**Go To: ****Exercise 1.1**** Exercise 1.2 Exercise 1.3 **

**NCERT Solutions Maths Class 10 Exercise 1.4**

**Q1. 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**

**Solution:
** **(i)**
13/3125

Factorizing the denominator of the given rational number, we get

3125 =5 × 5 × 5 × 5 × 5 = 5^{5}

Since the denominator is in the form of 5* ^{m}*,
therefore it is terminating.

**(ii)** 17/8

Factorizing the denominator of the given rational number, we get

8 = 2 × 2 × 2 = 2^{3}

Since the denominator is in the form of 2* ^{m}*,
therefore it is terminating.

**(iii)** 64/455

Factorizing the denominator of the given rational number, we get

455 = 5 × 7 × 13

Since there are 7 and 13 also in the denominator, so the denominator is not in
the form of 2* ^{m}* × 5

*.*

^{n}Hence,
the given rational number is non-terminating.

**(iv)** 15/1600

Factorizing the denominator of the given rational number, we get

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

Here, the denominator is in the form of 2* ^{m}*
× 5

*.*

^{n}Hence, the given rational number is terminating.

**(v)** 29/343

Factorizing the denominator of the given rational number, we get

343 = 7 × 7 × 7 = 7^{3}

Since 7 is also there in the denominator, so the denominator is not in the form
of 2* ^{m}* × 5

^{n}Hence, it is non-terminating.

**(vi)** 23/(2^{3} × 5^{2})

In this rational number, the denominator is in the form of 2* ^{m}* × 5

*.*

^{n}Hence, it is terminating.

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

Since 7 is also there in the denominator, so the denominator is not in the form
of 2* ^{m}* × 5

^{n}Hence, it is non-terminating.

**(viii)** 6/15

First convert the given rational number in its standard form.

Dividing the numerator and the denominator both by 3, we get 2/5.

Since the denominator is in the form of 5* ^{m}*,
therefore it is terminating.

**(ix)** 35/50

First
convert the given rational number in its standard form.

Dividing the numerator and the denominator both by 5, we get 7/10

Factorizing the denominator of the given rational number, we get

10 = 2 × 5

Since the denominator is in the form of 2* ^{m}*
× 5

*, therefore it is terminating.*

^{n}**(x)** 77/210

First convert the given rational number in its standard form.

Dividing the numerator and the denominator both by 7, we get 11/30.

Factorizing the denominator of the given rational number, we get

30 = 2 × 3 × 5

There is a 3 also in the denominator, so the denominator is not in form of 2* ^{m}* × 5

*.*

^{n}Hence, it is non-terminating.

**Q2.
Write down the decimal expansions of those rational numbers in Question 1 above
which have terminating decimal expansions.**

**Solution: **In question 1, (i) 13/3125, (ii) 17/8, (iv) 15/1600, (vi)
23/2352, (viii) 6/15 and (ix) 35/50 are terminating decimals. There decimal
expansions are as follow:

**(i)** 13/3125 = 13/5^{5} = 13 × 2^{5}/5^{5
}× 2^{5} = 416/10^{5} = 0.00416

**(ii)** 17/8 = 17/2^{3} = 17 × 5^{3}/2^{3}
× 5^{3} = 17 × 5^{3}/10^{3} = 2125/10^{3} =
2.125

**(iv)** 15/1600 = 15/2^{4 }× 10^{2} =
15 × 5^{4}/2^{4} × 5^{4} × 10^{2} = 9375/10^{6}
= 0.009375

**(vi)** 23/2^{3} × 5^{2} = 23 × 5^{3}
× 2^{2}/2^{3} × 5^{3} × 5^{2} × 2^{2} =
11500/10^{5} = 0.115

**(viii)** 6/15 = 2/5 = 2 × 2/5 × 2 = 4/10 = 0.4

**(ix)** 35/50 = 7/10 = 0.7

**Q3. 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) **

** **

**Solution:
****(i)** **43.123456789 **

Since
this number has a terminating decimal expansion, it is a rational number of the
form *p*/*q*, and *q* is of the form 2* ^{m}* × 5

*.*

^{n}**(ii)** **0.120120012000120000… **

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

**(iii) **

Since
the decimal expansion is non-terminating recurring, the given number is a
rational number of the form *p*/*q*, and *q* is not of the form 2* ^{m}*
× 5

*.*

^{n}