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 q ?

(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 .

NCERT solution class 10 chapter 1 Real numbers

Exercise 1.4

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