Question

Write three rational numbers that lie between the two given numbers.

(i) $\frac{2}{7},\frac{6}{7}$

(ii) $\frac{4}{5},\frac{2}{3}$

(iii) $-\frac{2}{3},\frac{4}{5}$

(iv) $\frac{7}{9},-\frac{5}{9}$

(v) $\frac{-3}{4},\frac{+5}{4}$

(vi) $\frac{7}{8},\frac{-5}{3}$

(vii) $\frac{5}{7},\frac{11}{7}$

(viii) $0,\frac{-3}{4}$

Answer 1

(i) The given numbers are $\frac{2}{7}$ and $\frac{6}{7}$.
We know that,
2 < 3 < 4 < 5 < 6
$\therefore \frac{2}{7}<\frac{3}{7}<\frac{4}{7}<\frac{5}{7}<\frac{6}{7}$
Hence, 3 rational numbers between $\frac{2}{7}$ and $\frac{6}{7}$ are :
$\frac{3}{7},\frac{4}{7}$ and $\frac{5}{7}$.

(ii) The given numbers are $\frac{4}{5}$ and $\frac{2}{3}$.
Let us convert these numbers into fractions with equal denominators.
$$\begin{gathered} \frac{4}{5}=\frac{4\times 6}{5\times 6}=\frac{24}{30} \\ \frac{2}{3}=\frac{2\times 10}{3\times 10}=\frac{20}{30} \end{gathered}$$
We know that,
20 < 21 < 22 < 23 < 24
$$\begin{gathered} \therefore \frac{20}{30}<\frac{21}{30}<\frac{22}{30}<\frac{23}{30}<\frac{24}{30} \\ \Rightarrow \frac{2}{3}<\frac{21}{30}<\frac{22}{30}<\frac{23}{30}<\frac{4}{5} \end{gathered}$$
Hence, 3 rational numbers between $\frac{2}{3}$ and $\frac{4}{5}$ are :
$\frac{21}{30},\frac{22}{30}$ and $\frac{23}{30}$.

(iii) The given numbers are $-\frac{2}{3}$ and $\frac{4}{5}$.
Let us convert each of given numbers into fractions with equal denominators.
$$\begin{gathered} -\frac{2}{3}=\frac{-2\times 5}{3\times 5}=-\frac{10}{15} \\ \frac{4}{5}=\frac{4\times 3}{5\times 3}=\frac{12}{15} \end{gathered}$$
We know that,
−10 < −9 < −8 < −7 <...........< 1 < 2 < 3 < 4 <..........< 12
$\Rightarrow -\frac{2}{3}<-\frac{9}{15}<-\frac{8}{15}<-\frac{7}{15}<.....<\frac{1}{15}<\frac{2}{15}<\frac{3}{15}<\frac{4}{15}<.....<\frac{4}{15}$
Hence, 3 rational numbers between $-\frac{2}{3}$ and $\frac{4}{5}$ are:
$-\frac{9}{15},-\frac{7}{15}$ and $\frac{4}{15}.$

(iv) The given numbers are $\frac{7}{9}$ and $-\frac{5}{9}.$
We know that,
−5 < −4 < −3 < −2 < −1 < 0 <.....< 6 < 7
$\therefore -\frac{5}{9}<-\frac{4}{9}<-\frac{3}{9}<-\frac{2}{9}<-\frac{1}{9}<0<.....<\frac{6}{9}<\frac{7}{9}$
Hence, 3 rational numbers between $-\frac{5}{9}$ and $\frac{7}{9}$ are:
$-\frac{4}{9},0$ and $\frac{6}{9}.$

(v) The given numbers are $-\frac{3}{4}$ and $\frac{5}{4}$.
We know that,
−3 < −2 < −1 < 0 < 1 < 2 < 3 < 4 < 5
$\therefore -\frac{3}{4}<-\frac{2}{4}<-\frac{1}{4}<0<\frac{1}{4}<\frac{2}{4}<\frac{3}{4}<\frac{4}{4}<\frac{5}{4}$
Hence, 3 rational numbers between $-\frac{3}{4}$ and $\frac{5}{4}$ are:
$-\frac{2}{4},-\frac{1}{4}$ and $\frac{3}{4}.$

(vi) The given numbers are $\frac{7}{8}$ and $-\frac{5}{3}$.
Let us convert each of the given numbers into fractions with equal denominators.
$$\begin{gathered} \frac{7}{8}=\frac{7\times 3}{8\times 3}=\frac{21}{24} \\ -\frac{5}{3}=\frac{-5\times 8}{3\times 8}=-\frac{40}{24} \end{gathered}$$
We know that,
−40 < −39 <....< −13 < −12 <......<11 < 12 <....17 <.... 21
$$\begin{gathered} \therefore -\frac{40}{24}<-\frac{39}{24}<.....<-\frac{13}{24}<-\frac{12}{24}<....<\frac{11}{24}<\frac{12}{24}<....<\frac{17}{24}<....<\frac{21}{24} \\ \Rightarrow -\frac{5}{3}<-\frac{39}{24}<......<-\frac{13}{24}<-\frac{12}{24}<....<\frac{11}{24}<\frac{12}{24}<....<\frac{17}{24}<....<\frac{7}{8} \end{gathered}$$
Hence, 3 rational numbers between $-\frac{5}{3}$ and $\frac{7}{8}$ are:
$-\frac{13}{24},\frac{11}{24}$ and $\frac{17}{24}.$

(vii) The given numbers are $\frac{5}{7}$ and $\frac{11}{7}$.
We know that,
5 < 6 < 7 < 8 < 9 < 10 < 11
$\therefore \frac{5}{7}<\frac{6}{7}<\frac{7}{7}<\frac{8}{7}<\frac{9}{7}<\frac{10}{7}<\frac{11}{7}$
Hence, 3 rational numbers between $\frac{5}{7}$ and $\frac{11}{7}$ are :
$\frac{6}{7},\frac{8}{7}$ and $\frac{9}{7}$

(viii) The given numbers are 0 and $-\frac{3}{4}$.
Let us convert each of the given numbers into fractions with equal denominators.
$$\begin{gathered} 0=\frac{0\times 8}{1\times 8}=\frac{0}{8} \\ -\frac{3}{4}=\frac{-3\times 2}{4\times 2}=-\frac{6}{8} \end{gathered}$$
We know that,
−6 < −5 < −4 < −3 < −2 < −1 < 0
$$\begin{gathered} \therefore -\frac{6}{8}<-\frac{5}{8}<-\frac{4}{8}<-\frac{3}{8}<-\frac{2}{8}<-\frac{1}{8}<\frac{0}{8} \\ \Rightarrow -\frac{3}{4}<-\frac{5}{8}<-\frac{4}{8}<-\frac{3}{8}<-\frac{2}{8}<-\frac{1}{8}<0 \end{gathered}$$
Hence, 3 rational numbers between $-\frac{6}{8}$ and 0 are:
$-\frac{5}{8},-\frac{2}{8}$ and $-\frac{1}{8}.$