### Rational Numbers - Solutions

CBSE Class –VII Mathematics

NCERT Solutions
Chapter 9 Rational Numbers (Ex. 9.1)

Question 1. List five rational numbers between:

(i) $-1$ and 0

(ii) $-2$ and $-1$

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

(iv) $\frac{-1}{2}$ and $\frac{2}{3}$

Answer: (i) $-1$ and 0

Let us write $-1$ and 0 as rational numbers with denominator 6.

$⇒$ $-1=\frac{-6}{6}$ and 0 = $\frac{0}{6}$

$\therefore$ $\frac{-6}{6}<\frac{-5}{6}<\frac{-4}{6}<\frac{-3}{6}<\frac{-2}{6}<\frac{-1}{6}<0$

$⇒$ $-1<\frac{-5}{6}<\frac{-2}{3}<\frac{-1}{2}<\frac{-1}{3}<\frac{-1}{6}<0$

Therefore, five rational numbers between $-1$ and 0 would be

$\frac{-5}{6},\frac{-2}{3},\frac{-1}{2},\frac{-1}{3},\frac{-1}{6}$

(ii) $-2$ and $-1$

Let us write $-2$ and $-1$ as rational numbers with denominator 6.

$⇒$ $-2=\frac{-12}{6}$ and $-1=\frac{-6}{6}$

$\therefore$ $\frac{-12}{6}<\frac{-11}{6}<\frac{-10}{6}<\frac{-9}{6}<\frac{-8}{6}<\frac{-7}{6}<\frac{-6}{6}$

$⇒$ $-2<\frac{-11}{6}<\frac{-5}{3}<\frac{-3}{2}<\frac{-4}{3}<\frac{-7}{6}<-1$

Therefore, five rational numbers between $-2$ and $-1$ would be

$\frac{-11}{6},\frac{-5}{3},\frac{-3}{2},\frac{-4}{3},\frac{-7}{6}$

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

Let us write $\frac{-4}{5}$ and $\frac{-2}{3}$ as rational numbers with the same denominators.

$⇒$ $\frac{-4}{5}=\frac{-36}{45}$ and $\frac{-2}{3}=\frac{-30}{45}$

$\therefore$ $\frac{-36}{45}<\frac{-35}{45}<\frac{-34}{45}<\frac{-33}{45}<\frac{-32}{45}<\frac{-31}{45}<\frac{-30}{45}$

$⇒$ $\frac{-4}{5}<\frac{-7}{9}<\frac{-34}{45}<\frac{-11}{15}<\frac{-32}{45}<\frac{-31}{45}<\frac{-2}{3}$

Therefore, five rational numbers between $\frac{-4}{5}$ and $\frac{-2}{3}$ would be

$\frac{-7}{9},\frac{-34}{45},\frac{-11}{15},\frac{-32}{45},\frac{-31}{45},\frac{-2}{3}$

(iv) $\frac{-1}{2}$ and $\frac{2}{3}$

Let us write $\frac{-1}{2}$ and $\frac{2}{3}$ as rational numbers with the same denominators.

$⇒$ $\frac{-1}{2}=\frac{-3}{6}$ and $\frac{2}{3}=\frac{4}{6}$

$\therefore$ $\frac{-3}{6}<\frac{-2}{6}<\frac{-1}{6}<0<\frac{1}{6}<\frac{2}{6}<\frac{3}{6}<\frac{4}{6}$

$⇒$ $\frac{-1}{2}<\frac{-1}{3}<\frac{-1}{6}<0<\frac{1}{6}<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}$

Therefore, five rational numbers between $\frac{-1}{2}$ and $\frac{2}{3}$ would be $\frac{-1}{3},\frac{-1}{6},0,\frac{1}{6},\frac{1}{3}$.

Question 2. Write four more rational numbers in each of the following patterns:

(i) $\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20},.........$

(ii) $\frac{-1}{4},\frac{-2}{8},\frac{-3}{12},..........$

(iii) $\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24},.........$

(iv) $\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9},..........$

Answer: (i) $\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20},.........$

$⇒$ $\frac{-3×1}{5×1},\frac{-3×2}{5×2},\frac{-3×3}{5×3},\frac{-3×4}{5×4},.........$

Therefore, the next four rational numbers of this pattern would be

$\frac{-3×5}{5×5},\frac{-3×6}{5×6},\frac{-3×7}{5×7},\frac{-3×8}{5×8}$ =