RD Sharma 2014 Solutions for Class 7 Math Chapter 4 Rational Numbers are provided here with simple step-by-step explanations. These solutions for Rational Numbers are extremely popular among class 7 students for Math Rational Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma 2014 Book of class 7 Math Chapter 4 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma 2014 Solutions. All RD Sharma 2014 Solutions for class 7 Math are prepared by experts and are 100% accurate.

Page No 4.12:

Question 1:

Determine whether the following rational numbers are in the lowest form or not:

(i) 6584
(ii) -1532
(iii) 24128
(iv) -56-32

Answer:

(i) We observe that 65 and 84 have no common factor i..e., their HCF is 1.

Thus, 6584 is in its lowest form.

(ii) We observe that -15 and 32 have no common factor i..e., their HCF is 1.

Thus, -1532is  in  its lowest form.

(iii) HCF of 24 and 128 is not 1.

Thus, given rational number is not in its simplest form.

(iv) HCF of 56 and 32 is 8.

Thus, given rational number is not in its simplest form.

Page No 4.12:

Question 2:

Express each of the following rational numbers to the lowest form:
(i) 422
(ii) -36180
(iii) 132-428
(iv) -32-56

Answer:

Lowest form of:
(i) 422 is:4 =2×222 = 2×11HCF of 4 and 22 is 2.Dividing the fraction by 2 , we get  211.(ii)  -36180 is:36 = 3×3×2×2180 = 5×3×3×2×2HCF of  36 and 180 is 36.Dividing the fraction by 36 , we get -15
(iii) 132-428 is:132 = 2×3×2×11 428 = 2×2×107HCF of 132 and 428 is 4.Dividing the fraction by 4 , we get   33-107(iv) -32-56 is:32 =2×2×2×2×256 = 2×2×2×7HCF of 32 and 56 is 8. Dividing the fraction by 8, we get47

Page No 4.12:

Question 3:

Fill in the blanks:
(i) -57=35=49
(ii) -4-9=18=12 
(iii) 6-13=-12 =24 
(iv) -6 =311=-55

Answer:

(i) Here, -5×57×5= -2535 Also , -5×77×7 = -3549.Therefore, -57=-2535=-3549(ii) Here, -4×-2-9×-2 = 818Also, -4×-3-9×-3 = 1227Therefore,  -4-9=818=1227(iii) Here, 6×-2-13×-2= -1226Also, 6×4-13×4 = 24-52Therefore, 6-13=-1226=24-52(iv) Here, -6/-2-22/-2 = 311Also, -6-22 = 3×-511×-5 =-15-55Therefore, -6-22=311=-15-55



Page No 4.15:

Question 1:

Write each of the following rational numbers in the standard form:
(i) 210
(ii) -836
(iii) 4-16
(iv) -15-35
(v) 299-161
(vi) -63-210
(vii) 68-119
(viii) -195275

Answer:

(i) The denominator is positive and HCF of 2 and 10 is 2.

 Dividing the numerator and denominator by 2, we get:

210= 2/210/2 = 15

(ii) The denominator is positive and HCF of 8 and 36 is 4.

 Dividing the numerator and denominator by 4, we get:

-836 =-8/436/4 = -29

(iii)

The denominator is negative.

4×-1-16×-1 = -416

HCF of 4 and 16 is 4.

 Dividing the numerator and denominator by 4, we get:

-4/416/4 = -14

(iv) The denominator is negative.

-15×-1-35×-1 = 1535

HCF of 15 and 35 is 5.

 Dividing the numerator and denominator by 5, we get:

15/535/5  = 37

(v) The denominator is negative.

299×-1-161×-1 = -299161

HCF of 299 and 161 is 23.

 Dividing the numerator and denominator by 23, we get:

 -299/23161/23 =-137

(vi) The denominator is negative.

-63×-1-210×-1 = 63210

HCF of 63 and 210 is 21.

 Dividing the numerator and denominator by 21, we get:

63/21210/21 = 310

(vii) The denominator is negative.

68×-1-119×-1 = -68119

HCF of 68 and 119 is 17.

 Dividing the numerator and denominator by 17, we get:

 -68/17119/17 = -47

(viii) The denominator is positive and HCF of 195 and 275 is 5.

 Dividing divide the numerator and denominator by 5, we get:
 
 -195/5275/5 = -3955



Page No 4.20:

Question 1:

Which of the following rational numbers are equal?
(i) -912 and 8-12
(ii) -1620 and 20-25
(iii) -721 and 3-9
(iv) -814 and 1321

Answer:

(i)
The standard form of -912is  -9/312/3 = -34The standard form of 8-12 is 8/-4-12/-4 = -23Since, the standard forms of two rational numbers are not same.Hence, they are not equal.

(ii)
Since, LCMof 20 and 25 is 100.Therefore making the denominators equal, -1620=-16×520×5=-80100 and 20-25=-20×425×4=-80100.Therefore, -1620=20-25.


(iii)
Since, LCMof 21 and 9 is  63.Therefore making the denominators equal, -721=-7×321×3=-2163 and 3-9=-3×79×7=-2163.Therefore, -721=3-9.


(iv)
Since, LCMof 14 and 21 is 42.Therefore making the denominators equal, -814=-8×314×3=-2442 and 1321=13×221×2=2642.Therefore, -814is not equal to 1321.

Page No 4.20:

Question 2:

If each of the following pairs represents a pair of equivalent rational numbers, find the values of x:
(i) 23 and 5x
(ii) -37 and x4
(iii) 35 and x-25
(iv) 136 and -65x

Answer:

(i) 23=5x, then x=5×32= 152
(ii) -37=x4, then x=-37×4= -127
(iii) 35=x-25, then x=35×(-25)=-755 =-15
(iv) 136=-65x, then x=613×(-65)=6×(-5)= -30

Page No 4.20:

Question 3:

In each of the following, fill in the blanks so as to make the statement true:
(i) A number which can be expressed in the form pq, where p and q are integers and q is not equal to zero, is called a .....

(ii) If the integers p and q have no common divisor other than 1 and q is positive, then the rational number pq is said to be in the ....
(iii) Two rational numbers are said to be equal, if they have the same .... form.

ab=a÷m....
(v) If p and q are positive integers, then pq is a ..... rational number and p-q is a ..... rational number.

(vi) The standard form of −1 is ...

(vii) If pq is a rational number, then q cannot be ....
(viii) Two rational numbers with different numerators are equal, if their numerators are in the same .... as their denominators.

Answer:

(i) rational number
(ii) standard rational number
(iii) standard form
(iv) ab=a÷mb÷m
(v) positive rational number, negative rational number
(vi) -11
(vii) zero
(viii) ratio

Page No 4.20:

Question 4:

In each of the following state if the statement is true (T) or false (F):
(i) The quotient of two integers is always an integer.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every fraction is a rational number.
(v) Every rational number is a fraction
(vi) If ab is a rational number and m any integer, then ab=a×mb×m
(vii) Two rational numbers with different numerators cannot be equal.
(viii) 8 can be written as a rational number with any integer as denominator.
(ix) 8 can be written as a rational number with any integer as numerator.
(x) 23 is equal to 46.

Answer:

(i) False; not necessary
(ii) True; every integer can be expressed in the form of p/q, where q is not zero.
(iii) False; not necessary
(iv) True; every fraction can be expressed in the form of p/q, where q is not zero.
(v) False; not necessary
(vi) True
(vii) False; they can be equal, when simplified further.
(viii) False
(ix) False
(x) True; in the standard form, they are equal.
 



Page No 4.26:

Question 1:

Draw the number line and represent the following rational numbers on it:
(i) 23
(ii) 34
(iii) 38
(vi) -58
(v) -316
(vi) -73
(vii) 22-7
(viii) -313

Answer:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Page No 4.26:

Question 2:

Which of the two rational numbers in each of the following pairs of rational numbers is greater?
(i) -38, 0
(ii) 52, 0
(iii) -411, 311
(iv) -712, 5-8
(v) 4-9, -3-7
(vi) -58, 3-4
(vii) 59, -3-8
(viii) 5-8, -712

Answer:

(i) We know that every positive rational number is greater than zero and every negative rational number is smaller than zero. Thus,

-38>0
(ii) 52>0. Because every positive rational number is greater than zero and every negative rational number is smaller than zero. 
(iii) -48<311. Because every positive rational number is greater than zero and every negative rational number is smaller than zero. 
(iv)

-712=-7×212×2=-1424 and 5-8=-5×38×3=-1524Therefore, -712>5-8

(v)

4-9=-4×79×7=-2863 and -3-7=3×77×9=2163Therefore, 4-9<-3-7
(vi)

-58 and 3-4=-3×24×2=-68Therefore, -58>3-4

(vii)

59=5×89×8=4072 and -3-8=3×98×9=2772Therefore,59>-3-8

(viii)
-712=-7×212×2=-1424 and 5-8=-5×38×3=-1524Therefore, -712>5-8

Page No 4.26:

Question 3:

Which of the two rational numbers in each of the following pairs of rational numbers is smaller?
(i) -6-13, 713
(ii) 16-5, 3
(iii) -43, 8-7
(iv) -125, -3

Answer:

(i) -6-13=613<713
(ii) 16-5<3
(iii)
-43=-4×73×7=-2821 and 8-7=-8×37×3=-2421Therefore, -43<8-7
(iv)
-125 and -3 = -3×51×5=-155Therefore -125>-3



Page No 4.27:

Question 4:

Fill in the blanks by the correct symbol out of >, =, or <:
(i) -67.....713
(ii) -35.....-56
(iii) -23.....5-8
(iv) 0 .....-25

Answer:

(i) Because every positive number is greater than a negative number, -67<713.(ii) On multiplying -35 by 66, we get -1830.On multiplying -56 by 55, we get -2530. Because -18> -25, -35>-56.(iii)  On multiplying -23 by 88, we get -1624.On multiplying 5-8 by 33, we get 15-24 = -1524.Because -15 >-16, -23<5-8.(iv) Because every positive number is greater than a negative number, 0>-25.

Page No 4.27:

Question 5:

Arrange the following rational numbers in ascending order:
(i) 35, -17-30, 8-15, -710
(ii) -49, 5-12, 7-18, 2-3

Answer:

(i) Ascending order:

Since, LCMof 5, -30, -15, 10 is 30.Multiplying the numerators and denominators to get the denominator equal to the LCM, 35=3×65×6=1830, 1730= 17×130×1 = 1730, 8-15=-8×215×2=-1630, -710=-7×310×3=-2130.Order is -21 <-16<17<8.Order is -710<8-15<1730<35.

(ii)

Since, LCMof 9, -12, -18, 3 is 36.Multiplying the numerators and denominators to get the denominator equal to the LCM, -49=-4×49×4=-1636 5-12=-5×312×3=-1536 7-18=-7×218×2=-1436 2-3=-2×123×12=-2436.Order is -24 <-16<-15<-14.Order is 2-3<-49<5-12<7-18.

Page No 4.27:

Question 6:

Arrange the following rational numbers in descending order:
(i) 78, 6416, 36-12, 5-4, 14028
(ii) -310, 17-30, 7-15, -1120 

Answer:

We have to arrange them in descending order.

(i)
Since, LCMof 8, 16, -12, -4, 28 is 336.Multiplying the numerators and denominators, to get the denominator equal to the LCM, 78=7×428×42=294336, 6416=64×2116×21=1344336,  36-12=-36×2812×28=-1008336,  5-4=-5×844×84=-420336, 14028=140×1228×12=1680336.Order is 1680> 1344 >294 >-420>-1008.Order is 14028>6416>78>5-4>36-12.
(ii)

Since, LCMof 10, -30, -15, 20 is 60.Multiplying the numerators and denominators, to get the denominator equal to the LCM, -310=-3×610×6=-1860, 17-30=-17×230×2=-3460,  7-15=-7×415×4=-2860,  -1120=-11×320×3=-3360, Order is, -18>-28>-33>-34.Order is -310>7-15>-1120>17-30.

Page No 4.27:

Question 7:

Which of the following statements are true:
(i) The rational number 2923 lies to the left of zero on the number line.
(ii) The rational number -12-17 lies to the left of zero on the number line.
(iii) The rational number 34 lies to the right of zero on the number line.
(iv) The rational numbers -12-5 and -717 are on the opposite side of zero on the number line.
(v) The rational numbers -21-5 and 7-31 are on the opposite side of zero on the number line.
(vi) The rational number -3-5 is one the right of -47 on the number line.

Answer:

(i) False; it lies to the right of zero because it is a positive number.
(ii) False; it lies to the right of zero because it is a positive number.
(iii) True
(iv) True; they are of opposite signs.
(v) False; they both are of same signs.
(v) True; they both are of opposite signs and positive number is greater than the negative number. Thus, it is on the right of the negative number.



Page No 4.3:

Question 1:

Write down the numerator of each of the following rational numbers:
(i) -75
(ii) 15-4
(iii) -17-21
(iv) 89
(v) 5

Answer:

Numerators are:
(i) -7
(ii) 15
(iii) -17
(iv) 8
(v) 5

Page No 4.3:

Question 2:

Write down the denominator of each of the following rational numbers:
(i) -45
(ii) 11-34
(iii) -15-82
(iv) 15
(v) 0

Answer:

Denominators are:
(i) 5
(ii) -34
(iii) -82
(iv) 1
(v) 1

Page No 4.3:

Question 3:

Write down the rational number whose numerator is (−3) × 4, and whose denominator is (34 − 23) × (7 − 4).

Answer:

According to the question:

Numerator = (-3) × 4 = -12

Denominator = (34 - 23) × (7 - â€‹4) = 11 × 3 = 33

∴ Rational number = -1233

Page No 4.3:

Question 4:

Write the following rational numbers as integers:
71, -121, 341, -731, 951

Answer:


Integers are 7, -12, 34, -73 and 95.

Page No 4.3:

Question 5:

Write the following integers as rational numbers with denominator 1:
-15, 17, 85, -100

Answer:

Rational numbers of given integers with denominator 1 are:

-151, 171,851,-1001

Page No 4.3:

Question 6:

Write down the rational number whose numerator is the smallest three digit number and denominator is the largest four digit number.

Answer:

Smallest three-digit number = 100

Largest four-digit number = 9999

∴ Required rational number = 1009999

Page No 4.3:

Question 7:

Separate positive and negative rational numbers from the following rational numbers:

-5-7, 12-5, 74, 13-9, 0, -18-7, -95116, -1-9

Answer:

Given rational numbers can be rewritten as:

57. -125, 74, -139, 0 , 187, -95116, 19

Thus, positive rational numbers are:

57, 74, 187,19
or, -5-7, 74, 0, -18-7. -6-9

Negative rational numbers are:

-125, -139, -95116
or, 12-5, 13-9, -95116



Page No 4.4:

Question 8:

Which of the following rational numbers are positive:
(i) -87
(ii) 98
(iii) -19-13
(iv) -2113

Answer:

The numbers can be rewritten as:

 (i)  -87(ii)  98(iii)  1913(iv) -2113

Positive rational numbers are (ii) and (iii), i.e., 98 and -19-13.

Page No 4.4:

Question 9:

Which of the following rational numbers are negative?
(i) -37
(ii) -5-8
(iii) 9-83
(iv) -115-197

Answer:

The  numbers can be rewritten as:
(i) -37(ii) 58(iii) -983(iv) 115197

Negative rational numbers are (i) and (iii).



Page No 4.8:

Question 1:

Express each of the following as a rational number with positive denominator:
(i) -15-28
(ii) 6-9
(iii) -28-11
(iv) 19-7

Answer:

Rational number with positive denominators:

(i) Multiplying the number by -1, we get:

-15-28=-15×-1-28×-1=1528

(ii) Multiplying the number by -1, we get: 

6-9=6×-1-9×-1=-69

(iii) Multiplying the number by -1, we get:

-28-11=-28×-1-11×-1=2811

(iv) Multiplying the number by -1, we get:

19-7=19×-1-7×-1=-197

Page No 4.8:

Question 2:

Express 35 as a rational number with numerator:
(i) 6
(ii) −15
(iii) 21
(iv) −27

Answer:

Rational number with numerator:

(i) 6 is:

3×25×2 = 610 (multiplying numerator and denominator by 2)

(ii)

-15 is:3×-55×-5 = -15-25 (multiplying numerator and denominator by -5)

(iii)

21 is:3×75×7 = 2135 (multiplying numerator and denominator by 7)

(iv)

-27 is:3×-95×-9 = -27-45 (multiplying numerator and denominator by -9)

Page No 4.8:

Question 3:

Express 57 as a rational number with denominator:
(i) −14
(ii) 70
(iii) −28
(iv) −84

Answer:

57 as a rational number with denominator:
(i) −14 is:

5×-27×-2  = -10-14 (Multiplying numerator and denominator by -2)
(ii) 70 is:

5×107×10 = 5070 (Multiplying numerator and denominator by 10)

(iii) −28 is:

5×-47×-4 = -20-28 (Multiplying numerator and denominator by -4)

(iv) −84 is:

5×-127×-12 = -60-84 (Multiplying numerator and denominator by -12)



Page No 4.9:

Question 4:

Express 34 as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) −80

Answer:

3/4 as rational number with denominator:

(i)
20 is:3×54×5=1520 (multiplying numerator and denominator by 5)


(ii)
36 is:3×94×9 = 2736 (multiplying numerator and denominator by 9)


(iii)
44 is:3×114×11 = 3344 (multiplying numerator and denominator by 11)

(iv)
-80 is:3×-204×-20 = -60-80 (multiplying numerator and denominator by -20)

Page No 4.9:

Question 5:

Express 25 as a rational number with numerator:
(i) −56
(ii) 154
(iii) −750
(iv) 500

Answer:

2/5 as a rational number with numerator:

(i)
-56 is:2×285×-28 =-56-140 (multiplying numerator and denominator by -28)

(ii)
154 is:2×775×77 = 154385 (multiplying numerator and denominator by 77)

(iii)
-750 is:2×-3755×-375 = -750-1875 (multiplying numerator and denominator by -375)

(iv)
500 is:2×2505×250 = 5001250 (multiplying numerator and denominator by 250)

Page No 4.9:

Question 6:

Express -192108 as a rational number with numerator:
(i) 64
(ii) −16
(iii) 32
(iv) −48

Answer:

Rational number with numerator:

(i) 64 as numerator: -192/-3108/-3 =64-36 (Dividing the numerator and denomintor by -3)(ii) -16 as numerator: -192/12108/12 = -169 (Dividing the numerator and denomintor by 12)(iii) 32 as numerator: -192/-6108/-6 =  32-18 (Dividing the numerator and denomintor by -6)(iv) -48 as numerator: -192/4108/4 = -4827 (Dividing the numerator and denomintor by 4)

Page No 4.9:

Question 7:

Express 168-294 as a rational number with denominator:
(i) 14
(ii) −7
(iii) −49
(iv) 1470

Answer:

Rational number with denominator:

(i) 14 as denominator: 168/-21-294/-21 = -814   (Dividing the numerator and denomintor by -21)(ii) -7  as denominator: 168/42-294/42 = 4-7   (Dividing the numerator and denomintor by 42)(iii) -49 as denominator: 168/6-294/6 = 28-49   (Dividing the numerator and denomintor by 6)(iv) 1470  as denominator: 168×-5-294×-5 = -8401470   (Multiplying the numerator and denomintor by -5)

Page No 4.9:

Question 8:

Write -1442 in a form so that the numerator is equal to:
(i) −2
(ii) 7
(iii) 42
(iv) −70

Answer:

Rational number with numerator:

(i) -2 is: -14/742/7= -26 ( Dividing numerator and denominator by 7)
(ii) 7 is: -14/-242/-2 = 7-21 ( Dividing numerator and denominator by -2)
(iii) 42 is: -14×-342×-3 =  42-126 ( Multiplying numerator and denominator by -3)
(iv) -70 is: -14×542×5 = -70210 ( Multiplying numerator and denominator by 5)

Page No 4.9:

Question 9:

Select those rational numbers which can be written as a rational number with numerator 6:

122, 23, 34, 4-5, 56, -67, -78

Answer:

Given rational numbers that can be written as a rational number with numerator 6 are:

122 (On multiplying by 6) = 613223 (On multiplying by 3) = 6934 (On multiplying by 2) = 68-67 (On multiplying by -1) = 6-7

Page No 4.9:

Question 10:

Select those rational numbers which can be written as a rational number with denominator 4:

78, 6416, 36-12, -1617, 5-4, 14028

Answer:

Given rational numbers that can be written as a rational number with denominator 4 are:

78 (On dividing by 2) = 3.546416 (On dividing by 4) =16436-12(On dividing by 3) =12-4 = -124-1617 can't be expressed with a denominator 4.5-4(On multiplying by -1) =-5414028(On dividing by 7) =204
122 (On multiplying by 6) = 613223 (On multiplying by 3) = 6934 (On multiplying by 2) = 6867 (On multiplying by 1) = 67

Page No 4.9:

Question 11:

In each of the following, find an equivalent form of the rational number having a common denominator:
(i) 34 and 512
(ii) 23, 76 and 1112
(iii) 57, 38, 914 and 2021

Answer:

Equivalent forms of the rational number having common denominator are:

(i)  34=3×34×3=912 and 512.

(ii)
23=2×43×4=812 and 76=7×26×2=1412 and 1112Forms are 812, 1412 and 1112


(iii)
57=5×247×24=120168 , 38=3×218×21=63168 , 914=9×1214×12=108168 and 2021=20×821×8=160168  Forms are 120168, 63168,108168 and 160168



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