Exercise 1.4

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 2^m ×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 2^m × 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 2^m × 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 2^m × 5ⁿ thus 29/343 has a non-terminating decimal expansion.

(vi)23/(2³5²)

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

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

(vii) 129/(2²5⁷7⁵)

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

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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.

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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 .