NCERT solutions for Class 8 Maths Chapter 16 Playing with Numbers

Access premium articles, webinars, resources to make the best decisions for career, course, exams, scholarships, study abroad and much more with

Plan, Prepare & Make the Best Career Choices

# NCERT solutions for Class 8 Maths Chapter 16 Playing with Numbers

Edited By Ramraj Saini | Updated on Mar 01, 2024 06:50 PM IST

Playing with Numbers Class 8 Questions And Answers provided here. These NCERT Solutions are created by expert team at craeers360 keeping in mind of the latest syllabus and pattern of CBSE 2023-23. You have studied various types of numbers such as Integer Numbers, Natural Numbers, Whole Numbers, and Rational Numbers. In NCERT solutions for Class 8 Maths chapter 16 Playing with Numbers, you will find some interesting tricks to find divisibility of numbers. Important topics like divisibility by 10, 5, 2, 9, and 3, numbers in general form, letters for digits are covered in this chapter.

Here you will get the detailed NCERT Solutions for Class 8 Maths by clicking on the link. In Playing with Numbers class 8 solutions, there are some games with numbers and puzzles on the arithmetic problems given in the textbook which makes this chapter very interesting. To strengthen the foundation of the maths you must develop the ability to play with the numbers. For that, you should practice questions given in the textbook. You can take help from NCERT solutions for Class 8 Maths chapter 16 Playing with Numbers that are prepared and explained in a detailed manner.

## Playing with Numbers Class 8 Solutions - Important Formulae

• Divisibility by 2: A number is divisible by 2 when its one's digit is 0, 2, 4, 6, or 8.

• Divisibility by 3: A number is divisible by 3 when the sum of its digits is divisible by 3.

• Divisibility by 4: A number is divisible by 4 when the number formed by its last two digits is divisible by 4.

• Divisibility by 5: A number is divisible by 5 when its one's digit is 0 or 5.

• Divisibility by 6: A number is divisible by 6 when it is divisible by both 2 and 3.

• Divisibility by 9: A number is divisible by 9 when the sum of its digits is divisible by 9.

• Divisibility by 10: A number is divisible by 10 when its one's digit is 0.

• Divisibility by 11: A number is divisible by 11 when the difference of the sum of its digits in odd places and the sum of its digits in even places is either 0 or a multiple of 11.

Free download NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers for CBSE Exam.

## Playing with Numbers Class 8 NCERT Solutions (Intext Questions and Exercise)

Class 8 maths chapter 16 NCERT solutions - Topic 16.2 Numbers in General Form

Question:

25

any two digit number ab made of digits a and b can be written as ab = 10 × a + b = 10a + b

hence,

25 = 10*2 + 5 = 20 + 5

Question:

73

any two digit number ab made of digits a and b can be written as ab = 10 × a + b = 10a + b

hence generalized form of the number

73 = 10*7 + 3

Question:1(iii)

129

A 3 digit number madeup of digits a, b, c will be written as,

abc = 100 × a + 10 × b + 1 × c = 100a + 10b + c

hence generalised form of number

129 = 100*1 + 10*2 + 1*9

Question:

302

A 3-digit number abc made up of digits a, b and c is written as abc = 100 × a + 10 × b + 1 × c = 100a + 10b + c

hence generalised form of

302 = 100*3 + 10*0 + 1*2

Question:2(i)

$10\times 5 + 6$

As we know ab = 10 × a + b = 10a + b

usual form of number

10*5 + 6 = 50 + 6 = 56

Question:

$100\times 7 + 10\times 1 + 8$

As we know abc = 100 × a + 10 × b + 1 × c

the usual form of number

100*7 + 10*1 + 8 = 700 + 10 + 8 = 718.

Question:2(iii)

$100\times a + 10\times c + b$

As we know abc = 100 × a + 10 × b + 1 × c

casual form of 100 × a + 10 × b + 1 × c = abc.

NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers - Topic 16.3 Games with Numbers

27

As we know

(10a + b) + (10b + a) = 11a + 11b = 11 (a + b)

here a = 2 and b = 7

27 + 72 = 99= 11 *9 = a multiple of 11

39

39 + 93 = 133 = 11*12 = a multiple of 11

this can be explained by

(10a + b) + (10b + a) = 11a + 11b = 11 (a + b)

Here a = 3 and b = 9

64

64 + 46 = 110 = 11*10= a multiple of 11

this can be explained by

(10a + b) + (10b + a) = 11a + 11b = 11 (a + b)

here a = 6 and b = 4

17

17 + 71 = 88 = 11*8 = a multiple of 11

this can be explained by

(10a + b) + (10b + a) = 11a + 11b = 11 (a + b)

here a = 1 and b = 7

Class 8 playing with numbers NCERT solutions - Topic 16.3 Games with Numbers

17

If the tens digit is larger than the ones digit (that is, a > b), then
(10a + b) – (10b + a) = 10a + b – 10b – a
= 9a – 9b = 9(a – b).
If the unit digit is larger than the tens digit (that is, b > a), he does:
(10b + a) – (10a + b) = 9(b – a).

here a = 1 and b = 7

71 - 17 = 54 = 9*6 = multiple of 9

21

If the tens digit is larger than the ones digit (that is, a > b), then
(10a + b) – (10b + a) = 10a + b – 10b – a
= 9a – 9b = 9(a – b).
If the unit digit is larger than the tens digit (that is, b > a), he does:
(10b + a) – (10a + b) = 9(b – a).

here a = 2 and b = 1

21 - 12 = 9 = 9*1 = multiple of 9

96

If the tens digit is larger than the ones digit (that is, a > b), then
(10a + b) – (10b + a) = 10a + b – 10b – a
= 9a – 9b = 9(a – b).
If the unit digit is larger than the tens digit (that is, b > a), he does:
(10b + a) – (10a + b) = 9(b – a).

here a = 9 and b = 6

96 - 69 = 27= 9*3= multiple of 9

37

If the tens digit is larger than the ones digit (that is, a > b), then
(10a + b) – (10b + a) = 10a + b – 10b – a
= 9a – 9b = 9(a – b).
If the unit digit is larger than the tens digit (that is, b > a), he does:
(10b + a) – (10a + b) = 9(b – a).

here a = 3 and b = 7

73 - 37 = 36 = 9*4 = multiple of 9

NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers - Topic 16.3 Games with Numbers

132

Let's assume the 3-digit number chosen by Minakshi = 100a + 10b + c.

After reversing the order of the digits, number = 100c + 10b + a.

On subtraction:

• If a > c, then the difference between the original number & reversed number

(100a + 10b + c) – (100c + 10b + a) = 100a + 10b + c – 100c – 10b – a = 99a – 99c = 99(a – c).

• If c > a, then the difference between the numbers is (100c + 10b + a) – (100a + 10b + c) = 99c – 99a = 99(c – a).

• If a & c are equal,then the difference is 0.

here a = 1, b = 3 and c = 2

231 - 132 = 99 = multiple of 99

469

Let the 3-digit number chosen by Minakshi = 100a + 10b + c.

After reversing the order of the digits, number = 100c + 10b + a.

On subtraction:

• If a > c, then the difference between the original numbedr & reversed numbers is

(100a + 10b + c) – (100c + 10b + a) = 100a + 10b + c – 100c – 10b – a = 99a – 99c = 99(a – c).

• If c > a, then the difference between the numbers is (100c + 10b + a) – (100a + 10b + c) = 99c – 99a = 99(c – a).

• if a & c are equal, then the difference is 0.

here a,b and c are 4, 6 & 9 respectively.

964 - 469 = 495 = 99*5 = multiple of 99

737

Let the 3-digit number chosen by Minakshi = 100a + 10b + c.

After reversing the order of the digits, number = 100c + 10b + a.

On subtraction:

• If a > c, then the difference between the original number & reversed number is

(100a + 10b + c) – (100c + 10b + a) = 100a + 10b + c – 100c – 10b – a = 99a – 99c = 99(a – c).

• If c > a, then the difference between the numbers is (100c + 10b + a) – (100a + 10b + c) = 99c – 99a = 99(c – a).

• If A 7 C are equal, the difference is 0.

here a = 7, b = 3 and c = 7

737- 737 = 0= multiple of 99

901

Let the 3-digit number chosen by Minakshi = 100a + 10b + c.

After reversing the order of the digits, number = 100c + 10b + a.

On subtraction:

• If a > c, then the difference between the original number & reversed number is

(100a + 10b + c) – (100c + 10b + a) = 100a + 10b + c – 100c – 10b – a = 99a – 99c = 99(a – c).

• If c > a, then the difference between the numbers is (100c + 10b + a) – (100a + 10b + c) = 99c – 99a = 99(c – a).

• If a & c are equal, the difference is 0.

here a = 9, b = 0 and c = 1

901- 109= 792= 99*8 = multiple of 99

quotient in each case = c - a.

NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers - Topic 16.3 Games with Numbers

417

Let choosen number be abc then,

abc = 100a + 10b + c

cab = 100c + 10a + b

bca = 100b + 10c + a

After adding all the above three, abc + cab + bca = 111(a + b + c) = 37 × 3(a + b + c),

It will be divisible by 37 becuase 37 is present in the equation.

here a = 4, b = 1, and c = 7

417 + 741 + 147 = 1332 = 37*36 i.e. divisible by 37.

632

Let choosen number be abc then,

abc = 100a + 10b + c

cab = 100c + 10a + b

bca = 100b + 10c + a

The addition of all the above three, abc + cab + bca = 111(a + b + c) = 37 × 3(a + b + c), which is divisible by 37

here a = 6, b = 3, and c = 2

632+ 263+ 362= 1221= 37*33 i.e. divisible by 37.

117

Let choosen number be abc then,

abc = 100a + 10b + c

cab = 100c + 10a + b

bca = 100b + 10c + a

The addition of the above all three, abc + cab + bca = 111(a + b + c) = 37 × 3(a + b + c), which is divisible by 37

here a = 1, b = 1, and c = 7

117 + 711 + 117 = 999 = 37*27 i.e. divisible by 37.

937

Let choosen number be abc then,

abc = 100a + 10b + c

cab = 100c + 10a + b

bca = 100b + 10c + a

The addition of all above three, abc + cab + bca = 111(a + b + c) = 37 × 3(a + b + c), which is divisible by 37

here a = 9, b = 3, and c = 7

937 + 793 + 397 = 2109 = 37*57 i.e. divisible by 37.

NCERT class 8 maths chapter 16 question answer - Exercise: 16.1

Here we are adding two numbers and unit place of the first number and the second number is A and 5 respectively. unit place of the answer is 2 so the way we can get this result is when we get 12 on adding unit places of both number i.e.

A + 5 = 12

which implies A = 12 - 5 = 7.

Ten's digit of both numbers are 3 and 2.remainder=1

so ten's digit of the answer(B) = 3 + 2 + 1 = 6

Hence A= 7 and B = 6.

Here

answer's unit place = 3 , possible addition of unit places digit = 13

A + 8 = 13

A = 13 - 8 = 5

remainder = 1

Ten's place of answer = 4 + 9 + 1 = 14

B = 4

remainder = 1

100's place = C = 1

Hence value of A = 5 , B = 4 and C = 1.

Here

first clue :

we have A = a number which when multiplied by itself gives the same number in the unit digit.

possible numbers = 1 and 6

Second Clue:

number when multiplied with 1 and added with the reminder of previous multiplication( A*A) = 9

both first and second clue implies that A = 6.

There can be two cases

1. when the addition of unit place digit doesn't produce Carry

A + 3 = 6

A = 3

However, to get 3 in unit place of our answer our B has to be 6 and that would produce carry hence this case is not possible.

2. when the addition of unit place digit produces Carry

A + 3 +1 = 6

A = 2

for getting 2 in unit place of answer we need the sum of unit digit of numbers = 12

B + 7 = 12

B = 5

Hence A = 2 and B = 5

Here multiplication of 3 and B gives a number whose unit place digit is B .

Possible value of B = 0 and 5

let B = 5

3 * A + 1 = CA

this is not possible for any value of A.

Hence B = 0

now A * 3 = CA ( a number whose unit place digit is A itself when multiplied by 3) hence

possible value of A = 5 and 0

since AB is a two digit number A can not equal to 0.

hence A = 5

A * 3 + 1 = CA

5 * 3 + 1 = 15

hence C = 1

A = 5, B = 5 and C = 1.

Here , multiplication of B and 5 gives a number whose ones digit is B. this is possible when B = 0 or 5

let B = 5

B * 5 = 5 * 5 = 25

Carry = 2

5*A + 2 = CA , This os possible only when A = 2 or 7

when A = 2 ,

5*2 + 2 = 12 which implies C = 1

when A = 7

5*7 + 2 = 37 which implies C = 3

now B = 0

B*5 = 0*5 = 0

Carry = 0 so

5 * A = CA which is possible when A = 0 or 5 Howerver A cannot equal to 0 since AB is a two digit number

so A = 5

5*5 = 25 which implies C = 2

hence possible values of A B and C are

A = 5, 2, 7 , B = 0, 5, 5 and C = 2, 1, 3

The product of 6 & B gives a number whose unit digit is B again.

possible value of B = 0. 2 , 4, 6 or 8

If B = o then our product will be zero. hence this value of B is not possible.

If B = 2, then B x 6 = 12 . Carry for next step = 1.

6A + 1 = BB = 22

implies A = 21/6 = not any integer value hence this case is also not possible.

If B = 6 then B *6 = 36 and 3 will be carry for next step.

6A + 3 = BB = 66

implies A = 63/6 = not an integer value hence this case is also not possible.

If B = 8 then B * 6 = 48 and 4 will be carry for next step.

6A + 4 = BB = 88

implies A = 14 however A is a single digit number hence this case is also not possible.

If B = 4 then B*6 = 24 and 2 will be carry for next step.

6A + 2 = BB = 22

implies A = 7 .

Hence A = 7 and B = 4 is the correct answer.

The addition of 1 and B gives a number whose ones digit is 0. this is possible when digit B = 9 .

1 + B = 10 and 1 is the Carry for the next step

Now, A + 1 + 1 = B => 9

Implies A = 7.

Hence A = 7 and B = 9 is Correct answer

The additiion of B and 1 is 8 is giving a number whose ones digit is 8. this means digit B is 7 .

B + 1 = 8 and no carry for next step.

next step :

Now, A + B = 1 => A + 7

which implies A = 4

A + B = 11 and 1 is carry for next step

1 + 2 + A = B

1 + 2 + 4 = 7

Hence A = 4 and B = 7 is correct answer.

The addition of A and B is giving a number whose ones digit is 9. The sum can only be 9 not 10 as a sum of two single digits cannot exceed 18. hence there will not be any carry for the next step

2 + A = 0

implies A = 8

2 + 8 = 10 and 1 is the carry for next step.

1 + 1 + 6 = A = 8

it satisfies hence A = 8 and B = 1 is the correct answer.

NCERT playing with numbers class 8 solutions - Topic 16.5.2 Divisibility By 5

The detailed solution for the above-written problem is as follows,

The unit digit, when divided by 5, must be leaving a remainder of 3. So the unit digit must be either 3 or 8.

The detailed solution for the above-written question is as follows

If a number is divisible by 5 then it's unit digit must be 0 or 5. so if we need the remainder of 1 when divided by 5 then the numbers unit digit must be 1 or 6.

The detailed solution for the above-written question is as follows

If the unit digit of a number is 0 or 5, then it is divisible by 5. hence if we need the remainder of 4 then unit digit of number should be 4 or 9.

Playing with numbers class 8 NCERT solutions - Topic 16.5.3 Divisibility By 2

The detailed solution for the above-written question is as follows

N is odd; so it's unit digit is odd. Therefore, the unit digit must be 1, 3, 5, 7 or 9.

The detailed solution for the above-written question is as follows

N is Even; so it's unit digit is even. Therefore, the unit digit must be 2, 4, 6, 8 or 0.

Since N leaves the remainder of 4 when divided by 5. the possible values in ones place of number N are 4 or 9.

now, since it leaves a remainder of 1 when divided by 2, the N would be an odd number. hence ones digit of N is also an odd number. which means ones digit of our number N is 9.

NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers - Topic 16.5.4 Divisibility By 9 And 3

108

Any number will be divisible by 9 only if the sum of all the digits in that number will be divisible by 9.

Sum of digit of 108 = 1 + 0 + 8 = 9 which is divisible by 9 i.e (9/9 = 1).

hence we conclude 108 is divisible by 9.

616

Any number will be divisible by 9 only if the sum of all the digits in that number will be divisible by 9.

Sum of digits of 616 = 6 + 1 + 6 = 13 which is not divisible by 9,

Hence we conclude 616 is not divisible by by 9.

294

Any number will be divisible by 9 only if the sum of all the digits in that number will be divisible by 9.

Sum of digits of 294 = 2 + 9 + 4 = 15 which is not divisible by 9.

Hence we conclude 294 is not divisible by 9.

432

Any number will be divisible by 9 only if the sum of all the digits in that number will be divisible by 9.

Sum of digit 432 = 4 + 3 + 2 = 9 which is divisible by 9.

Hence we conclude 432 is divisible by 9.

927

Any number will be divisible by 9 only if the sum of all the digits in that number will be divisible by 9.

Sum of digits of 927 = 9 + 2 + 7 = 18 which is divisible by 18.

Hence we conclude number 927 is divisible by 9.

NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers - Topic Think, Discuss And Write

Yes, it has been prooved that if a number is divisible by any number m , then it will also be divisible by each of the factors of m.

Let's Assume n is divisible by m, and m is divisible by k.This means

n=pm and

m = qk where all are integers

Now,

n = p(qk) =( pq)k which means n is divisible by k.

Hence a number is divisible by any number m , then it will also be divisible by each of the factors of m.

Is it necessary that $(a - b + c)$ should be divisible by 11?

let the number abc be 132

Here a = 1, b = 3 and c = 2

132= 100*1 + 10*3 + 2 = 99 + 11*3 + (1 - 3 + 2)

= 11(9*1+3) + (1 - 3 + 2 )

if number is divisible by 11 then (a - b + c ) must be divisible by 11.

as in above case of number 132 the a - b + c = 1 -3 + 2 = 0 whichis divisible by 11.

Hence we conclude ( a - b + c ) should be divisible by 11 if abc is divisible by 11.

If the number abcd is divisible by 11 then [ (b + d) - (a + c) ] also must be divisible by 11.

let the number be 1089

here a = 1, b = 0, c = 8 and d = 9

1089 = 1000*1 + 100*0 + 10*8 + 9

= (1001*1 + 99*0 + 11*8) + [(0 + 9) - (1 + 8)]

= 11(91*1 + 9*0 + 8) + [ 9 - 9 ]

here [ (b + d) - (a + c) ] = [9 - 9 ] = 0 which is divisible by 11.

hence If the number abcd is divisible by 11 then [ (b + d) - (a + c) ] also must be divisible by 11.

Yes,

A number will always be divisible by 11 if the difference between the sum of digits at its odd places and that of digits at the even places is divisible by 11.

So, for instance, 2728 has the alternating sum of digits 2-7+2-8 = -11. Here -11 is divisible by 11, so is 2728.

Similarly, for 31415, the alternating sum of digits is 3-1+4-1+5 = 10. This would not divisible by 11, so neither is 31415.

NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers - Topic 16.5.4 Divisibility By 9 And 3

108

Any number will be divisible by 3 only if the sum of all the digits in that number will be divisible by 3.

Sum of digits of number 108 = 1 + 0 +8 = 9 which is divisible by 3.

Hence we conclude number 108 is divisible by 3.

616

Any number will be divisible by 3 only if the sum of all the digits in that number will be divisible by 3.

Sum of digits of number 616 = 6 + 1 + 6 = 13 which is not divisible by 3.

Hence we conclude number 616 is not divisible by 3.

294

Any number will be divisible by 3 only if the sum of all the digits in that number will be divisible by 3.

Sum of digits of number 294= 2 + 9 + 4 = 15 which is divisible by 3.

Hence we conclude number 294 is divisible by 3.

432

Any number will be divisible by 3 only if the sum of all the digits in that number will be divisible by 3.

Sum of digits of number 432 = 4 + 3 + 2 = 9 which is divisible by 3.

Hence we conclude number 432 is divisible by 3.

927

Any number will be divisible by 3 only if the sum of all the digits in that number will be divisible by 3.

Sum of digits of number 927 = 9 + 2 + 7 = 18 which is divisible by 3.

Hence we conclude number 108 is divisible by 3.

NCERT Solutions for Class 8 Maths Chapter 16 Playing with Numbers - Exercise: 16.2

If a number is a multiple of 9 , then the sum of its digit will be divisible by 9.

Sum of digits of 21y5= 2 + 1 + y + 5 = 8 + y

8 + y is a multiple of 9 when y = 1 (since y can only be single digit )

hence y = 1 is correct answer.

If a number is a multiple of 9, then the sum of its digit will be divisible by 9 .

Sum of digits of 31z5 = 3 + 1 + z + 5 = 9 + z

hence 9 + z have to be multiple of 9

this possible when 9 + z = 0, 9 , 18, 27 ...

since z is a single digit number this sum can only be 9 or 18 . therefore, z should be 0 or 9 .

hence two possible values of z = 0 or 9.

(Since $24x$ is a multiple of 3, its sum of digits $6 + x$ is a multiple of 3; so $6 + x$ is one of these numbers: 0, 3, 6, 9, 12, 15, 18, ... . But since $x$ is a digit, it can only be that $6 + x = 6 \;or\; 9 \;or \;12 \;or\; 15$ . Therefore, $x = 0 \;or\; 3 \;or \;6 \;or\;9$ . Thus, $x$ can have any of four different values.)

if a number is multiple of 3 then the sum of its digits is also a multiple of 3.

Summ of digits of 24x = 2 + 4 + x = 6 + x

6 + x is a multiple of 3 which means

6 + x = 0 or 3 or 9 or 12....

but since x is a single digit the possible sum is 6 or 9 or 12 or 15 and hence value of x are 0 or 3 or 6 or 9 respectively.

hence possible value of x = 0, 3, 6, or 9.

If a number is multiple of 3, the sum of its digits will be multiple of 3.

Sum of digits of 31z5 = 3 + 1 + z + 5 = 9 + z

9 + z is a multiple of 3 and since z is a single digit

9 + z is any one of 9, 12, 15 or 18

thus value of z is 0, 3, 6 or 9 respectively.

Hence the possible value of z is 0,3,6 or 9.

## Playing with numbers class 8 NCERT solutions - Topics

• Numbers in General Form
• Games with Numbers
• Letters for Digits
• Tests of Divisibility

## NCERT Solutions for Class 8 Maths - Chapter Wise

 Chapter -1 Rational Numbers Chapter -2 Linear Equations in One Variable Chapter-3 Understanding Quadrilaterals Chapter-4 Practical Geometry Chapter-5 Data Handling Chapter-6 Squares and Square Roots Chapter-7 Cubes and Cube Roots Chapter-8 Comparing Quantities Chapter-9 Algebraic Expressions and Identities Chapter-10 Visualizing Solid Shapes Chapter-11 Mensuration Chapter-12 Exponents and Powers Chapter-13 Direct and Inverse Proportions Chapter-14 Factorization Chapter-15 Introduction to Graphs Chapter-16 Playing with Numbers

## Key Features Of Playing With Numbers Class 8 NCERT Solutions

Comprehensive Coverage: Maths chapter 16 class 8 solutions cover all topics and concepts related to playing with numbers as per the Class 8 syllabus.

Step-by-Step Solutions: Class 8 maths ch 16 question answer are detailed, step-by-step explanations for each problem, making it easy for students to understand and apply mathematical concepts related to number patterns, divisibility, and prime factors.

Variety of Problems: A wide range of problems, including exercises and additional questions, to help students practice and test their understanding of number patterns and properties.

## Check NCERT Books and NCERT Syllabus

1. What are the important topics of ch 16 maths class 8 Playing with Numbers ?

Numbers in general form, letters for digits, and divisibility test are the important topics of this chapter.

2. Does CBSE class maths is tough ?

In CBSE class 8 you will study basic and very simple maths where most of the topics related to the previous classes.

3. How many chapters are there in the CBSE class 8 maths ?

There are 16 chapters starting from rational number to playing with numbers in the CBSE class 8 maths.

4. Where can I find the complete solutions of NCERT for class 8 ?

Here you will get the detailed NCERT solutions for class 8 by clicking on the link.

5. Where can I find the complete solutions of NCERT for class 8 maths ?

Here you will get the detailed NCERT solutions for class 8 maths by clicking on the link.

6. Which is the official website of NCERT ?

NCERT official is the official website of the NCERT where you can get NCERT textbooks and syllabus from class 1 to 12.

## Upcoming School Exams

#### National Institute of Open Schooling 12th Examination

Admit Card Date:28 March,2024 - 22 May,2024

#### National Institute of Open Schooling 10th examination

Admit Card Date:28 March,2024 - 22 May,2024

#### Punjab Board of Secondary Education 12th Examination

Exam Date:05 April,2024 - 27 April,2024

#### Bihar Board 12th Examination

Admit Card Date:19 April,2024 - 11 May,2024

#### Nagaland Board High School Leaving Certificate Examination

Result Date:26 April,2024 - 26 April,2024

Get answers from students and experts

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

 Option 1) Option 2) Option 3) Option 4)

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

 Option 1) 2.45×10−3 kg Option 2)  6.45×10−3 kg Option 3)  9.89×10−3 kg Option 4) 12.89×10−3 kg

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

 Option 1) Option 2) Option 3) Option 4)

A particle is projected at 600   to the horizontal with a kinetic energy . The kinetic energy at the highest point

 Option 1) Option 2) Option 3) Option 4)

In the reaction,

 Option 1)   at STP  is produced for every mole   consumed Option 2)   is consumed for ever      produced Option 3) is produced regardless of temperature and pressure for every mole Al that reacts Option 4) at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?

 Option 1) 0.02 Option 2) 3.125 × 10-2 Option 3) 1.25 × 10-2 Option 4) 2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

 Option 1) decrease twice Option 2) increase two fold Option 3) remain unchanged Option 4) be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

 Option 1) Molality Option 2) Weight fraction of solute Option 3) Fraction of solute present in water Option 4) Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

 Option 1) twice that in 60 g carbon Option 2) 6.023 × 1022 Option 3) half that in 8 g He Option 4) 558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

 Option 1) less than 3 Option 2) more than 3 but less than 6 Option 3) more than 6 but less than 9 Option 4) more than 9

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

4 Jobs Available
##### Bio Medical Engineer

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.

4 Jobs Available
##### Ethical Hacker

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

3 Jobs Available
##### GIS Expert

GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.

3 Jobs Available
##### Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
##### Geothermal Engineer

Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

3 Jobs Available
##### Database Architect

If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi

3 Jobs Available
##### Remote Sensing Technician

Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.

Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.

3 Jobs Available
##### Budget Analyst

Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.

4 Jobs Available
##### Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
##### Underwriter

An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Operations Manager

Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

3 Jobs Available
##### Stock Analyst

Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.

2 Jobs Available
##### Researcher

A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.

2 Jobs Available
##### Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.

5 Jobs Available
##### Transportation Planner

A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

3 Jobs Available
##### Environmental Engineer

Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.

2 Jobs Available
##### Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available
##### Conservation Architect

A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

2 Jobs Available
##### Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

2 Jobs Available
##### Highway Engineer

Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.

2 Jobs Available
##### Field Surveyor

Are you searching for a Field Surveyor Job Description? A Field Surveyor is a professional responsible for conducting field surveys for various places or geographical conditions. He or she collects the required data and information as per the instructions given by senior officials.

2 Jobs Available
##### Orthotist and Prosthetist

Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

6 Jobs Available
##### Pathologist

A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

5 Jobs Available
##### Gynaecologist

Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.

4 Jobs Available
##### Audiologist

The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.

3 Jobs Available
##### Oncologist

An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

3 Jobs Available
##### Anatomist

Are you searching for an ‘Anatomist job description’? An Anatomist is a research professional who applies the laws of biological science to determine the ability of bodies of various living organisms including animals and humans to regenerate the damaged or destroyed organs. If you want to know what does an anatomist do, then read the entire article, where we will answer all your questions.

2 Jobs Available
##### Actor

For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.

4 Jobs Available
##### Acrobat

Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

3 Jobs Available
##### Video Game Designer

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.

Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
##### Choreographer

The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.

2 Jobs Available
##### Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available
##### Photographer

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available
##### Producer

An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

2 Jobs Available
##### Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.

3 Jobs Available
##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
##### Reporter

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available
##### Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.

5 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.

2 Jobs Available
##### Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

2 Jobs Available
##### Process Development Engineer

The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.

2 Jobs Available
##### AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.

4 Jobs Available

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.

4 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Automation Test Engineer

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.

2 Jobs Available