R d sharma solutions mathematics class 8 Solutions for Chapter 1 rational numbers

Add the following rational numbers:

(i) -5/7 and 3/7

(ii) -15/4 and 7/4

(iii) -8/11 and -4/11

(iv) 6/13 and -9/13

Add the following rational numbers:

(i) 3/4 and -5/8

(ii) 5/-9 and 7/3

(iii) -3 and 3/5

(iv) -7/27 and 11/18

(v) 31/-4 and -5/8

(vi) 5/36 and -7/12

(vii) -5/16 and 7/24

(viii) 7/-18 and 8/27

Simplify:

(i) 8/9 + -11/6

(ii) 3 + 5/-7

(iii) 1/-12 + 2/-15

(iv) -8/19 + -4/57

(v) 7/9 + 3/-4

(vi) 5/26 + 11/-39

(vii) -16/9 + -5/12

(viii) -13/8 + 5/36

(ix) 0 + -3/5

(x) 1 + -4/5

Add and express the sum as a mixed fraction:

(i) -12/5 and 43/10

(ii) 24/7 and -11/4

(iii) -31/6 and -27/8

(iv) 101/6 and 7/8

Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers:

(i) -11/5 and 4/7

(ii) 4/9 and 7/-12

(iii) -3/5 and -2/-15

(iv) 2/-7 and 12/-35

(v) 4 and -3/5

(vi) -4 and 4/-7

Verify associativity of addition of rational numbers i.e., (x + y) + z = x + (y + z), when:

(i) x = ½, y = 2/3, z = -1/5

(ii) x = -2/5, y = 4/3, z = -7/10

(iii) x = -7/11, y = 2/-5, z = -3/22

(iv) x = -2, y = 3/5, z = -4/3

Write the additive of each of the following rational numbers:

(i) -2/17

(ii) 3/-11

(iii) -17/5

(iv) -11/-25

Write the negative(additive) inverse of each of the following:

(i) -2/5

(ii) 7/-9

(iii) -16/13

(iv) -5/1

(v) 0

(vi) 1

Using commutativity and associativity of addition of rational numbers, express each of the following as a rational number:

(i) 2/5 + 7/3 + -4/5 + -1/3

(ii) 3/7 + -4/9 + -11/7 + 7/9

(iii) 2/5 + 8/3 + -11/15 + 4/5 + -2/3

(iv) 4/7 + 0 + -8/9 + -13/7 + 17/21

Re-arrange suitably and find the sum in each of the following:

(i) 11/12 + -17/3 + 11/2 + -25/2

(ii)-6/7 + -5/6 + -4/9 + -15/7

(iii) 3/5 + 7/3 + 9/ 5+ -13/15 + -7/3

(iv) 4/13 + -5/8 + -8/13 + 9/13

(v) 2/3 + -4/5 + 1/3 + 2/5

(vi) 1/8 + 5/12 + 2/7 + 7/12 + 9/7 + -5/16

Subtract the first rational number from the second in each of the following:

(i) 3/8, 5/8

(ii) -7/9, 4/9

(iii) -2/11, -9/11

(iv) 11/13, -4/13

(v) ¼, -3/8

(vi) -2/3, 5/6

(vii) -6/7, -13/14

(viii) -8/33, -7/22

Evaluate each of the following:

(i) 2/3 – 3/5

(ii) -4/7 – 2/-3

(iii) 4/7 – -5/-7

(iv) -2 – 5/9

(v) -3/-8 – -2/7

(vi) -4/13 – -5/26

(vii) -5/14 – -2/7

(viii) 13/15 – 12/25

(ix) -6/13 – -7/13

(x) 7/24 – 19/36

(xi) 5/63 – -8/21

The sum of the two numbers is 5/9. If one of the numbers is 1/3, find the other.

The sum of the two numbers is -1/3. If one of the numbers is -12/3, find the other

The sum of the two numbers is -4/3. If one of the numbers is -5, find the other.

The sum of the two rational numbers is -8. If one of the numbers is -15/7, find the other.

What should be added to -7/8 so as to get 5/9?

What number should be added to -5/11 so as to get 26/33?

What number should be added to -5/7 to get -2/3?

What number should be subtracted from -5/3 to get 5/6?

What number should be subtracted from 3/7 to get 5/4?

What should be added to (2/3 + 3/5) to get -2/15?

What should be added to (1/2 + 1/3 + 1/5) to get 3?

What number should be subtracted from (3/4 – 2/3) to get -1/6?

Fill in the blanks:

(i) -4/13 – -3/26 = ….

(ii) -9/14 + …. = -1

(iii) -7/9 + …. =3

(iv) … + 15/23 = 4

Simplify each of the following and write as a rational number of the formp/q:

(i) 3/4 + 5/6 + -7/8

(ii) 2/3 + -5/6 + -7/9

(iii) -11/2 + 7/6 + -5/8

(iv) -4/5 + -7/10 + -8/15

(v) -9/10 + 22/15 + 13/-20

(vi) 5/3 + 3/-2 + -7/3 +3

Express each of the following as a rational number of the form p/q:

(i) -8/3 + -1/4 + -11/6 + 3/8 – 3

(ii) 6/7 + 1 + -7/9 + 19/21 + -12/7

(iii) 15/2 + 9/8 + -11/3 + 6 + -7/6

(iv) -7/4 +0 + -9/5 + 19/10 + 11/1

(v) -7/4 +5/3 + -1/2 + -5/6 + 2

Simplify:

(i) -3/2 + 5/4 – 7/4

(ii) 5/3 – 7/6 + -2/3

(iii) 5/4 – 7/6 – -2/3

(iv) -2/5 – -3/10 – -4/7

(v) 5/6 + -2/5 – -2/15

(vi) 3/8 – -2/9 + -5/36

Multiply:

(i) 7/11 by 5/4

(ii) 5/7 by -3/4

(iii) -2/9 by 5/11

(iv) -3/17 by -5/-4

(v) 9/-7 by 36/-11

(vi) -11/13 by -21/7

(vii) -3/5 by -4/7

(viii) -15/11 by 7

Multiply:

(i) -5/17 by 51/-60

(ii) -6/11 by -55/36

(iii) -8/25 by -5/16

(iv) 6/7 by -49/36

(v) 8/-9 by -7/-16

(vi) -8/9 by 3/64

Simplify each of the following and express the result as a rational number in standard form:

(i) (-16/21) × (14/5)

(ii) (7/6) × (-3/28)
(iii) (-19/36) × 16

(iv) (-13/9) × (27/-26)

(v) (-9/16) × (-64/-27)

(vi) (-50/7) × (14/3)

(vii) (-11/9) × (-81/-88)

(viii) (-5/9) × (72/-25)

Simplify:

(i) ((25/8) × (2/5)) – ((3/5) × (-10/9))

(ii) ((1/2) × (1/4)) + ((1/2) × 6)

(iii) (-5 × (2/15)) – (-6 × (2/9))

(iv) ((-9/4) × (5/3)) + ((13/2) × (5/6))

(v) ((-4/3) × (12/-5)) + ((3/7) × (21/15))

(vi) ((13/5) × (8/3)) – ((-5/2) × (11/3))

(vii) ((13/7) × (11/26)) – ((-4/3) × (5/6))

(viii) ((8/5) × (-3/2)) + ((-3/10) × (11/16))

Simplify:

(i) ((3/2) × (1/6)) + ((5/3) × (7/2) – (13/8) × (4/3))

(ii) ((1/4) × (2/7)) – ((5/14) × (-2/3) + (3/7) × (9/2))

(iii) ((13/9) × (-15/2)) + ((7/3) × (8/5) + (3/5) × (1/2))

(iv) ((3/11) × (5/6)) – ((9/12) × (4/3) + (5/13) × (6/15))

Verify the property: x × y = y × x by taking:

(i) x = -1/3, y = 2/7

(ii) x = -3/5, y = -11/13

(iii) x = 2, y = 7/-8

(iv) x = 0, y = -15/8

Verify the property: x × (y × z) = (x × y) × z by taking:

(i) x = -7/3, y = 12/5, z = 4/9

(ii) x = 0, y = -3/5, z = -9/4

(iii) x = 1/2, y = 5/-4, z = -7/5

(iv) x = 5/7, y = -12/13, z = -7/18

Verify the property: x × (y + z) = x × y + x × z by taking:

(i) x = -3/7, y = 12/13, z = -5/6

(ii) x = -12/5, y = -15/4, z = 8/3

(iii) x = -8/3, y = 5/6, z = -13/12
(iv) x = -3/4, y = -5/2, z = 7/6

Use the distributivity of multiplication of rational numbers over their addition to simplify:

(i) 3/5 × ((35/24) + (10/1))

(ii) -5/4 × ((8/5) + (16/5))

(iii) 2/7 × ((7/16) – (21/4))
(iv) 3/4 × ((8/9) – 40)

Find the multiplicative inverse (reciprocal) of each of the following rational numbers:

(i) 9

(ii) -7

(iii) 12/5

(iv) -7/9

(v) -3/-5

(vi) 2/3 × 9/4

(vii) -5/8 × 16/15

(viii) -2 × -3/5

(ix) -1

(x) 0/3

(xi) 1

Name the property of multiplication of rational numbers illustrated by the following statements:

(i) -5/16 × 8/15 = 8/15 × -5/16

(ii) -17/5 ×9 = 9 × -17/5

(iii) 7/4 × (-8/3 + -13/12) = 7/4 × -8/3 + 7/4 × -13/12

(iv) -5/9 × (4/15 × -9/8) = (-5/9 × 4/15) × -9/8

(v) 13/-17 × 1 = 13/-17 = 1 × 13/-17

(vi) -11/16 × 16/-11 = 1

(vii) 2/13 × 0 = 0 = 0 × 2/13

(viii) -3/2 × 5/4 + -3/2 × -7/6 = -3/2 × (5/4 + -7/6)

Fill in the blanks:

(i) The product of two positive rational numbers is always…

(ii) The product of a positive rational number and a negative rational number is always….

(iii) The product of two negative rational numbers is always…

(iv) The reciprocal of a positive rational numbers is…

(v) The reciprocal of a negative rational numbers is…

(vi) Zero has …. Reciprocal.

(vii) The product of a rational number and its reciprocal is…

(viii) The numbers … and … are their own reciprocals.

(ix) If a is reciprocal of b, then the reciprocal of b is.

(x) The number 0 is … the reciprocal of any number.

(xi) reciprocal of 1/a, a ≠ 0 is …

(xii) (17×12)-1 = 17-1 × …

Fill in the blanks:

(i) -4 × 7/9 = 79 × …

(ii) 5/11 × -3/8 = -3/8 × … (iii) 1/2 × (3/4 + -5/12) = 1/2 × … + … × -5/12

(iv) -4/5 × (5/7 + -8/9) = (-4/5 × …) + -4/5 × -8/9

Divide:

(i) 1 by 1/2

(ii) 5 by -5/7

(iii) -3/4 by 9/-16

(iv) -7/8 by -21/16

(v) 7/-4 by 63/64

(vi) 0 by -7/5

(vii) -3/4 by -6

(viii) 2/3 by -7/12

(ix) -4 by -3/5

(x) -3/13 by -4/65

Find the value and express as a rational number in standardform:

(i) 2/5 ÷ 26/15

(ii) 10/3 ÷ -35/12

(iii) -6 ÷ -8/17

(iv) -40/99 ÷ -20

(v) -22/27 ÷ -110/18

(vi) -36/125 ÷ -3/75

The product of two rational numbers is 15. If one of the numbers is -10, find the other.

The product of two rational numbers is -8/9. If one of the numbers is -4/15, find the other.

By what number should we multiply -1/6 so that the product may be -23/9? Solution:

By what number should we multiply -15/28 so that the product may be -5/7? Solution:

By what number should we multiply -8/13 so that the product may be 24?

By what number should -3/4 be multiplied in order to produce 2/3?

Find (x+y) ÷ (x-y), if

(i) x= 2/3, y= 3/2

(ii) x= 2/5, y= 1/2

(iii) x= 5/4, y= -1/3

(iv) x= 2/7, y= 4/3

(v) x= 1/4, y= 3/2

The cost of 7 2/3 meters of rope is Rs 12 ¾. Find the cost per meter.

The cost of 2 1/3 meters of cloth is Rs 75 ¼. Find the cost of cloth per meter.

By what number should -33/16 be divided to get -11/4?

Divide the sum of -13/5 and 12/7 by the product of -31/7 and -1/2.

Divide the sum of 65/12 and 12/7 by their difference.

If 24 trousers of equal size can be prepared in 54 meters of cloth, what length of cloth is required for each trouser?

Find a rational number between -3 and 1.

Find any five rational numbers less than 2.

Find two rational numbers between -2/9 and 5/9

Find two rational numbers between 1/5 and 1/2

Find ten rational numbers between 1/4 and 1/2.

Find ten rational numbers between -2/5 and 1/2.

Find ten rational numbers between 3/5 and 3/4.