question_answer 16)

                Find five rational numbers between:                 (i) \[\frac{2}{3}\] and \[\frac{4}{5}\]                 (ii) \[\frac{-3}{2}\] and \[\frac{5}{3}\]                 (iii) \[\frac{1}{4}\] and \[\frac{1}{2}\].

Answer:

                (i) \[\frac{2}{3}\times \frac{2\times 5}{3\times 5}\,=\frac{10}{15}\]                 \[\frac{4}{5}=\frac{4\times 3}{5\times 3}\,=\frac{12}{15}\]                           | Converting them into rational numbers with the same denominators                 Now, \[\frac{10}{15}\,=\frac{10\times 4}{15\times 4}\,=\frac{40}{60}\] | Multiplying the numerator and denominator both by 4 \[\frac{12}{15}=\frac{12\times 4}{15\times 4}\,=\frac{48}{60}\]                  | Multiplying the numerator and denominator both by 4 Therefore, five rational numbers between \[\frac{2}{3}\] and \[\frac{4}{5}\] may be taken as \[\frac{41}{60},\,\frac{42}{60},\,\frac{43}{60},\,\frac{44}{60},\,\frac{45}{60}\] (ii) \[\frac{-3}{2}=\frac{-3\times 3}{2\times 3}=\frac{-9}{6}\] \[\frac{5}{3}=\frac{5\times 2}{3\times 2}=\frac{10}{6}\]                                 | Converting them to rational numbers with the same denominators. Therefore, five rational numbers between \[\frac{-3}{2}\] and \[\frac{5}{3}\] may be taken as \[\frac{-8}{6}\,,\,\frac{-7}{6},0,\,\frac{1}{6},\,\frac{2}{6}\] (iii) \[\frac{1}{4}=\frac{1}{4}\] \[\frac{1}{2}=\frac{1\times 2}{2\times 2}\,=\frac{2}{4}\]                                                | Converting them into rational numbers with the same denominators.                 Now, \[\frac{1}{4}=\frac{1\times 8}{4\times 8}=\frac{8}{32}\]                     | Multiplying the numerator and  denominator both by 8 \[\frac{1}{2}=\frac{1\times 16}{2\times 16}=\frac{16}{32}\]                          | Multiplying the numerator and denominator both by 16 Thus, five rational numbers between \[\frac{1}{4}\]  and \[\frac{1}{2}\] may be taken as \[\frac{9}{32},\,\frac{10}{32},\,\frac{11}{32},\,\frac{12}{32},\,\frac{13}{32}\].