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## Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 2 Real Numbers Ex 2.1

Question 1.
Which arrow best shows the position of $$\frac{11}{3}$$ on the number line?

Solution:
D represent $$\frac{11}{3}$$ on the number line.

Question 2.
Find any three rational numbers between $$\frac{-7}{11}$$ and $$\frac{2}{11}$$
Solution:
Three rational numbers between $$\frac{-7}{11}$$ and $$\frac{2}{11}$$
$$\frac{-6}{11}$$, $$\frac{-5}{11}$$, $$\frac{-4}{11}$$, ……… $$\frac{1}{11}$$

Question 3.
Find any five rational numbers between
(i) $$\frac{1}{4}$$ and $$\frac{1}{5}$$
Solution:
Converting the given rational numbers with the same denominators.
$$\frac{1}{4}$$ = $$\frac{1×30}{4×30}$$ = $$\frac{30}{120}$$
$$\frac{1}{5}$$ = $$\frac{1×24}{5×24}$$ = $$\frac{24}{120}$$
Five rational numbers between $$\frac{30}{120}$$ and $$\frac{24}{120}$$ are $$\frac{25}{120}$$, $$\frac{26}{120}$$, $$\frac{27}{120}$$, $$\frac{28}{120}$$ and $$\frac{29}{120}$$
Five rational numbers between $$\frac{1}{4}$$ and $$\frac{1}{5}$$ are $$\frac{25}{120}$$, $$\frac{26}{120}$$, $$\frac{27}{120}$$, $$\frac{28}{120}$$ and $$\frac{29}{120}$$
Other Method:
A rational numbers between $$\frac{1}{4}$$ and $$\frac{1}{5}$$ = $$\frac{1}{2}$$($$\frac{1}{4}$$+$$\frac{1}{5}$$) = $$\frac{1}{2}$$($$\frac{5+4}{20}$$) = $$\frac{1}{2}$$ × $$\frac{9}{20}$$ = $$\frac{9}{40}$$
A rational numbers between $$\frac{1}{4}$$ and $$\frac{9}{40}$$ = $$\frac{1}{2}$$($$\frac{1}{4}$$+$$\frac{9}{40}$$) = $$\frac{1}{2}$$($$\frac{10+9}{40}$$) = $$\frac{19}{80}$$
A rational numbers between $$\frac{1}{4}$$ and $$\frac{19}{80}$$ = $$\frac{1}{2}$$($$\frac{1}{4}$$+$$\frac{19}{20}$$) = $$\frac{1}{2}$$($$\frac{20+19}{80}$$) = $$\frac{39}{160}$$
A rational numbers between $$\frac{1}{4}$$ and $$\frac{39}{160}$$ = $$\frac{1}{2}$$($$\frac{1}{4}$$+$$\frac{39}{160}$$) = $$\frac{1}{2}$$($$\frac{40+39}{160}$$) = $$\frac{79}{320}$$
A rational numbers between $$\frac{1}{4}$$ and $$\frac{79}{320}$$ = $$\frac{1}{2}$$($$\frac{1}{4}$$+$$\frac{79}{320}$$) = $$\frac{1}{2}$$($$\frac{80+79}{320}$$) = $$\frac{159}{640}$$
∴ Five rational numbers are between $$\frac{9}{40}$$, $$\frac{19}{80}$$, $$\frac{39}{160}$$, $$\frac{79}{320}$$ and $$\frac{159}{640}$$

(ii) 0.1 and 0.11
Solution:
$$\frac{1×100}{10×100}$$ = $$\frac{100}{1000}$$
$$\frac{11×10}{100×10}$$ = $$\frac{110}{1000}$$
The five rational numbers are $$\frac{101}{1000}$$, $$\frac{102}{1000}$$, $$\frac{103}{1000}$$, $$\frac{104}{1000}$$, $$\frac{105}{1000}$$, …….. (or)
The five rational numbers are 0.101, 0.102, 0.103, 0.104 and 0.105.

(iii) -1 and -2
Solution:
Converting to rational numbers, – 1 = $$-\frac{10}{11}$$ and – 2 = $$-\frac{20}{10}$$
So five rational numbers between -2 and -1 are $$-\frac{11}{10}$$, $$-\frac{12}{10}$$, $$-\frac{13}{10}$$, $$-\frac{14}{10}$$, $$-\frac{15}{10}$$.
Other Method:
A rational number between -1 and -2 = $$\frac{1}{2}$$[-1-2] = $$\frac{1}{2}$$[-3] = $$-\frac{3}{2}$$
A rational number between -1 and $$-\frac{3}{2}$$ = $$\frac{1}{2}$$[-1 – $$\frac{3}{2}$$] = $$\frac{1}{2}$$($$\frac{-2-3}{2}$$) = $$-\frac{5}{4}$$
A rational number between -1 and $$-\frac{5}{4}$$ = $$\frac{1}{2}$$[-1 – $$\frac{5}{4}$$] = $$\frac{1}{2}$$($$\frac{-4-5}{4}$$) = $$-\frac{9}{8}$$
A rational number between -1 and $$-\frac{9}{8}$$ = $$\frac{1}{2}$$[-1 – $$\frac{9}{8}$$] = $$\frac{1}{2}$$($$\frac{-8-9}{8}$$) = $$-\frac{17}{16}$$
A rational number between -1 and $$-\frac{17}{16}$$ = $$\frac{1}{2}$$[1 – $$\frac{17}{16}$$] = $$\frac{1}{2}$$($$\frac{-16-17}{16}$$) = $$\frac{1}{2}$$ ($$\frac{-33}{16}$$) = $$\frac{-33}{32}$$
The five rational numbers are $$-\frac{3}{2}$$, $$-\frac{5}{4}$$, $$-\frac{9}{8}$$, $$-\frac{17}{16}$$, and $$\frac{-33}{32}$$