#### Exercise 4.5

*Q 1 .Which of the following numbers are equal ?*

*(i) . -9/12 and 8/-12*

*(ii) . -16/20 and 20/-25*

*(iii) . -7/21 and 3/-9*

*(iv) . -8/-14 and 13/21*

SOLUTION :

(i) . The standard form of -9/12 is -9/3 , 12/3 = -34

The standard form of 8/-12 is 8/-4 , 12/-4 = -2/3

Since , the standard forms of two rational numbers are not same . Hence , they are not equal .

(ii) Since , LCM of 20 and 25 is 100 .

Therefore making the denominators equal , -16/20 = (-16×5)/(20×50 ) = -80/100 and 20/- 25 = (-20×4)/(25×4) = -80/100 .

Therefore , -16/20 = 20/-25 .

(iii) . Since , LCM of 21 and 9 is 63 .

Therefore making the denominators equal , -7/21 = (-7×3)/(21×3) = -21/63 and 3/-9 = (-3 x7)/(9 x7) = -21/63 .

Therefore , -7/21 = 3/-9 .

(iv) . Since , LCM of 14 and 21 is 42 .

Therefore making the denominators equal , -8/-14 = (-8×3)/(-14×3)=-24/-42 and 13/21 = (13 x2)/(21 x2) = 26/42 .

Therefore , -8/14 is not equal to 13/21 .

*Q 2 . If each of the following pairs represents a pair of equivalent rational numbers , find the values of x :*

*(i) . 2/3 and 5/x *

*(ii) . -3/7 and x/4*

*(iii) . 3/5 and x/-25*

*(iv) . 13/6 and -65/x *

SOLUTION :

(i). 2/3 = 5/x , then x = 5×3/2 = 15/2

(ii) . -3/7 = x/4 , then x = -3/7×4 = -12/7

(iii) . 3/5 = x/-25 , then x = 3/5x(-25) = -75/5 = -15

(iv) . 13/6 = -65/x , then x = 6/13x(-65) = 6x(-5)= -30

*Q 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 p/q , where p and q are integers and q is not equal to zero , is called a â€¦â€¦â€¦.. *

*(ii) . If the integers p and q have no common divisor other than 1 and q is positive , then the rational number p/q is said to be in the …. *

* (iii) . Two rational numbers are said to be equal , if they have the same …. form .*

* (iv) . If m is a common divisor of a and b , then \(\frac{a}{b}=\frac{a\div m}{….}\) *

*(v) . If p and q are positive Integers , then p/q is a â€¦..rational number and p/-q is a …… rational *

*number .*

* (vi) . The standard form of -1 is …*

* (vii) . If p/q 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 .*

SOLUTION :

(i) . rational number

(ii) . standard rational number

(iii) . standard form

(iv) . a/b = (aÃ·m)/(bÃ·m )

(v). positive rational number , negative rational number

(vi) . -1/1

(vii). Zero

(viii). ratio

*Q 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 traction is a rational number .*

* (v) . Every rational number is a fraction . *

*(vi) . If a/b is a rational number and m any integer , then \(\frac{a}{b}=\frac{a\times m}{b\times m}\) .*

*(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) . 2/3 is equal to 4/6 .*

SOLUTION :

(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 .