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(i) \(\frac{11}{12}-\frac{17}{3}-\frac{14}{2}\)

= \(\frac{11-68-84}{12}\)

= \(\frac{-141}{12}\)

(ii) \(\frac{-6}{7}-\frac{-15}{7}-\frac{5}{6}-\frac{4}{9}\)

= \(\frac{-21}{7}-\frac{5}{6}-\frac{4}{9}\)

= \(\frac{-3\times 18-5\times 3-4\times 2}{18}\)

= \(\frac{-77}{18}\)

(iii) \(\frac{3}{5}+\frac{9}{5}+\frac{7}{3}-\frac{7}{3}-\frac{-13}{15}\)

= \(\frac{12}{5}-\frac{13}{15}\)

= \(\frac{12\times 3}{5\times 3}-\frac{13}{15}\)

= \(\frac{36-13}{15}\)

= \(\frac{23}{15}\)

(iv) \(\frac{4}{13}+\frac{9}{13}-\frac{8}{13}-\frac{5}{8}+\frac{9}{5}\)

= \(\frac{4+9-8}{13}-\frac{5}{8}+\frac{9}{5}\)

= \(\frac{5}{13}-\frac{5}{8}+\frac{9}{5}\)

= \(\frac{200-325+936}{520}\)

= \(\frac{811}{520}\)

(v) \(\frac{2}{3}+\frac{1}{3}+\frac{2}{5}-\frac{4}{5}\)

= \(\frac{2+1}{3}+\frac{2-4}{5}\)

= \(\frac{3}{3}-\frac{2}{5}\)

= \(\frac{15-6}{15}\)

= \(\frac{9}{15}\)

= \(\frac{3}{5}\)

(vi) \(\frac{5}{12}+\frac{7}{12}+\frac{2}{7}+\frac{9}{7}+\frac{1}{8}-\frac{5}{16}\)

= \(\frac{12}{12}+\frac{11}{7}+\frac{1}{8}-\frac{5}{16}\)

= \(\frac{336+528+42-105}{336}\)

= \(\frac{801}{336}\)

(i) \(\frac{2}{3}-\frac{3}{5}\)

= \(\frac{2\times 5-3\times 3}{15}\)

= \(\frac{10-9}{15}\)

= \(\frac{1}{15}\)

(ii)\(\frac{-4}{7}+\frac{2}{3}\)

= \(\frac{-4\times 3+2\times 7}{21}\)

= \(\frac{-12+14}{21}\)

= \(\frac{2}{21}\)

(iii) \(\frac{4}{7}-\frac{5}{7}\)

= \(\frac{4\times 1-5\times 1}{7}\)

= \(\frac{4-5}{7}\)

= \(\frac{-1}{7}\)

(iv) \(-2-\frac{5}{9}\)

= \(\frac{-2\times 9-5\times 1}{9}\)

= \(\frac{-18-5}{9}\)

= \(\frac{-23}{9}\)

(v) \(\frac{3}{8}+\frac{2}{7}\)

= \(\frac{3\times 7+2\times 8}{56}\)

= \(\frac{21+16}{56}\)

= \(\frac{37}{56}\)

(vi) \(\frac{-4}{13}+\frac{5}{26}\)

= \(\frac{-4\times 2+5\times 1}{26}\)

= \(\frac{-8+5}{26}\)

= \(\frac{-3}{26}\)

(vii) \(\frac{-5}{14}+\frac{2}{7}\)

= \(\frac{-5\times 1+2\times 2}{14}\)

= \(\frac{-5+4}{14}\)

= \(\frac{-1}{14}\)

(viii) \(\frac{13}{15}-\frac{12}{25}\)

= \(\frac{13\times 5-12\times 3}{75}\)

= \(\frac{65-36}{75}\)

= \(\frac{29}{75}\)

(ix) \(\frac{-6}{13}+\frac{7}{13}\)

= \(\frac{-6\times 1+7\times 1}{13}\)

= \(\frac{-6+7}{13}\)

= \(\frac{1}{13}\)

(x) \(\frac{7}{24}-\frac{19}{36}\)

= \(\frac{7\times 3-19\times 2}{72}\)

= \(\frac{21-38}{72}\)

= \(\frac{-17}{72}\)

(xi) \(\frac{5}{63}+\frac{8}{21}\)

= \(\frac{5\times 1+8\times 3}{63}\)

= \(\frac{5+24}{63}\)

= \(\frac{29}{63}\)

(i) \(\frac{5}{8}-\frac{3}{8}\) = \(\frac{5-3}{8}\)

= \(\frac{2}{8}\) (Therefore, L.C.M of 8 and 8 is 8)

= \(\frac{1}{4}\)

(ii) \(\frac{4}{9}-\frac{-7}{9}\) = \(\frac{4+7}{9}\)

= \(\frac{11}{9}\) (Therefore, L.C.M of 9 and 9 is 9)

(iii) \(\frac{-9}{11}-\frac{-2}{11}\) = \(\frac{-9+2}{11}\)

 = \(\frac{-7}{11}\) (Therefore, L.C.M of 11 and 11 is 11) 

 (iv) \(\frac{-4}{13}-\frac{11}{13}\) = \(\frac{-4-11}{13}\)

= \(\frac{-15}{13}\) (Therefore, L.C.M of 13 and 13 is 13)

 (v) \(\frac{-3}{8}-\frac{1}{4}\) = \(\frac{-3-2}{8}\)

 = \(\frac{-5}{8}\) (Therefore, L.C.M of 8 and 4 is 8)

 (vi) \(\frac{5}{6}-\frac{-2}{3}\) = \(\frac{5+4}{6}\)

 = \(\frac{9}{6}\) (Therefore, L.C.M of 6 and 3 is 6)

 (vii) \(\frac{-13}{14}-\frac{-6}{7}\) = \(\frac{-13+12}{14}\)

 = \(\frac{-1}{14}\) (Therefore, L.C.M of 14 and 7 is 14)

 (viii) \(\frac{-7}{22}-\frac{-8}{33}\) = \(\frac{-21+16}{66}\)

 = \(\frac{-55}{66}\) (Therefore, L.C.M of 22 and 33 is 66)

(i) False

Explanation:

Between any two distinct integers not necessary to be one integer.

(ii) True

Explanation:

According to the properties of rational numbers between any two distinct rational numbers there is always a rational number.

(iii) True

Explanation:

According to the properties of rational numbers between any two distinct rational numbers there are infinitely many rational numbers.

We know that between 7 and -4, below mentioned numbers will lie

-3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

According to definition of rational numbers are in the form of (p/q) where q not equal to zero.

Therefore six rational numbers between (7/13) and (-4/13) are

(-3/13), (-2/13), (-1/13), (0/13), (1/13), (2/13), (3/13), (4/13), (5/13), (6/13)

We know that between -4 and -8, below mentioned numbers will lie

-3, -2, -1, 0, 1, 2.

According to definition of rational numbers are in the form of (p/q) where q not equal to zero.

Therefore six rational numbers between (-4/8) and (3/8) are

(-3/8), (-2/8), (-1/8), (0/8), (1/8), (2/8), (3/8)

According to question,

= \(\frac{\frac{16}{12}+\frac{12}{7}}{\frac{65}{12}-\frac{12}{7}}\)

= \(\frac{\frac{455}{84}+\frac{144}{84}}{\frac{455}{84}-\frac{144}{84}}\)

= \(\frac{599}{311}\)

Given material required for 24 trousers = 54 m

Cloth required for 1 trouser = (54/24)

= (9/4) meters

According to question total number trousers = 24

Total length of the cloth = 54

Length of the cloth required for each trouser = total length of the cloth/number of trousers

= 54/24

= 9/2

∴ 9/2 meters is required for each trouser.

D) N ⊂ W ⊂ Z ⊂ Q

Correct option is  C) √5

D) All the above

From the question it is given that,

Distance travelled by train = 1445/2 km

Time taken by the train to cover distance 1445/2 = 17/2 hours

The speed of the train = (1445/2) ÷ (17/2)

= (1445/2) × (2/17)

= (85/1) × (1/1)

= 85 km/h

∴The speed of the train is 85 km/h.

If ‘0’ is subtracted from the set of integers then it becomes Z – {0}. 

Closure property under division on integers.

Ex: -4 ÷ 2 = -2 is an integer.

 3 ÷ 5 = 3/5  is not an integer. 

∴ Set of integers doesn’t satisfy closure property under division. 

Closure property under division on natural numbers. 

Ex: 2 ÷ 4 = 1/2 is not a natural number. 

∴ Set of natural numbers doesn’t satisfy closure property under division.

(-7) from (-4/7)

We have:

= (-4/7) – (-7/1)

= (-4/7) + (additive inverse of -7/1)

= (-4/7) + (7/1)

LCM of 7 and 1 is 7

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(-4×1)/ (7×1)] = (-4/7)

= [(7×7)/ (1×7)] = (49/7)

Then,

= (-4/7) + (49/7)

= (-4+49)/7

= (45/7)

(5/9) from (-2/3)

We have:

= (-2/3) – (5/9)

= (-2/3) + (additive inverse of 5/9)

= (-2/3) + (-5/9)

LCM of 3 and 9 is 9

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(-2×3)/ (3×3)] = (-6/9)

= [(-5×1)/ (9×1)] = (-5/9)

Then,

= (-6/9) + (-5/9)

= (-6-5)/9

= (-11/9)

(5) from (-3/5)

We have:

= (5/1) – (-3/5)

= (5/1) + (additive inverse of -3/5)

= (5/1) + (3/5)

LCM of 1 and 5 is 5

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(5×5)/ (1×5)] = (25/5)

= [(3×1)/ (5×1)] = (3/5)

Then,

= (25/5) + (3/5)

= (25+3)/5

= (28/5)

(3/4) – (4/5)

We have:

= (3/4) – (4/5)

= (3/4) + (additive inverse of 4/5)

= (3/4) + (-4/5)

LCM of 4 and 5 is 20

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(3×5)/ (4×5)] = (15/20)

= [(-4×4)/ (5×4)] = (-16/20)

Then,

= (15/20) + (-16/20)

= (15-16)/20

= (-1/20)

(-3) – (4/7)

We have:

= (-3/1) – (4/7)

= (-3/1) + (additive inverse of 4/7)

= (-3/1) + (-4/7)

LCM of 1 and 7 is 7

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(-3×7)/ (1×7)] = (-21/7)

= [(-4×1)/ (7×1)] = (-4/7)

Then,

= (-21/7) + (-4/7)

= (-21-4)/7

= (-25/7)

(7/24) – (19/36)

We have:

= (7/24) – (19/36)

= (7/24) + (additive inverse of 19/36)

= (7/24) + (-19/36)

LCM of 24 and 36 is 72

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(7×3)/ (24×3)] = (21/72)

= [(-19×2)/ (36×2)] = (-38/72)

Then,

= (21/72) + (-38/72)

= (21-38)/72

= (-17/72)

(4/9) – (2/-3)

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

(2/-3) = [(2× (-1))/ (-3×-1)]

= (-2/3)

We have:

= (4/9) – (-2/3)

= (4/9) + (additive inverse of -2/3)

= (4/9) + (2/3)

LCM of 9 and 3 is 9

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(4×1)/ (9×1)] = (4/9)

= [(2×3)/ (3×3)] = (6/9)

Then,

= (4/9) + (6/9)

= (4+6)/9

= (10/9)

(-5/14) – (-2/7)

We have:

= (-5/14) – (-2/7)

= (-5/14) + (additive inverse of -2/7)

= (-5/14) + (2/7)

LCM of 14 and 7 is 14

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(-5×1)/ (14×1)] = (-5/14)

= [(2×2)/ (7×2)] = (4/14)

Then,

= (-5/14) + (4/14)

= (-5+4)/14

= (-1/14)

(7/11) – (-4/-11)

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

(-4/-11) = [(-4× (-1))/ (-11×-1)]

= (4/11)

We have:

= (7/11) – (4/11)

= (7/11) + (additive inverse of 4/11)

= (7/11) + (-4/11)

Then,

= (7-4)/11

= (3/11)

In the question is given to verify the property = x × y = y × x

Where, x = -5/7, y = 14/15

Then, (-5/7) × (14/15) = (14/15) × (-5/7)

LHS = (-5/7) × (14/15)

= (-1/1) × (2/3)

= -2/3

RHS = (14/15) × (-5/7)

= (2/3) × (-1/1)

= -2/3

By comparing LHS and RHS

LHS = RHS

∴ -2/3 = -2/3

Hence x × y = y × x

In the question is given to verify the property = x × y = y × x

Where, x = 2/3, y = 9/4

Then, (2/3) × (9/4) = (9/4) × (2/3)

LHS = (2/3) × (9/4)

= (1/1) × (3/2)

= 3/2

RHS = (9/4) × (2/3)

= (3/2) × (1/1)

= 3/2

By comparing LHS and RHS

LHS = RHS

∴ 3/2 = 3/2

Hence x × y = y × x

The arrangement of the given rational number is as per the rule of distributive law over subtraction.

Now take out 1/5 as common.

Then,

= 1/5 [(2/15) – (2/5)]

The LCM of the denominators 15 and 5 is 15

(2/15) = [(2×1)/ (15×1)] = (2/15)

and (2/5) = [(2×3)/ (5×3)] = (6/15)

= 1/5 [(2 – 6)/15]

= 1/5 [-4/15]

= (1/5) × (-4/15)

= -4/75

Yes, -3/-5 is a rational number because every fraction is a rational number.

Associativity

The arrangement of the given rational number is as per the rule of distributive law over addition.

= (-3/5) × {(3/7) + (-5/6)}

The LCM of the denominators 7 and 6 is 42

(3/7) = [(3×6)/ (7×6)] = (18/42)

and (-5/6) = [(-5×7)/ (6×7)] = (-35/42)

= -3/5 [(18 – 35)/42]

= -3/5 [-17/42]

= (-3/5) × (-17/42)

= 51/210 … [divide both denominator and numerator by 3]

= 17/30

In the question is given to verify the property = x × y = y × x

Where, x = 7, y = ½

Then, 7 × ½ = ½ × 7

LHS = 7 × ½

= 7/2

RHS = ½ × 7

= 7/2

By comparing LHS and RHS

LHS = RHS

∴ 7/2 = 7/2

Hence x × y = y × x

Yes, -3/7 is a negative rational number.

The arrangement of the given rational number is as per the rule of Associative property for Multiplication.

Rational numbers with denominator 80 and numerator from 1 to 

79(like \(\frac{1}{80}, \frac{2}{20}, - ......- \frac{79}{80}\))

There are 79 such rational numbers.

(c)(14/9)

Because,

Let the missing number be X,

(-5/9) + (X) = 1

By sending (-5/9) from the left hand side to the right side it becomes (5/9)

(X) = 1 + (5/9)

(X) = (9+5)/9

(X) = (14/9)

(a)-14

Because,

(X/6) = (7/-3)

(X) = (7× 6)/-3

(X) = 42/-3

(X) = -14

(b) (-6/7)

Because,

= (-102/119) … [÷17]

= (-6/7)

\(\frac{-5}{9}\div\frac{2}{3}\)

\(=\frac{-5}{9}\times\frac{3}{2}\)

\(=\frac{-5\times3}{9\times2}\)

\(=\frac{-15}{18}=\frac{-15\div3}{18\div3}=\frac{-5}{6}\)

Rational number between \(\frac{-2}{3}\) and \(\frac{1}{4}\)

\(=\frac{1}{2}(\frac{-2}{3}+\frac{1}{4})\)

\(=\frac{1}{2}(\frac{-2\times4+1\times3}{12})\)

\(=\frac{1}{2}(\frac{-8+3}{12})\)

\(=\frac{1}{2}\times\frac{-5}{12}\)

\(=\frac{-5}{24}\)

\(\frac{4}{9}\div\text{x}=\frac{-8}{15}\)

\(\Rightarrow\) \(\text{x}=\frac{4}{9}\div\frac{-8}{15}\)

\(\Rightarrow\) \(\text{x}=\frac{4}{9}\times\frac{15}{-8}\)

\(\Rightarrow\) \(\text{x}=\frac{4\times15}{9\times-8}\)

\(\Rightarrow\) \(\text{x}=\frac{60}{-72}=\frac{60\times-1}{-72\times-1}=\frac{-60}{72}\)

\(\Rightarrow\) \(\text{x}=\frac{-60}{72}=\frac{-60\div6}{72\div6 }=\frac{-5}{6}\)

Reciprocal of \(\frac{-3}{4}\) = \(\frac{4}{-3}\)

\(\frac{4}{-3}=\frac{4\times-1}{-3\times-1}=\frac{-4}{3}\)

Additive inverse of a number \(\frac{a}{b}\) is the number \(-\frac{a}{b}\) such that, \(\frac{a}{b}+(\frac{-a}{b})=0\)

Therefore,

Additive inverse of \(\frac{-5}{9}\) is \(\frac{5}{9}\)

(A) 1/8 − 1/8 = 0

(D) does not exist

 (-18/11) from (1)

We have:

= (1/1) – (-18/11)

= (1/1) + (additive inverse of -18/11)

= (1/1) + (18/11)

LCM of 1 and 11 is 11

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(1×11)/ (1×11)] = (11/11)

= [(18×1)/ (11×1)] = (18/11)

Then,

= (11/11) + (18/11)

= (11+18)/11

= (29/11)

D) 2 × (3 + 4) = (2 × 3) + (2 × 4)

(d) We can find infinite many rational numbers between two given rational numbers.

(-9/7) from (-1)

We have:

= (-1/1) – (-9/7)

= (-1/1) + (additive inverse of -9/7)

= (-1/1) + (9/7)

LCM of 1 and 7 is 7

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

= [(-1×7)/ (1×7)] = (-7/7)

= [(9×1)/ (7×1)] = (9/7)

Then,

= (-7/7) + (9/7)

= (-7+9)/7

= (2/7)

(a) – (1/4) × {(2/3) + (-4/7)} = [-(1/4) × (2/3)] + [(-1/4) × (-4/7)]

Because, we know the rule of distributive law, i.e. a × (b + c)] = [(a × b) + (a × c)

(a) x

If y be the reciprocal of rational number x, i.e. y = 1/x

x = 1/y

Then,

Reciprocal of y = x

(a) 104/21

= (-3 × -7) / (8 × 13)

= (21/104)

Reciprocal of 21/104 is 104/21