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

In the earlier classes, students would have learnt about the decimal expansion of a rational number that is either terminating or non-terminating repeating. In this exercise, the problems focus on helping students understand exactly when the decimal expansion of a rational number is terminating and when it is non-terminating repeating. RD Sharma Solutions Class 10 is the best study resource available to learn the correct methods to solve such problems and to secure high marks in the board exam. Also, students can download the RD Sharma Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.6 PDF, which is provided below.

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

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Access Answers to RD Sharma Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.6

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 = 2³ x 5

We see that the denominator 8 of 23/8 is of the form 2^m x 5ⁿ, 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 = 3² x 7²

We see that the denominator 441 of 125/441 is not of the form 2^m x 5ⁿ, 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 5²

We see that the denominator 50 of 35/50 is of the form 2^m x 5ⁿ, 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 5ⁿ, where m and n are non-negative integers.

Hence, 77/210 has a non-terminating repeating decimal expansion.

(v) 129/(2² x 5⁷ x 7¹⁷)

Solution:

We have 129/(2² x 5⁷ x 7¹⁷), and here the denominator is 2² x 5⁷ x 7¹⁷.

Clearly,

We see that the denominator is not of the form 2^m x 5ⁿ, 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 = 2² x 5³

We see that the denominator 500 of 47/500 is of the form 2^m x 5ⁿ, 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 5ⁿ, 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 = 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) 13/125

Solution:

The given rational number is 13/125.

It’s seen that, 125 = 5³ 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 2³)/ (125 x 2³) = 104/1000 = 0.104

(iii) 7/80

Solution:

The given rational number is 7/80.

It’s seen that, 80 = 2⁴ x 5 is of the form 2^m x 5ⁿ, 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 = 5⁴ is of the form 2^m x 5ⁿ, where m = 0 and n = 4.

So, the given number has a terminating decimal expansion.

(v) 129/(2² x 5⁷)

Solution:

The given number is 129/(2² x 5⁷).

It’s seen that, 2² x 5⁷ is of the form 2^m x 5ⁿ, 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 × 5ⁿ, 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 = 2³ x 5⁴

It’s seen that, 2³ x 5⁴ is of the form 2^m x 5ⁿ, 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 5ⁿ, 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 5ⁿ, where m and n are non-negative integers.

Further,

327.7081 can be expressed as 3277081/10000 = p/q

⇒ q = 10000 = 2³ x 5³

Hence, the prime factors of q have only factors of 2 and 5.