RD Sharma Solutions Class 10 Maths Chapter 1 Exercise 1.6 Real Numbers

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

Solution:

We have 23/8, and here the denominator is 8. ⇒ 8 = 2 ³ x 5 We see that the denominator 8 of 23/8 is of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers. Hence, 23/8 has a terminating decimal expansion. And the decimal expansion of 23/8 terminates after three places of decimal.

(ii) 125/441

Solution:

We have 125/441, and here the denominator is 441. ⇒ 441 = 3 ² x 7 ² We see that the denominator 441 of 125/441 is not of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers. Hence, 125/441 has a non-terminating repeating decimal expansion.

(iii) 35/50

Solution:

We have 35/50, and here the denominator is 50. ⇒ 50 = 2 x 5 ² We see that the denominator 50 of 35/50 is of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers. Hence, 35/50 has a terminating decimal expansion. And the decimal expansion of 35/50 terminates after two places of decimal.

(iv) 77/210

Solution:

We have 77/210, and here the denominator is 210. ⇒ 210 = 2 x 3 x 5 x 7 We see that the denominator 210 of 77/210 is not of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers. Hence, 77/210 has a non-terminating repeating decimal expansion.

(v) 129/(2 ² x 5 ⁷ x 7 ¹⁷ )

Solution:

We have 129/(2 ² x 5 ⁷ x 7 ¹⁷ ), and here the denominator is 2 ² x 5 ⁷ x 7 ¹⁷ . Clearly, We see that the denominator is not of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers. And hence, 125/441 has a non-terminating repeating decimal expansion.

(vi) 987/10500

Solution:

We have 987/10500 But, 987/10500 = 47/500 (reduced form) And now the denominator is 500. ⇒ 500 = 2 ² x 5 ³ We see that the denominator 500 of 47/500 is of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers. Hence, 987/10500 has a terminating decimal expansion. And the decimal expansion of 987/10500 terminates after three places of decimal.

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

(i) 3/8

Solution:

The given rational number is 3/8 It’s seen that, 8 = 2 ³ is of the form 2 ^m x 5 ⁿ , where m = 3 and n = 0. So, the given number has a terminating decimal expansion. ∴

(ii) 13/125

Solution:

The given rational number is 13/125. It’s seen that, 125 = 5 ³ is of the form 2 ^m x 5″, where m = 0 and n = 3. So, the given number has a terminating decimal expansion. ∴ 13/ 125 = (13 x 2 ³ )/ (125 x 2 ³ ) = 104/1000 = 0.104

(iii) 7/80

Solution:

The given rational number is 7/80.

It’s seen that, 80 = 2 ⁴ x 5 is of the form 2 ^m x 5 ⁿ , where m = 4 and n = 1. So, the given number has a terminating decimal expansion.

(iv) 14588/625

Solution:

The given rational number is 14588/625. It’s seen that, 625 = 5 ⁴ is of the form 2 ^m x 5 ⁿ , where m = 0 and n = 4. So, the given number has a terminating decimal expansion.

(v) 129/(2 ² x 5 ⁷ )

Solution:

The given number is 129/(2 ² x 5 ⁷ ). It’s seen that, 2 ² x 5 ⁷ is of the form 2 ^m x 5 ⁿ , where m = 2 and n = 7. So, the given number has a terminating decimal expansion.

3. Write the denominator of the rational number 257/5000 in the form 2 ^m × 5 ⁿ , where m and n are non-negative integers. Hence, write the decimal expansion without actual division.

Solution:

The denominator of the given rational number is 5000. ⇒ 5000 = 2 ³ x 5 ⁴ It’s seen that, 2 ³ x 5 ⁴ is of the form 2 ^m x 5 ⁿ , where m = 3 and n = 4. ∴ 257/5000 = (257 x 2)/(5000 x 2) = 514/10000 = 0.0514 is its decimal expansion.

4. What can you say about the prime factorisation of the denominators of the following rational numbers:

(i) 43.123456789

Solution:

Since 43.123456789 has a terminating decimal expansion, its denominator is of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers.

(ii)

Solution:

Since the given rational has a non-terminating decimal expansion, its denominator has factors other than 2 or 5.

(iii)

Solution:

Since the given rational number has a non-terminating decimal expansion, its denominator has factors other than 2 or 5.

(iv) 0.120120012000120000….

Solution:

Since 0.120120012000120000…. has a non-terminating decimal expansion, its denominator has factors other than 2 or 5.

5. A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q when this number is expressed in the form p/q? Give reasons.

Solution:

Since 327.7081 has a terminating decimal expansion, its denominator should be of the form 2 ^m x 5 ⁿ , where m and n are non-negative integers. Further, 327.7081 can be expressed as 3277081/10000 = p/q ⇒ q = 10000 = 2 ³ x 5 ³ Hence, the prime factors of q have only factors of 2 and 5.