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

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I want NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.4 to complete my hometask. Provide Class 10 Mathematics NCERT Solutions to solve difficult questions.

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Here you will find CBSE NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 PDF through which you can complete you hometask on time. These NCERT Solutions for Class 10 Maths can be used to solve difficult problems in the examinations.

 Book Name Class 10 Mathematics NCERT Textbook Chapter Chapter 1 Real Numbers Exercise Ex 1.4

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) 17/18
(iii) 64/455
(iv) 15/1600
(v) 29/343
(vi) 23/2353
(vii) 129/22577
(viii) 16/15
(ix) 35/50
(x)  77/ 210

Solution

(i) 13/3125
3125 =  5 × 5 × 5 × 5 × 5 = 55
=  20 × 55,    |
20 = 1
which is of the form (2n · 5m)

13/3125 is a terminating decimal.
i.e., 13/3125 will have a terminating decimal expansion.

(ii) 17/8
8 = 2 × 2 × 2 = 23 = 1 × 23
= 50 × 23, which is of the form 5m · 2n

17/8 will have a terminating decimal expansion.

(iii) 64/455
455 = 5 × 7 × 13, which is not of the form 2n · 5m
64/455 will have a non-terminating repeating decimal expansion.

(iv) 15/1600
1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
= 26 × 52, which is of the form 2n · 5m

15/1600 will have a terminating decimal expansion.

(v) 29/343
343 = 7 × 7 × 7 = 73, which is not of the form 2n · 5m.
29/343 will have a non-terminating repeating decimal expansion.

(vi)  23/2352
Let p/q = 23/2352
i.e., q =  23 · 52, which is of the form 2n · 5m.

23/2352 will have a terminating decimal expansion.

(vii) 129/225775
Let p/q = 129/225775
i.e., q = 22 · 57 · 75, which is not of the form 2n · 5m.

129/225775   will have a non-terminating repeating decimal expansion.

(viii) 6/15

∴  6/15 will have a terminating decimal expansion.

(ix) 35/50

∵ 50 = 2 × 5 × 5 = 21 × 52, which is of the form 2n · 5m.

∴ 35/50 will have a terminating decimal expansion.

(x) 77/210

∵  210 =  2 × 3 × 5 × 7 = 21 · 31 · 51 · 71,

which is not of the form of 2n · 5m.

∴  77/ 210 will have a non-terminating repeating decimal expansion.

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

(i)  13/ 3125
(ii) 17/18
(iii) 64/455
(iv) 15/1600
(v) 29/343

(viii) 6/15
(ix) 35/50
(x) 77/ 210

Solution

(i) 13/3125

(ii) 17/18

(iii) 64/455 represents non-terminating repeating decimal expansion.

(iv) 15/1600

(v) 29/343, represents a non-terminating repeating decimal expansion.

(viii) 6/15

6/15 = 2/5

= 2/5×2/2

= 4/10   = 0.4

(ix) 35/50

35/50 = 0.7

(x) 77/ 210, represents a non-terminating repeating decimal expansion.

Given Class 10 Maths Chapter 1 Real numbers NCERT Solutions can be used to score good marks in the examinations and prepare your own answers.

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

(i) 43.123456789

(ii) 0.120120012000120000...

Solution

(i) 43.123456789

∵ The given decimal expansion terminates.

∴ It is a rational of the form p/q

⇒ p/q = 43.123456789

Here, p  = 43.123456789  and
q = 29 × 59

Prime factors of q are 29 and 59.

(ii) 0.120120012000120000 .....
The given decimal expansion is neither terminating nor non-terminating repeating,
It is not a rational number.

The given decimal expansion is non-terminating repeating.
It is a rational number.
Let p/q = x = 43.123456789 ......(1)
Multiplying both sides by 1000000000, we have
1000000000 x = 43.123456789  ...(2)
Subtracting (1) from (2), we have:
(1000000000 x) – x = (43.123456789) – 43.123456789

999999999 x  = 43.123456746

Here, p =  4791495194
and  q  = 111111111, which is not of form 2n· 5m
i.e., the prime factors of q are not of form 2n· 5m.