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You are here: Home / NCERT Solutions / Class 10 / Maths / Chapter 1 Real Numbers / Exercise 1.4

Exercise 1.4

Last Updated on April 1, 2021 By Mrs Shilpi Nagpal Leave a Comment

NCERT Solutions for Class 10 Maths
Chapter 1 Real Numbers Exercise 1.4

Page 17


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) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/(2352) (vii) 129/(225775) (viii) 6/15 (ix) 35/50 (x) 77/210

Answer :

Note: If the denominator has only factors of 2 and 5 or in the form of 2m ×5n then it has terminating decimal expansion.  If the denominator has factors other than 2 and 5 then it has a non-terminating decimal expansion.

(i) 13/3125

Factorizing the denominator, we get,

3125 = 5 × 5 × 5 = 55

Since, the denominator has only 5 as its factor, 13/3125 has a terminating decimal expansion.

(ii) 17/8

Factorizing the denominator, we get,

8 = 2×2×2 = 23

Since, the denominator has only 2 as its factor, 17/8 has a terminating decimal expansion.

(iii) 64/455

Factorizing the denominator, we get,

455 = 5×7×13

Since, the denominator is not in the form of 2m × 5n, thus 64/455 has a non-terminating decimal expansion.

(iv) 15/ 1600

Factorizing the denominator, we get,

1600 = 2652

Since, the denominator is in the form of 2m × 5n, thus 15/1600 has a terminating decimal expansion.

(v) 29/343

Factorizing the denominator, we get,

343 = 7×7×7 = 73 Since, the denominator is not in the form of 2m × 5n thus 29/343 has a non-terminating decimal expansion.

(vi)23/(2352)

Clearly, the denominator is in the form of 2m × 5n.

Hence, 23/ (2352) has a terminating decimal expansion.

(vii) 129/(225775)

As you can see, the denominator is not in the form of 2m × 5n.

Hence, 129/ (225775) has a non-terminating decimal expansion.

(viii) 6/15

6/15 = 2/5

Since, the denominator has only 5 as its factor, thus, 6/15 has a terminating decimal expansion.

(ix) 35/50

35/50 = 7/10

Factorising the denominator, we get,

10 = 2 5

Since, the denominator is in the form of 2m × 5n thus, 35/50 has a terminating decimal expansion.

(x) 77/210

77/210 = (7× 11)/ (30 × 7) = 11/30

Factorising the denominator, we get,

30 = 2 × 3 × 5

As you can see, the denominator is not in the form of 2m × 5n .Hence, 77/210 has a non-terminating decimal expansion.


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

Answer :

$
\begin{equation}
\begin{aligned}
&\text { (i) } \frac{13}{3125}\\\\
&\frac{13}{3125}=\frac{13}{5 \times 5 \times 5 \times 5 \times 5}=\frac{13}{5^{5}} \times \frac{2^{5}}{2^{5}}\\\\
&=\frac{13 \times 32}{(5 \times 2)^{5}}=\frac{416}{10^{5}}=0.00416
\end{aligned}
\end{equation}
$

$
\begin{equation}
\begin{aligned}
&\text { (ii) } \frac{17}{8}\\\\
&\frac{17}{8}=\frac{17}{2 \times 2 \times 2}=\frac{17}{2^{3}} \times \frac{5^{3}}{5^{3}}\\\\
&=\frac{17 \times 125}{(2 \times 5)^{3}}=\frac{2125}{10^{3}}=2.125
\end{aligned}
\end{equation}
$

$
\begin{equation}
\begin{aligned}
&\text { (iii) } \frac{64}{455}\\
\end{aligned}
\end{equation}
$

Decimal expansion is non-terminating and repeating.

$
\begin{equation}
\begin{aligned}
&\text { (iv) } \frac{15}{1600}\\\\
&=\frac{3 \times 5}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5}\\\\
&=\frac{3}{2^{6} \times 5} \times \frac{5^{5}}{5^{5}}=\frac{3 \times 3125}{(2 \times 5)^{6}}\\\\
&=\frac{9375}{10^{6}}=0.009375
\end{aligned}
\end{equation}
$

$
\begin{equation}
\begin{aligned}
&\text { (v) } \frac{29}{343}\\
\end{aligned}
\end{equation}
$

Decimal expansion is non-terminating and repeating.

$
\begin{equation}
\begin{aligned}
&\text { (vi) } \frac{23}{2^{3} 5^{2}}\\\\
&\frac{23}{2^{3} 5^{2}}=\frac{23}{2^{3} \times 5^{2}} \times \frac{5}{5}\\\\
&=\frac{23 \times 5}{(2 \times 5)^{3}}=\frac{115}{10^{3}}=0.115
\end{aligned}
\end{equation}
$

$
\begin{equation}
\begin{aligned}
&\text { (vii) } \frac{129}{2^{2}5^{7}7^{5}}\\\\
\end{aligned}
\end{equation}
$

Decimal expansion is non-terminating and repeating.

$
\begin{equation}
\begin{aligned}
&\text { (viii) } \frac{6}{15}\\\\
&\frac{6}{15}=\frac{2 \times 3}{3 \times 5}=\frac{2}{5} \times \frac{2}{2}\\\\
&=\frac{2 \times 2}{2 \times 5}=\frac{4}{10}=0.4
\end{aligned}
\end{equation}
$

$
\begin{equation}
\begin{aligned}
&\text { (ix) } \frac{35}{50}\\\\
&\frac{35}{50}=\frac{5 \times 7}{2 \times 5 \times 5}=\frac{7}{2 \times 5}\\\\
&=\frac{7}{10}=0.7
\end{aligned}
\end{equation}
$

$
\begin{equation}
\begin{aligned}
&\text { (x) } \frac{77}{210}\\\\
\end{aligned}
\end{equation}
$

Decimal expansion is non-terminating and repeating.


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, p/q what can you say about the prime factors of q?

(i) 43.123456789
(ii) 0.120120012000120000. . .
(iii) 43.123456789

Answer :

(i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form $\frac{p}{q}$ 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. . .

Since, it has non-terminating and non- repeating decimal expansion, it is an irrational number.

(iii) $43 . \overline{123456789}$

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form $\frac{p}{q}$ 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 .

Filed Under: Chapter 1 Real Numbers, Class 10, Maths, NCERT Solutions

About Mrs Shilpi Nagpal

Author of this website, Mrs Shilpi Nagpal is MSc (Hons, Chemistry) and BSc (Hons, Chemistry) from Delhi University, B.Ed (I. P. University) and has many years of experience in teaching. She has started this educational website with the mindset of spreading Free Education to everyone.

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