**NCERT Solutions for Class 10 Maths**

Chapter 1 Real Numbers Exercise 1.4

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/(2 ^{3}5^{2}) (vii) 129/(2^{2}5^{7}7^{5}) (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} ×5^{n} 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 = 5^{5}

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 = 2 ^{3}**

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} × 5^{n}, thus 64/455 has a non-terminating decimal expansion.

**(iv) 15/ 1600**

Factorizing the denominator, we get,

1600 = 2^{6}5^{2}

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

**(v) 29/343**

Factorizing the denominator, we get,

343 = 7×7×7 = 7^{3} Since, the denominator is not in the form of 2^{m} × 5^{n} thus 29/343 has a non-terminating decimal expansion.

**(vi)23/(2 ^{3}5^{2})**

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

Hence, 23/ (2^{3}5^{2}) has a terminating decimal expansion.

**(vii) 129/(2 ^{2}5^{7}7^{5})**

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

Hence, 129/ (2^{2}5^{7}7^{5}) 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} × 5^{n} 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} × 5^{n} .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 .

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