, R is true.
(3) Both A and R are true.
(4) Both A and R are false.
(i) 1
(ii) 2
(iii) 3 ✅
(iv) 4
