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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 \(\frac{-33}{32}\)