Question

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

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


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

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


3. Compare the following rational numbers.


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


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

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


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

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

Answer 1

Answer 1 :-

{Answer is attached}

Steps of construction –

• Firstly, we’ll solve the given fraction to make it in decimal form.

• The quotient we’ll get will determine the position of the fraction on the number line. For example- 0.25 [It will lie between 0 and 1 because it is bigger than 0 but smaller than 1]

• Then, we will divide the gap between those numbers in parts [No. of parts = Denominator of fraction]

• The position of the fraction in the parts will be the numerator of the fraction.

Answer 2 :-

~Here, we need to make the rational in the standard form. For this we’ll divide both the numerator and denominator by their highest common factors.

$\sf \bullet \;\;\; \dfrac{5}{15}$

$\sf \dashrightarrow \dfrac{5 \div 5}{15 \div 5}$

$\boxed{\bf{ \leadsto \;\; \dfrac{1}{3} }}$

$\sf \bullet \;\;\; \dfrac{-24}{40}$

$\sf \dashrightarrow \dfrac{ -24 \div 8}{40 \div 8 }$

$\boxed{\bf{ \leadsto \;\; \dfrac{-3}{5}}}$

$\sf \bullet \;\;\; \dfrac{33}{-77}$

$\sf \dashrightarrow \dfrac{33 \div 11}{-77 \div 11}$

$\boxed{\bf{ \leadsto \;\; \dfrac{3}{-7} }}$

$\sf \bullet \;\;\; \dfrac{-45}{-105}$

$\sf \dashrightarrow \dfrac{-45 \div -15}{-105 \div -15}$

$\boxed{\bf{ \leadsto \;\; \dfrac{3}{7} }}$

Answer 3 :-

Here, we have to compare two rational numbers. Firstly, we’ll make them like fraction and then we can compare by comparing their numerators.

$\sf \bullet \;\;\; \dfrac{-9}{27} , \dfrac{6}{-18}$

$\sf \implies \dfrac{-9 \times 2}{27 \times 2} , \dfrac{6 \times 3}{18 \times 3}$

$\boxed{\bf{ \leadsto \;\; \dfrac{-18}{54} = \dfrac{-18}{54} }}$

$\sf \bullet \;\;\; \dfrac{-5}{7} , \dfrac{10}{-6}$

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

$\boxed{\bf{ \leadsto \;\; \dfrac{-30}{42} > \dfrac{-70}{42}}}$

$\sf \bullet \;\;\; \dfrac{3}{-8} , \dfrac{-15}{40}$

$\sf \implies - \dfrac{3 \times 5}{8 \times 5} , \dfrac{-15}{40}$

$\boxed{\bf{ \leadsto \;\; \dfrac{-15}{40} = \dfrac{-15}{40}}}$

$\sf \bullet \;\;\; \dfrac{-11}{7} , \dfrac{33}{21}$

$\sf \implies \dfrac{-11 \times 3}{7 \times 3} , \dfrac{33}{21}$

$\boxed{\bf{ \leadsto \;\; \dfrac{-33}{21} < \dfrac{33}{21}}}$

Answer 4 :-

Here, we have to arrange the given fractions in ascending order [smallest to biggest].For this we need to make all rational numbers to like fractions and then we can arrange them by comparing their numerators.

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

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

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

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

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

$\boxed{\bf{ \leadsto \;\; \dfrac{2}{-3} , \dfrac{-4}{9} , \dfrac{-5}{12} , \dfrac{7}{18}}}$

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

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

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

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

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

$\boxed{\bf{ \leadsto \;\; \dfrac{3}{-4}, \dfrac{-7}{16}, \dfrac{-5}{12} , \dfrac{9}{-24}}}$

Answer 5 :-

Here, we have to arrange the given fractions in descending order [ biggest to smallest]. For this we need to make all rational numbers to like fractions and then we can arrange them by comparing their numerators.

$\sf \bullet \; \dfrac{2}{5} , \dfrac{1}{3} , \dfrac{3}{4} , \dfrac{1}{6}$

$\sf \implies \dfrac{2}{5} = \dfrac{2 \times 12}{ 5 \times 12} = \dfrac{24}{60}$

$\sf \implies \dfrac{1}{3} = \dfrac{1 \times 20}{3 \times 20} = \dfrac{20}{60}$

$\sf \implies \dfrac{3}{4} = \dfrac{ 3 \times 15}{ 4 \times 15 } = \dfrac{45}{60}$

$\sf \implies \dfrac{1}{6} = \dfrac{ 1 \times 10}{ 6 \times 10} = \dfrac{10}{60}$

$\boxed{\bf{ \leadsto \;\; \dfrac{3}{4} , \dfrac{2}{5} , \dfrac{1}{3} , \dfrac{1}{6} }}$

$\sf \bullet \; \dfrac{5}{6} , \dfrac{7}{8} , \dfrac{11}{12} , \dfrac{3}{10}$

$\sf \implies \dfrac{5}{6} = \dfrac{ 5 \times 20}{ 6 \times 20} = \dfrac{100}{120}$

$\sf \implies \dfrac{7}{8} = \dfrac{7 \times 15}{ 8 \times 15} = \dfrac{105}{120}$

$\sf \implies \dfrac{11}{12} = \dfrac{ 11 \times 10}{ 12 \times 10} = \dfrac{110}{120}$

$\sf \implies \dfrac{3}{10} = \dfrac{ 3 \times 12}{ 10 \times 12} = \dfrac{36}{120}$

$\boxed{\bf{ \leadsto \;\; \dfrac{7}{8} , \dfrac{11}{12} , \dfrac{5}{6} , \dfrac{3}{10} }}$