Class 7 Rational Numbers Exercise 4.5

\begin{array}{l} \text {Q1. Which of the following rational numbers are equal? } \\ \text {(i) } \quad \frac{-9}{12} \text { and } \frac{8}{-12} \quad \text {(ii) } \quad \frac{-16}{20} \text { and } \frac{20}{-25} \\ \text {(iii) } \quad \frac{-7}{21} \text { and } \frac{3}{-9} \quad \text {(iv) } \quad \frac{-8}{-14} \text { and } \frac{13}{21} \end{array}\begin{array}{l} \text {sol. To test the equality of the given rational numbers, we first express them in the standard form.} \\\text {(i) Standard form of } \frac{-9}{12} =\frac{-9 \div 3}{12 \div 3} = \frac{-3}{4} \\ \\\text {Standard form of } \frac{8}{-12} =\frac{8 \div -4}{-12 \div -4} =\frac{-2}{3} \\ \\ \text {As we can clearly see that the standard forms of two rational numbers are not same. } \\ \text {Hence, they are not equal.} \\ \\\text {(ii) Standard form of } \frac{-16}{20} =\frac{-16 \div 4}{20 \div 4} = \frac{-4}{5} \\ \\\text {Standard form of }\frac{20}{-25} =\frac{20 \div -5}{-25 \div -5} =\frac{-4}{5} \\ \\ \text {Since standard forms of two rational numbers are same. Hence, they are equal.} \\ \\\text {(iii) Standard form of } \frac{-7}{21} = \frac{-7 \div 7}{21 \div 7} = \frac{-1}{3} \\ \\\text {Standard form of } \frac{3}{-9} =\frac{3 \div -3}{-9 \div -3} =\frac{-1}{3} \\ \\ \text {Since standard forms of two rational numbers are same. Hence, they are equal.} \\ \\\text {(iv) Standard form of } \frac{-8}{-14} =\frac{-8 \div -2}{-14 \div -2} = \frac{4}{7} \\ \\\text {Standard form of } \frac{13}{21} = \frac{13}{21} \\ \\ \text {As we can clearly see that the standard forms of two rational numbers are not same. } \\ \text {Hence, they are not equal.} \end{array}\begin{array}{l} \text {Q2. If each of the following pairs represents a pair of equivalent rational numbers, find the values x } \\ \text {(i) } \quad \frac{2}{3} \text { and } \frac{5}{x} \quad \text {(ii) } \quad \frac{-3}{7} \text { and } \frac{x}{4} \\ \text {(iii) } \quad \frac{3}{5} \text { and } \frac{x}{-25} \quad \text {(iv) } \quad \frac{13}{6} \text { and } \frac{-65}{x} \end{array}\begin{array}{l} \text {sol. } \\ \text {(i) } \quad \frac{2}{3}=\frac{5}{x} \\ \Rightarrow \quad x=5 \times \frac{3}{2}=\frac{15}{2} \\ \\\text {(ii) } \quad \frac{-3}{7}=\frac{x}{4} \\ \Rightarrow \quad x=\frac{-3}{7} \times 4=\frac{-12}{7} \\ \\\text {(iii) } \quad \frac{3}{5}=\frac{x}{-25} \\ \Rightarrow \quad x=\frac{3}{5} \times (-25)=\frac{-75}{5}=-15 \\ \\\text {(iv) } \quad \frac{13}{6}=\frac{-65}{x} \\ \Rightarrow \quad x=(-65) \times \frac{6}{13} = -30 \end{array}\begin{array}{l} \text {Q3. In each of the following, fill in the blanks so as to make the statement true } \\ \text {(i) A number which can be expressed in the form } \frac{p}{q} \text {, where p and q are integers } \\ \text { and q is not equal to zero, is called a ………..} \\\text {(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} \text { is said to be in the …………….} \\ \text {(iii) Two rational numbers are said to be equal, if they have the same ……… form. } \\\text {(iv) If m is a common divisor of a and b, then } \\ \frac{a}{b}=\frac{a \div m}{\ldots} \\\text {(v) If p and q are positive integers, then } \frac{p}{q} \text { is a ……… rational number and } \\ \frac{p}{-q} \text { is a ………… rational number.} \\\text {(vi) The standard form of -1 is ………..} \\ \text {(vii) If } \frac{p}{q} \text { is a rational number, then q cannot be …………} \\ \text {(viii) Two rational numbers with different numerators are equal, if their numerators are in } \\ \text { the same …….. as their denominators. } \\ \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) rational number } \\ \\\text {(ii) standard form } \\ \\\text {(iii) standard form } \\ \\\text {(iv) } \quad \frac{a}{b}=\frac{a \div m}{b \div m} \\ \\\text {(v) positive, negative } \\ \\\text {(vi) } \quad \frac{-1}{1} \\ \\\text {(vii) Zero } \\ \\ \text {(viii) ratio } \\ \end{array}\begin{array}{l} \text {Q4. In each of the following state if the statement is true (T) or false (F) } \\ \text {(i) The quotient of two integers is always an integer. } \\ \text {(ii) Every integer is a rational number. } \\ \text {(iii) Every rational number is an integer.} \\ \text {(iv) Every fraction is a rational number.} \\ \text {(v) Every rational number is a fraction. } \\ \text {(vi) If } \frac{a}{b} \text { is a rational number and m any integer, then } \frac{a}{b}=\frac{a \times m}{b \times m} \\ \text {(vii) Two rational numbers with different numerators cannot be equal. } \\ \text {(viii) 8 can be written as a rational number with any integer as denominator. } \\ \text {(ix) 8 can be written as a rational number with any integer as numerator. } \\ \text {(x) } \quad \frac{2}{3} \text { is equal to } \frac{4}{6} \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) False. Its not always necessary. } \\ \\\text {(ii) True. Every integer can be expressed in the form of } \frac{p}{q} \text {, where q is not zero. } \\ \\\text {(iii) False. Its not always necessary. } \\ \\\text {(iv) True. Every fraction can be expressed in the form of } \frac{p}{q} \text {, where q is not zero. } \\ \\\text {(v) False. Its not always necessary. } \\ \\\text {(vi) True } \\ \\\text {(vii) False. They can be equal after bringing it to standard form.} \\ \\\text {(viii) False } \\ \\\text {(ix) False } \\ \\\text {(x) True } \\ \end{array}