RD Sharma solution class 7 chapter 4 Rational numbers Exercise 4.5

Exercise 4.5

Page-4.20

Question 1:

Which of the following rational numbers are equal?
(i) $\frac{-9}{12} \mathrm{and} \frac{8}{-12}$
(ii) $\frac{-16}{20} \mathrm{and} \frac{20}{-25}$
(iii) $\frac{-7}{21} \mathrm{and} \frac{3}{-9}$
(iv) $\frac{-8}{14} \mathrm{and} \frac{13}{21}$

Answer 1:

(i)
$$\begin{gathered} \mathrm{The} \mathrm{standard} \mathrm{form} \mathrm{of} \frac{-9}{12}\mathrm{is} \frac{-9/3}{12/3} = \frac{-3}{4} \\ \mathrm{The} \mathrm{standard} \mathrm{form} \mathrm{of} \frac{8}{-12} \mathrm{is} \frac{8/-4}{-12/-4} = \frac{-2}{3} \\ \mathrm{Since}, \mathrm{the} \mathrm{standard} \mathrm{forms} \mathrm{of} \mathrm{two} \mathrm{rational} \mathrm{numbers} \mathrm{are} \mathrm{not} \mathrm{same}. \\ \mathrm{Hence}, \mathrm{they} \mathrm{are} \mathrm{not} \mathrm{equal}. \end{gathered}$$

(ii)
$$\begin{gathered} \mathrm{Since}, \mathrm{LCM}\hspace{0.17em}\mathrm{of} 20 \mathrm{and} 25 \mathrm{is} 100. \\ \mathrm{Therefore} \mathrm{making} \mathrm{the} \mathrm{denominators} \mathrm{equal}, \\ \frac{-16}{20}=\frac{-16\times 5}{20\times 5}=\frac{-80}{100} \mathrm{and} \frac{20}{-25}=\frac{-20\times 4}{25\times 4}=\frac{-80}{100}. \\ \mathrm{Therefore}, \frac{-16}{20}=\frac{20}{-25}. \end{gathered}$$


(iii)
$$\begin{gathered} \mathrm{Since}, \mathrm{LCM}\hspace{0.17em}\mathrm{of} 21 \mathrm{and} 9 \mathrm{is} 63. \\ \mathrm{Therefore} \mathrm{making} \mathrm{the} \mathrm{denominators} \mathrm{equal}, \\ \frac{-7}{21}=\frac{-7\times 3}{21\times 3}=\frac{-21}{63} \mathrm{and} \frac{3}{-9}=\frac{-3\times 7}{9\times 7}=\frac{-21}{63}. \\ \mathrm{Therefore}, \frac{-7}{21}=\frac{3}{-9}. \end{gathered}$$


(iv)
$$\begin{gathered} \mathrm{Since}, \mathrm{LCM}\hspace{0.17em}\mathrm{of} 14 \mathrm{and} 21 \mathrm{is} 42. \\ \mathrm{Therefore} \mathrm{making} \mathrm{the} \mathrm{denominators} \mathrm{equal}, \\ \frac{-8}{14}=\frac{-8\times 3}{14\times 3}=\frac{-24}{42} \mathrm{and} \frac{13}{21}=\frac{13\times 2}{21\times 2}=\frac{26}{42}. \\ \mathrm{Therefore}, \frac{-8}{14}\mathrm{is} \mathrm{not} \mathrm{equal} \mathrm{to} \frac{13}{21}. \end{gathered}$$

Question 2:

If each of the following pairs represents a pair of equivalent rational numbers, find the values of x:
(i) $\frac{2}{3} \mathrm{and} \frac{5}{x}$
(ii) $\frac{-3}{7} \mathrm{and} \frac{x}{4}$
(iii) $\frac{3}{5} \mathrm{and} \frac{x}{-25}$
(iv) $\frac{13}{6} \mathrm{and} \frac{-65}{x}$

Answer 2:

(i) $\frac{2}{3}=\frac{5}{x}, \mathrm{then} x=5\times \frac{3}{2}= \frac{15}{2}$
(ii) $\frac{-3}{7}=\frac{x}{4}, \mathrm{then} x=\frac{-3}{7}\times 4= \frac{-12}{7}$
(iii) $\frac{3}{5}=\frac{x}{-25}, \mathrm{then} x=\frac{3}{5}\times (-25)=\frac{-75}{5} =-15$
(iv) $\frac{13}{6}=\frac{-65}{x}, \mathrm{then} x=\frac{6}{13}\times (-65)=6\times (-5)= -30$

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 $\frac{p}{q}$, 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 $\frac{p}{q}$ is said to be in the ....
(iii) Two rational numbers are said to be equal, if they have the same .... form.

$\frac{a}{b}=\frac{a÷m}{....}$

(v) If p and q are positive integers, then $\frac{p}{q}$ is a ..... rational number and $\frac{p}{-q}$ is a ..... rational number.

(vi) The standard form of −1 is ...

(vii) If $\frac{p}{q}$ 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.

Answer 3:

(i) rational number
(ii) standard rational number
(iii) standard form
(iv) $\frac{a}{b}=\frac{a÷m}{b÷m}$
(v) positive rational number, negative rational number
(vi) $\frac{-1}{1}$
(vii) zero
(viii) ratio

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 $\frac{a}{b}$ 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) $\frac{2}{3} \mathrm{is} \mathrm{equal} \mathrm{to} \frac{4}{6}.$

Answer 4:

(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.