Ex-1.6 Real Numbers, Class 10, Maths RD Sharma Solutions | Extra Documents, Videos & Tests for Class 10 PDF Download

Q.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) 23/8

(ii) 125/441

(iii) 35/50

(iv) 77/210

(v) 

Sol: (i) The given number is 23/8

Here, 8 = 2³ and 2 is not a factor of 23.

So, the given number is in its simplest form.

Now, 8 = 2³ is of the form 2^m x 5ⁿ, where m = 3 and n = 0.

So, the given number has a terminating decimal expansion.

(ii) The given number is 125/441

Here, 441 = 3² x 7² and none of 3 and 7 is a factor of 125.

So, the given number is in its simplest form.

Now, 441 = 3² x 7² is not of the form 2^m x 5ⁿ

So, the given number has a non-terminating repeating decimal expansion.

(iii) The given number is 35/50 and HCF(35, 50) = 5.

Here, 7/10 is in its simplest form.

Now, 10 = 2 x 5 is of the form 2^m x 5ⁿ, where in = 1 and n = 1.

So, the given number has a terminating decimal expansion.

(iv) The given number is 77/210 and HCF(77, 210) = 7.

∴ 77:7/210:7 = 11/30

Here, 11/30 is in its simplest form. 30

Now, 30 = 2 x 3 x 5 is not of the form 2^m x 5ⁿ.

So, the given number has a non-terminating repeating decimal expansion.

(v) The given number is 

Clearly, none of 2, 5 and 7 is a factor of 129.

So, the given number is in its simplest form.

Q.2: Write down the decimal expansions of the following rational numbers by writing their denominators in the form of 2^m x 5ⁿ, where m, and n, are the non- negative integers.

(i) 3/8

(ii) 13/125

(iii) 7/80

(iv) 14588/625

(v) 

Sol: (i) The given number is 3/8

Clearly, 8 = 2³ is of the form 2^m x 5ⁿ, where m = 3 and n = 0.

So, the given number has terminating decimal expansion.

(ii) The given number is 13/125.

Clearly, 125 = 5³ is of the form 2m x 5", where m = 0 and n = 3.

So, the given number has terminating decimal expansion.

(iii) The given number is 7/80.

Clearly, 80 = 2⁴ x 5 is of the form 2^m X 5ⁿ, where m = 4 and n = 1.

So, the given number has terminating decimal expansion.

(iv) The given number is 14588/625

Clearly, 625 = 5⁴ is of the form 2^m x 5ⁿ, where m = 0 and n = 4.

So, the given number has terminating decimal expansion.

(v) The given number is 

Clearly, 2² x 5⁷ is of the form 2^m x 5ⁿ, where in = 2 and n = 7.

So, the given number has terminating decimal expansion.

 = 0.0004182

Q.4: what can you say about the prime factorization of the denominators of the following rational:

(i) 43.123456789

(ii) 

(iii) 

(iv) 0.120120012000120000

Sol: (i) Since 43.123456789 has terminating decimal expansion. So, its denominator is of the form 2^m x 5ⁿ, where m, n are non-negative integers.

(ii) Since  has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

(iii) Since  has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

(iv) Since 0.120120012000120000 … has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.