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The LCM of the denominators 20 and 40 is 40

∴ (15/20) = [(15×2)/ (20×2)] = (30/40)

and (35/40) = [(35×1)/ (40×1)] = (35/40)

Now, 30 < 35

⇒ (30/40) < (35/40)

Hence, (15/20) < (35/40)

∴ 35/40 is greater.

So, between the numbers (15/20) and (35/40), the greater number is (35/40).

False.

For every rational numbers a, b and c, [a × (b + c) = (a × b) + (a × c)]

(i) 0
(ii) 1 and -1
(iii) 0

False.

Because, for every rational numbers a, b and c, [a × (b + c) = (a × b) + (a × c)]

Form the question it is given that equivalent of 7/9 = Numerator/45

To get 45 in the denominator multiply both numerator and denominator by 5

Then,

= (7 × 5)/ (9 × 5)

= 35/45

So, the equivalent rational number of 7/9, whose denominator is 45 is (35/45)

False.

Because, the reciprocal of -1 is -1 and reciprocal of 1 is 1.

(i) No
(ii) 1, -1
(iii) -1/5
(iv) X
(v) Rational number
(vi) Positive

(2 × -4)/ (5 × 9)

= -8/45

Reciprocal = -45/8

Hence, the reciprocal of (2/5) × (-4/5) is -45/8.

The reciprocal of a negative rational number is negative rational number.

Let us take negative rational number -3/4

The reciprocal of a negative rational number is 4/-3 = -4/3

If y be the reciprocal of x, then the reciprocal of y² in terms of x will be x².

From the question, (1/x) = y

Then,

Reciprocal of y² = 1/y²

Substitute (1/x) in the place of y,

= 1/ (1/x)²

= x²/1

= x²

The numbers 1 and -1 are their own reciprocal.

Reciprocal of 1 = 1/1 = 1

Reciprocal of -1 = 1/-1 = -1

Zero has no reciprocal.

The reciprocal of 0 = 1/0

= Undefined

Let us assume the missing number be x

Then,

= 1 / (213 × 657) = (1/213) × (x)

X = 213/ (213/657)

X = 1/657

X = 657^-1

So, (213 × 657)^-1 = 213^-1 × 657^-1

The rational number 10.11 in the from p/q is 1011/100.

The negative of 1 is -1.

The multiplicative inverse of 4/3 is ¾.

For rational numbers (a/b), (c/d) and (e/f) we have (a/b) × ((c/d) + (e/f)) = ((a/b) × (c/d)) + ((a/b) × (e/f))

Infinite

There are infinite rational numbers between any two rational numbers.

-5/7 is more than -3.

0 ÷ (-5/6) = 0.

We now that, division of zero by any number is zero.

0/(5/6) = 0

Since any number divided by zero is zero only

(-3/7) ÷ (-7/3) = 9/49.

(-3/7) ÷ (-7/3)

= (-3/7) × (-3/7)

= (-3 × -3)/ (7 × 7)

= 9/49

The reciprocal of 1 is 1.

Reciprocal of 1/1 = 1/1 = 1

The reciprocal of 0 does not exist.

The reciprocal of zero does not exist, as reciprocal of 0 is 1/0, which is not defined.

(5/6) [<] (8/4)

The LCM of 6 and 4 is 12

∴ (5/6) = [(5 × 2)/ (6 × 2)] = (10/12)

and (8/4) = [(8 × 3)/(4 × 3)] = (24/12)

Now, 10 < 24

⇒ (10/12) < (24/12)

Hence, (5/6) > (-3/8)

(3/7) [>] (-5/6)

Negative rational number is less than positive rational number.

(7/-8) [<] (8/9)

Negative rational number is less than positive rational number.

(-9/7) [<] (4/-7)

First we write each of the given numbers with a positive denominator.

= (4/-7)

= [(4 × (-1))/ (-7 × -1)]

= (-4/7)

So, (-9/7) < (-4/7)

(8/8) [=] (2/2)

(8/8) divide both denominator and numerator by 8

Then, 1/1 = 1

(2/2) divide both denominator and numerator by 2

Then, 1/1 = 1

Therefore, 1 = 1

(i) True 

\(\frac{-3}{5}\) is a negative number. 

All negative numbers are less than 0 and therefore, lie to the left of 0 on the number line.

Hence,\(\frac{-3}{5}\) lies to the left of 0 on the number line.

(ii) False 

\(\frac{-12}{7}\) is a negative number. All negative numbers are less than 0 and therefore, lie to the left of 0 on the number line. 

Hence,\(\frac{-12}{7}\) lies to the left of 0 on the number line.

(iii) True 

\(\frac{1}{3}\)is a positive number.

All positive numbers are greater than 0 and therefore, lie to the right of 0 on the number line.

Hence,\(\frac{1}{3}\) lies to the right of 0 on the number line. 

\(\frac{-5}{12}\) is a negative number.

All negative numbers are less than 0 and therefore, lie to the left of 0 on the number line.

Hence,\(\frac{-5}{12}\) lies to the left of 0 on the number line. 

Therefore, the rational numbers, \(\frac{1}{3}\)and \(\frac{-5}{12}\)are on opposite sides of 0 on the number line.

(iv) False 

\( \frac{-18}{-13}=\frac{-18\times-1}{-13\times-1}=\frac{18}{13}\)

\(\frac{18}{13}\) is a positive number. 

All positive numbers are greater than 0 and therefore, lie to the right of 0 on the number line. 

Hence,\(\frac{18}{13}\) lies to the right of 0 on the number line.

(3/-8) and (1/8)

First we write each of the given numbers with a positive denominator.

(3/-8) = [(3× (-1))/ (-8×-1)]

= (-3/8)

Then,

(-3/8)+ (1/8)

We have:

= [(-3 + 1)/8] … [∵ denominator is same in both the rational numbers]

= (-2/8)

(-2/5) and (1/5)

We have:

= [(-2 + 1)/5] … [∵ denominator is same in both the rational numbers]

= (-1/5)

(12/7) and (3/7)

We have:

= [(12 + 3)/7] … [∵ denominator is same in both the rational numbers]

= (15/7)

(-3/4), (5/-12), (-7/16), (9/-24)

First we write each of the given numbers with a positive denominator.

(5/-12) = [(5× (-1))/ (-12×-1)]

= (-5/12)

(9/-24) = [(9× (-1))/ (-24×-1)]

= (-9/24)

LCM of 4, 12, 16 and 24 is 48

Now,

(-3/4)= [(-3×12)/ (4×12)] = (-36/48)

(-5/12)= [(-5×4)/ (12×4)] = (-20/48)

(-7/16)= [(-7×3)/ (16×3)] = (-21/48)

(-9/24)= [(-9×2)/ (24×2)] = (-18/48)

Clearly,

(-36/48)< (-21/48) < (-20/48) < (-18/48)

Hence,

(-3/4)< (-7/16) < (5/-12) < (9/-24)

(2/3), (3/4), (5/-6), (-7/12)

First we write each of the given numbers with a positive denominator.

(5/-6) = [(5× (-1))/ (-6×-1)]

= (-5/6)

LCM of 3, 4, 6 and 12 is 12

Now,

(2/3)= [(2×4)/ (3×4)] = (8/12)

(3/4)= [(3×3)/ (4×3)] = (9/12)

(-5/6)= [(-5×2)/ (6×2)] = (-10/12)

(-7/12)= [(-7×1)/ (12×1)] = (-7/12)

Clearly,

(-10/12)< (-7/12) < (8/12) < (9/12)

Hence,

(-5/6)< (-7/15) < (2/3) < (3/4)

(-2/3) < (5/-8)

Because,

First we write each of the given numbers with a positive denominator.

= (5/-8)

= [(5× (-1))/ (-8×-1)]

= (-5/8)

The LCM of the denominators 3 and 8 is 24

∴ (-2/3) = [(-2×8)/ (3×8)] = (-16/24)

and (-5/8) = [(-5×3)/ (8×3)] = (-15/24)

Now, -16 = -15

⇒ (-16/24) < (-15/24)

⇒ (-2/3) < (-5/8)

Hence, (-2/3) < (5/-8)

(2/5), (7/10), (8/15), (13/30)

LCM of 5, 10, 15 and 30 is 30

Now,

(2/5) = [(2×6)/ (5×6)] = (12/30)

(7/10) = [(7×3)/ (10×3)] = (21/30)

(8/15) = [(8×2)/ (15×2)] = (16/30)

(13/30) = [(13×1)/ (30×1)] = (13/30)

Clearly,

(12/30) < (13/30) < (16/30) < (21/30)

Hence,

(2/5) < (13/30) < (16/30) < (21/30)

0 < (-3/-15)

Because,

First we write each of the given numbers with a positive denominator.

= (-3/-15)

= [(-3× (-1))/ (-15×-1)]

= (3/15)

The LCM of the denominators 1 and 15 is 15

∴ (0/1) = [(0×15)/ (1×15)] = (0/15)

and (3/15) = [(3×1)/ (15×1)] = (3/15)

Now, 0 < 3

⇒ (0/15) < (3/15)

⇒ 0 < (3/15)

Hence, 0 < (-3/-15)

-10

To get -10 in the numerator multiply by -2 for both numerator and denominator.

Then we get,

= [(5× (-2))/ (8× (-2))]

= (-10/-16)

To get 15 in the numerator multiply by 3 for both numerator and denominator.

Then we get,

= [(5×3)/ (8×3)]

= (15/24)

(i) Given (2/5)

To get numerator -56 we have to multiply both numerator and denominator by -28

Then we get, (2/5) × (-28/-28)

= (-56/-140)

Therefore (2/5) as a rational number with numerator -56 is (-56/-150)

(ii) Given (2/5)

To get numerator 154 we have to multiply both numerator and denominator by 77

Then we get, (2/5) × (77/77)

= (154/385)

Therefore (2/5) as a rational number with numerator 154 is (154/385)

(iii) Given (2/5)

To get numerator -750 we have to multiply both numerator and denominator by -375

Then we get, (2/5) × (-375/-375)

= (-750/-1875)

Therefore (2/5) as a rational number with numerator -750 is (-750/-1875)

(iv) Given (2/5)

To get numerator 500 we have to multiply both numerator and denominator by 250

Then we get, (2/5) × (250/250)

= (500/1250)

Therefore (2/5) as a rational number with numerator 500 is (500/1250)

-2 > (-13/5)

Because,

The LCM of the denominators 1 and 5 is 5

∴ (-2/1) = [(-2×5)/ (1×5)] = (-10/5)

and (-13/5) = [(-13×1/ (5×1)] = (-13/5)

Now, -10 > -13

⇒ (-10/5) > (-13/5)

Hence, -2 > (-13/5)

(-8/9) > (-9/10)

The LCM of the denominators 9 and 10 is 90

∴ (-8/9) = [(-8×10)/ (9×10)] = (-80/90)

and (-9/10) = [(-9×9)/ (10×9)] = (-81/90)

Now, -80 > -81

⇒ (-80/90) > (-81/90)

Hence, (-8/9) > (-9/10)

(i) The additive inverse of \(\frac{-2}{17}\) is \(\frac{2}{17}\)

(ii) The additive inverse of \(\frac{3}{-11}\) is \(\frac{3}{11}\)

(iii) The additive inverse of \(\frac{-17}{5}\) is \(\frac{17}{5}\)

(iv) The additive inverse of \(\frac{-11}{-25}\) is \(\frac{-11}{25}\)

60

To get 60 in the numerator multiply by -5 for both numerator and denominator.

Then we get,

= [(-12×-5)/ (13×-5)]

= (60/-65)

(i) (5/6) or 0

Since every positive rational number is greater than 0.

We have:

= (5/6) > (0)

(ii) (-3/5) or 0

Since every negative rational number is less than 0.

We have:

= (-3/5) < 0

(iii) (5/8) or (3/8)

Since both denominators are same therefore compare the numerators.

We have,

= 5 > 3

∴ (5/8) > (3/8)

(-3/7) > (6/-13)

Because,

First we write each of the given numbers with a positive denominator.

= (6/-13)

= [(6× (-1))/ (-13×-1)]

= (-6/13)

The LCM of the denominators 7 and 13 is 91

∴ (-3/7) = [(-3×13)/ (7×13)] = (-39/91)

and (-6/13) = [(-6×7)/ (13×7)] = (-42/91)

Now, -39 > -42

⇒ (-39/91) > (-42/91)

⇒ (-3/7) > (-6/13)

Hence, (-3/7) > (6/-13)

(4/-3) or (-8/7)

First we write each of the given numbers with a positive denominator.

= (4/-3)

= [(4× (-1))/ (-3×-1)]

= (-4/3)

The LCM of the denominators 3 and 7 is 21

∴ (-4/3) = [(-4×7)/ (3×7)] = (-28/21)

and (-8/7) = [(-8×3)/ (7×3)] = (-24/21)

Now, -28 < -24

⇒ (-28/21) < (-24/21)

⇒ (-4/3) < (-8/7)

Hence, (4/-3) < (-8/7)

∴ -8/7 is greater.

(4/-5) or (-7/8)

First we write each of the given numbers with a positive denominator.

= (4/-5)

= [(4× (-1))/ (-5×-1)]

= (-4/5)

The LCM of the denominators 5 and 8 is 40

∴ (-4/5) = [(-4×8)/ (5×8)] = (-32/40)

and (-7/8) = [(-7×5)/ (8×5)] = (-35/40)

Now, -32 > -35

⇒ (-32/40) > (-35/40)

⇒ (-4/5) > (-7/8)

Hence, (4/-5) > (-7/8)

∴ (4/-5) is greater.

(9/-13) or (7/-12)

First we write each of the given numbers with a positive denominator.

One number = (9/-13)

= [(9× (-1))/ (-13×-1)]

= (-9/13)

One number = (7/-12)

= [(7× (-1))/ (-12×-1)]

= (-7/12)

The LCM of the denominators 13 and 12 is 156

∴ (-9/13) = [(-9×12)/ (13×12)] = (-108/156)

and (-7/12) = [(-7×13)/ (12×13)] = (-91/156)

Now, -108 < -91

⇒ (-108/156) < (-91/156)

⇒ (-9/13) < (-7/12)

Hence, (9/-13) < (7/-12)

∴ (7/-12) is greater.

(7/-9) or (-5/8)

First we write each of the given numbers with a positive denominator.

= (7/-9)

= [(7× (-1))/ (-9×-1)]

= (-7/9)

The LCM of the denominators 9 and 8 is 72

∴ (-7/9) = [(-7×8)/ (9×8)] = (-56/72)

and (-5/8) = [(-5×9)/ (8×9)] = (-45/72)

Now, -56 < -45

⇒ (-56/72) < (-45/72)

⇒ (-7/9) < (-5/8)

Hence, (7/-9) < (-5/8)

∴ (-5/8) is greater.