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## Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 2 Real Numbers Ex 2.2

Question 1.

Express the following rational numbers into decimal and state the kind of decimal expression.

(i) \(\frac{2}{7}\)

(ii) -5\(\frac{3}{11}\)

(iii) \(\frac{22}{3}\)

(iv) \(\frac{327}{200}\)

Solution:

(i) \(\frac{2}{7}\) = 0.2857142….

= 0.\(\overline {285714}\)

Non-terminating and recurring decimal expansion.

(ii) -5\(\frac{3}{11}\) = -5 + 0.272 = -5.272……..

= -5.\(\overline {27}\)

Non-terminating and recurring decimal expansion.

(iii) \(\frac{22}{3}\) = 7.333……..

= 7.\(\overline {3}\)

Non-terminating and recurring decimal expansion.

(iv) \(\frac{327}{200}\) = \(\frac{327}{2×100}\)

= \(\frac{3.27}{2}\)

= 1.635

Terminating decimal expansion.

Question 2.

Express \(\frac{1}{13}\) in decimal form. Find the length of the period of decimals.

Solution:

\(\frac{1}{13}\) = 0.07692307

= 0.\(\overline {076923}\)

Length of the period of decimal is 6.

Question 3.

Express the rational number \(\frac{1}{33}\) in recurring decimal form by using the recurring decimal expansion of \(\frac{1}{11}\). Hence write \(\frac{71}{33}\) in recurring decimal form.

Solution:

\(\frac{1}{11}\) = 0.0909……… = 0.\(\overline {09}\)

∴ \(\frac{1}{33}\) = \(\frac{1}{3}\) × \(\frac{1}{11}\)

= \(\frac{1}{3}\) × 0.0909 ……..

= 0.0303 …… = 0.\(\overline {03}\)

\(\frac{71}{33}\) = 2\(\frac{5}{33}\) = 2 + \(\frac{5}{33}\) = 2 + 5 × \(\frac{1}{33}\)

= 2 + 5 × 0.\(\overline {03}\)

2 + (5 × 0.030303 ……..)

2 + 0.151515 ………

2+ 0.\(\overline {15}\)

2.\(\overline {15}\)

Question 4.

Express the following decimal expression into rational numbers.

(i) 0.24

Solution:

Let x = 0.242424 ………. →(1)

100 x = 24.2424 ……… →(2)

(2) – (1) ⇒ 100 x – x = 24.2424 ……….. (-)

0.2424 ……..

99 x = 24.0000

x = \(\frac{24}{99}\)

(or)

\(\frac{8}{33}\)

(ii) 2.327

Solution:

Let x = 2.327327327 ………. →(1)

1000 x = 2327.327327 ……… →(2)

(2) – (1) ⇒ 1000 x – x = 2327.327327 ……….. (-)

2.327327 ……..

999 x = 2325.000

x = \(\frac{2325}{999}\)

(or)

\(\frac{775}{333}\)

(iii) – 5.132

Solution:

– 5.132 = -5 + \(\frac{1}{10}\) + \(\frac{3}{100}\) + \(\frac{2}{1000}\)

= \(\frac{-5000 + 100 +30 + 2}{1000}\) = \(\frac{-4868}{1000}\)

(or)

\(\frac{-1217}{250}\)

(iv) 3.17

Solution:

Let x = 3.1777 ………. →(1)

10 x = 31.777 ……… →(2)

100 x = 317.77 …….. →(3)

(3) – (2) ⇒ 100 x – 10 x = 317.77 ……….. (-)

31.777 ……..

90 x = 286.000

x = \(\frac{286}{90}\)

(or)

\(\frac{143}{45}\)

(v) 17.215

Solution:

Let x = 17.2151515 ………. →(1)

10 x = 172.151515 ……… →(2)

100 x = 17215.1515 …….. →(3)

(3) – (2) ⇒ 1000 x – 10 x = 17215.1515 ……….. (-)

17215.1515 ……..

990 x = 17043

x = \(\frac{17043}{990}\)

(or)

\(\frac{5681}{330}\)

(vi) -21.2137

Solution:

Let x = -21.213777 ………. →(1)

1000 x = -21213.777 ……… →(2)

100 x = -212137.77 …….. →(3)

(3) – (2) ⇒ 10000 x – 1000 x = -21213.777 ……….. (-)

-21213.777 ……..

9000 x = -190924

x = \(\frac{-190924}{9000}\)

(or)

\(\frac{-47731}{2250}\)

Question 5.

Without actual division, find which of the following rational numbers have terminating decimal expression.

(i) \(\frac{7}{128}\)

Solution:

\(\frac{7}{128}\) = \(\frac{7}{2^{7}}\)

∴ \(\frac{7}{128}\) has terminating decimal expression.

(ii) \(\frac{21}{15}\)

Solution:

\(\frac{21}{15}\) = \(\frac{7}{5}\) = \(\frac{7}{5^1}\)

\(\frac{21}{15}\) has terminating decimal expression.

(iii) 4\(\frac{9}{35}\)

Solution:

4\(\frac{9}{35}\) = \(\frac{149}{35}\)

4\(\frac{149}{5×7}\) (It is not in the form of \(\frac{P}{2^{m} × 5^{n}}\)

∴ 4\(\frac{9}{35}\) has non-terminating recurring decimal expression.

(iv) \(\frac{219}{2200}\)

Solution:

\(\frac{219}{2200}\) = \(\frac{219}{2^{3} × 5^{2} × 11}\) (It is not in the form of \(\frac{P}{2^{m} × 5^{n}}\)

∴ \(\frac{219}{2200}\) has non-terminating recurring decimal expression.