Question

1. Represent the following rational numbers on the number line.

$\begin{gathered}\sf (a) \: \frac{ - 3}{4} \\ \\ \sf (b) \: \frac{31}{ - 6} \\ \\ \sf (c) \: \frac{ - 1}{2} \\ \\ \sf (d) \: \frac{3}{4} \end{gathered}$


2. Write the following rational numbers in the standard form.

$\begin{gathered}\sf \: (a) \: \frac{5}{15} \\ \\ \sf (b) \: \frac{ - 24}{40} \\ \\ \sf (c) \: \frac{33}{ - 77} \\ \\ \sf (d) \: \frac{ - 45}{ - 105} \end{gathered}$


3. Compare the following rational numbers.

$\begin{gathered}\sf (a) \: \frac{ - 9}{27} , \frac{6}{ - 18} \\ \\ \sf (b) \: \frac{-5}{7} , \frac{10}{-6} \\ \\ \sf (c) \: \frac{3}{-8} , \frac{-15}{40} \\ \\ \sf (d) \: \frac{-11}{7} , \frac{33}{21} \end{gathered}$


4. Arrange the following rational numbers in the descending order.

$\begin{gathered}\sf (a) \: \frac{2}{-3} , \frac{-4}{9} , \frac{-5}{12} , \frac{7}{-18} \\ \\ \sf (b) \: \frac{3}{-4} , \frac{-5}{12} , \frac{-7}{16} , \frac{9}{-24} \end{gathered}$


5. Arrange the following rational numbers in the ascending order.

$\begin{gathered}\sf (a) \: \frac{2}{5} , \frac{1}{3} , \frac{3}{4} , \frac{1}{6} \\ \\ \sf (b) \: \frac{5}{6} , \frac{7}{8} , \frac{11}{12} , \frac{3}{10}\end{gathered}$
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Answer 1

1. Represent the following rational numbers on the number line.

Solution : [See the attached images]

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2. Write the following rational numbers in the standard form.

Note : To write the standard form of any rational numbers, we have to divide both the numerator and denominator by their highest common factors.

$\bf \: (a) \: \: \sf\dfrac{5}{15} \: = \dfrac{5 \div 5}{15 \div 5}$

$\sf \implies \: \: \cancel\dfrac{5}{15} \: = \boxed{\bf \dfrac{1}{3} }$

$\bf \: (b) \: \sf\dfrac{ - 24}{40} \: = \dfrac{ - 24 \div 8}{40 \div 8}$

$\sf \implies \: \: \cancel\dfrac{ - 24}{8} \: = \boxed{\bf \dfrac{ - 3}{5} }$

$\bf \: (c) \: \sf\dfrac{33}{ - 77} \: = \dfrac{ 33 \div 11}{ - 77 \div 11}$

$\sf \implies \: \: \cancel\dfrac{ 33}{ - 77} \: = \dfrac{3}{ - 7} = \boxed{\bf \dfrac{ - 3}{7} }$

$\bf \: (d) \: \sf \dfrac{ - 45}{ - 105} \: = \dfrac{ - 45 \div 15}{ - 105 \div 15}$

$\sf \implies \: \: \cancel\dfrac{ - 45}{ - 105} \: = \dfrac{ - 3}{ - 7} = \boxed{\bf \dfrac{3}{7} }$

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3. Compare the following rational numbers.

$\bf (a) \: \: \sf \dfrac{ - 9}{27} \: , \: \dfrac{6}{ - 18}$

$\sf \implies \: \: \dfrac{ - 9}{27} \: , \: \dfrac{ - 6}{18}$

$\sf \implies \: \: - 9 \times 18 \: \: \: , \: \: - 27 \times 6$

$\sf \implies \: \: - 162 \: \: \: = \: \: - 162$

$\sf \implies \: \: \dfrac{ - 9}{27} \: \: \: = \: \: \dfrac{6}{ - 18}$

Hence, both are equal.

$\bf (b) \: \: \sf \: \dfrac{-5}{7} \: \: \: , \: \: \dfrac{10}{-6}$

$\sf \implies \: \: \dfrac{ - 5}{7} \: , \: \dfrac{ -10}{6}$

$\sf \implies \: \: - 5 \times 6 \: \: \: , \: \: 7 \times - 10$

$\sf \implies \: \: -30 \: \: \: > \: \: - 70$

$\sf \: Hence, \: \bf\dfrac{ - 5}{7} \: \: \sf is \: greater \: than \: \: \bf\dfrac{10}{-6}$

$\bf (c) \: \: \sf \: \: \dfrac{3}{-8} \: \: \: , \: \dfrac{-15}{40}$

$\sf \implies \: \: \dfrac{ - 3}{8} \: , \: \dfrac{ -15}{40}$

$\sf \implies \: \: - 3 \times 40 \: \: \: , \: \: 8 \times - 15$

$\sf \implies \: \: - 120 \: \: \: = \: \: - 120$

$\sf \implies \: \: \dfrac{3}{-8} \: \: \: = \: \: \dfrac{-15}{40}$

Hence, both are equal.

$\bf (d) \: \: \sf \: \dfrac{-11}{7} \: \: , \: \dfrac{33}{21}$

$\sf \implies \: \: - 11 \times 21 \: \: \: , \: \: 7 \times 33$

$\sf \implies \: \: - 231 \: \: \: < \: \: 231$

$\sf \implies \: \: \dfrac{-11}{7} \: \: \: < \: \: \dfrac{33}{21}$

$\sf \: Hence, \: \: \bf\dfrac{33}{21} \: \: \sf is \: greater \: than \: \: \bf\dfrac{-11}{7}$

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4. Arrange the following rational numbers in the descending order.

Note : Descending order means largest to smallest. For this we need to make all rational numbers to like fractions and then we can arrange them by comparing their numerators.

$\bf(a) \: \: \: \sf\dfrac{2}{-3} , \: \dfrac{-4}{9} , \: \dfrac{-5}{12} , \: \dfrac{7}{-18}$

$\longrightarrow \: \: \sf\dfrac{ - 2}{3} , \: \dfrac{-4}{9} , \: \dfrac{-5}{12} , \: \dfrac{ - 7}{18}$

$\bf \underline{Now},$

$\sf \implies \dfrac{ - 2}{3} = \dfrac{-2 \times 12}{3 \times 12} = \dfrac{-24}{36}$

$\sf \implies \dfrac{-4}{9} = \dfrac{-4 \times 4}{9 \times 4} = \dfrac{-16}{36}$

$\sf \implies \dfrac{-5}{12} = \dfrac{-5 \times 3}{12 \times 3} = \dfrac{-15}{36}$

$\sf \implies \dfrac{ - 7}{18} = \dfrac{-7 \times 2}{18 \times 2} = \dfrac{-14}{36}$

$\bf \underline{Answer} = \boxed{ \sf \dfrac{-7}{18} \: > \: \dfrac{-5}{12} \: > \: \dfrac{-4}{9} \: > \: \dfrac{-2}{3}}$

$\sf \: (b) \: \: \: \dfrac{3}{-4} , \: \dfrac{-5}{12} , \: \dfrac{-7}{16} , \: \dfrac{9}{-24}$

$\sf \longrightarrow \: \: \dfrac{ - 3}{4} , \: \dfrac{-5}{12} , \: \dfrac{-7}{16} , \: \dfrac{ - 9}{24}$

$\bf \underline{Now},$

$\sf \implies \dfrac{ - 3}{4} = \dfrac{ -3 \times 24}{ 4 \times 24} = \dfrac{-72}{96}$

$\sf \implies \dfrac{-5}{12} = \dfrac{-5 \times 8}{ 12 \times 8} = \dfrac{-40}{96}$

$\sf \implies \dfrac{-7}{16} = \dfrac{-7 \times 6}{16 \times 6} = \dfrac{-42}{96}$

$\sf \implies \dfrac{-9}{24} = \dfrac{-9 \times 4}{24 \times 4} = \dfrac{-36}{96}$

$\bf \underline{Answer} = \boxed{ \sf \dfrac{-9}{4} \: > \: \dfrac{-5}{12} \: > \: \dfrac{-7}{16} \: > \: \dfrac{-3}{4}}$

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5. Arrange the following rational numbers in the ascending order.

Note : Ascending order means smallest to largest. For this we need to make all rational numbers to like fractions and then we can arrange them by comparing their numerators.

$\bf (a) \:\: \:\sf \dfrac{2}{5} , \:\dfrac{1}{3} ,\: \dfrac{3}{4} ,\: \dfrac{1}{6}$

Solution : [See the attached images]

$\bf (b)\: \: \sf \dfrac{5}{6} , \:\dfrac{7}{8} ,\: \dfrac{11}{12},\: \dfrac{3}{10}$

Solution : [See the attached images]