NCERT solution class 10 chapter 1 Real numbers

Exercise 1.1    Exercise 1.2   Exercise 1.3   Exercise 1.4

Exercise 1.4

Question 1

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i) \dfrac{13}{3125}

Sol :

3125=5^5

The denominator is in the form of 5^m

Hence, the decimal expansion of \dfrac{13}{3125} is terminating.

 

(ii) \dfrac{17}{8}

Sol :

8=2^3

The denominator is in the form of 2^m

Hence, the decimal expansion of \dfrac{17}{8} is terminating.

 

(iii) \dfrac{64}{455}

Sol :

455=5\times 7\times 13

Since,the denominator is not in the form of 2^m\times 5^n and it also contains 7 and 13 as its factors , its decimal expansion will be non-terminating repeating .

 

(iv) \dfrac{15}{1600}

Sol :

1600=2^6\times 5^2

The denominator is of the form 2^m\times 5^n .

Hence , the decimal expansion of \dfrac{15}{1600} is terminating .

 

(v) \dfrac{29}{343}

Sol :

343=7^3

Since,the denominator is not in the form of 2^m\times 5^n and it also contains 7 as its factors , its decimal expansion will be non-terminating repeating .

 

(vi) \dfrac{23}{2^3\times 5^2}

Sol :

The denominator is of the form 2^m\times 5^n .

Hence , the decimal expansion of \dfrac{23}{2^3\times 5^2} is terminating .

 

(vii) \dfrac{129}{2^2\times 5^7\times 7^5}

Sol :

The denominator is of the form 2^m\times 5^n but it also has 75 as its factor, the decimal expansion of \dfrac{129}{2^2\times 5^7\times 7^5} is non-terminating repeating .

 

(viii) \dfrac{6}{15}

Sol :

\dfrac{6}{15}=\dfrac{2\times 3}{3\times 5}=\dfrac{2}{5}

The denominator is of the form 5n

Hence ,the decimal expansion of \dfrac{6}{15} is terminating .

 

(ix) \dfrac{35}{50}

Sol :

\dfrac{35}{50}=\dfrac{7\times 5}{10\times 5}=\dfrac{7}{10}

10=2\times 5

The denominator is of the form 2^m\times 5^n

Hence , the decimal expansion of \dfrac{35}{50} is terminating .

 

(x) \dfrac{77}{210}

Sol :

\dfrac{77}{210}=\dfrac{11\times 7}{30\times 7}=\dfrac{11}{30}

30=2\times 3\times 5

The denominator is in the form of 2^m\times 5^n but it contains 3 as its factors , the decimal expansion of \dfrac{77}{210} is non-terminating repeating .

 

Question 2

Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Sol :

(i) \dfrac{13}{3125}=0.00416

 \begin{array}{rll} 0.00416\\3125\showdiv{13.00000}\\ \underline{\phantom{0}0\phantom{.00000}}\\ 130\phantom{0000}\\ \underline{\phantom{00}0\phantom{0000}}\\ 1300\phantom{000}\\ \underline{\phantom{000}0\phantom{000}}\\ 13000\phantom{00}\\ \underline{12500\phantom{00}}\\5000\phantom{0}\\ \underline{\phantom{00}3125\phantom{0}}\\ \phantom{00}18750\\ \underline{\phantom{00}18750}\\ \underline{\phantom{000000}0} \end{array}

 

(ii) \dfrac{17}{8}=2.125 (working)

(iii) \dfrac{64}{455}=

(iv) \dfrac{15}{1600}=0.009375

(v) \dfrac{29}{343}

(vi) \dfrac{23}{2^3\times 5^2}

(vii) \dfrac{129}{2^2\times 5^7\times 7^5}

(viii) \dfrac{6}{15}

(ix) \dfrac{35}{50}

(x) \dfrac{77}{210}

Question 3

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form \dfrac{p}{q}, what can you say about the prime factor of ?

(i) 43.123456789

Sol :

Since this number has a terminating decimal expansion, it is a rational number of the form \dfrac{p}{q} and q is of the form 2^m\times 5^n .

i.e., the prime factors of q will be either 2 or 5 or both .

 

(ii) 0.120120012000120000\dots

Sol :

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

 

(iii) 43.\overline{123456789}

Sol :

Since, the decimal expansion is non-terminating recurring, the given number is a rational number of the form \dfrac{p}{q} and q is not of the form 2^m\times 5^n

i.e., the prime factors of q will also have a factor other than 2 or 5 .

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