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

Chapter 1 of RD Sharma's Class 10 Mathematics book focuses on the Real Numbers. This chapter is foundational laying the groundwork for understanding different types of numbers and their properties. It covers the concept of the real numbers, their classification, and key operations performed with them. Exercise 1.6 in this chapter offers practice problems to solidify students' grasp of the topic.

Real Numbers

The Real numbers encompass all the numbers that can be found on the number line. They include rational numbers and irrational numbers that cannot be expressed as fractions but have non-repeating, non-terminating decimal expansions. Real numbers are crucial in mathematics as they represent quantities that can be measured or counted making them fundamental to the various mathematical operations and real-world applications.

Question 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:

Denominator = 8 

⇒ 8 = 2³ x 5

The denominator 8 of the fraction 23/8 is of the form 2^m x 5ⁿ, where m, n are non-negative integers.

Therefore, 23/8 has terminating decimal expansion which terminates after three places of decimal.

(ii) 125/441

Solution:

Denominator = 441.

⇒ 441 = 3² x 7²

The denominator 441 of 125/441 is not of the form 2^m x 5ⁿ, where m, n are non-negative integers.

Therefore, the fraction 125/441 has a non-terminating repeating decimal expansion.

(iii) 35/50

Solution:

Denominator = 50.

⇒ 50 = 2 x 5²

The denominator 50 of the fraction 35/50 is of the form 2^m x 5ⁿ, where m, n are non-negative integers.

Therefore, 35/50 has a terminating decimal expansion which terminates after two places of decimal.

(iv) 77/210

Solution:

Denominator = 210.

⇒ 210 = 2 x 3 x 5 x 7

The denominator 210 of the fraction 77/210 is not of the form 2^m x 5ⁿ, where m, n are non-negative integers.

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

(v) 129/(2² x 5⁷ x 7¹⁷)

Solution:

The denominator = 2² x 5⁷ x 7¹⁷.

The denominator of the fraction cannot be expressed in the form 2^m x 5ⁿ, where m, n are non-negative integers.

Therefore, 125/441 has a non-terminating repeating decimal expansion.

(vi) 987/10500

Solution:

On reducing the above fraction, we have,

987/10500 = 47/500 (reduced form)

Denominator = 500.

⇒ 500 = 2² x 5³

The denominator 500 of 47/500 can be expressed in the form 2^m x 5ⁿ, where m, n are non-negative integers.

Therefore, 987/10500 has a terminating decimal expansion which terminates after three places of decimal.

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

(i) 3/8

Solution:

Rational number is 3/8.

We can see that 8 = 2³ is of the form 2^m x 5ⁿ, where m = 3 and n = 0.

Therefore, the given number has terminating decimal expansion.

(ii) 13/125

Solution:

We can see that  125 = 5³ is of the form 2^m x 5ⁿ, where m = 0 and n = 3.

Therefore, the given rational number has terminating decimal expansion.

∴ 13/ 125 = (13 x 2³)/(125 x 2³) 

= 104/1000 

= 0.104

(iii) 7/80

Solution:

We can see, 80 = 2⁴ x 5 is of the form 2^m x 5ⁿ, where m = 4 and n = 1.

Therefore, the given number has terminating decimal expansion.

∴ 7/ 80 = (7 x 5³)/ (2⁴ x 5  x 5³)

 = 7 x  125 / (5 x  2)⁴ 

= 875/10000 

= 0.0875

(iv) 14588/625

Solution:

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

So, the given number has terminating decimal expansion.

∴ 14588/ 625 = (14588 x 2⁴)/ (2⁴ x 5⁴ ) = 233408/10⁴ 

= 233408/10000 = 23.3408

(v) 129/(2² x 5⁷)

Solution:

We can see, 2² x 5⁷ is of the form 2^m x 5ⁿ, where m = 2 and n = 7.

So, the given number has terminating decimal expansion.

∴ 129/ 2² x 5⁷ = 129 x 2⁵ / 2² x 5⁷ x 2⁵

= 4182/10⁷

=4182/10000000

=0.0004182

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

Solution:

Denominator = 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 the required decimal expansion.

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

(i) 43.123456789

Solution:

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

(ii) 43.\overline{123456789} 

Solution:

The given rational has a non-terminating decimal expansion. Therefore, the denominator of the number has factors other than the numbers 2 or 5.

(iii) 27.\overline{142857} 

Solution:

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

(iv) 0.120120012000120000….

Solution:

Since 0.120120012000120000…. has a non-terminating decimal expansion. Therefore, the denominator of the number has factors other than 2 or 5.

Read more:

• Real Numbers
• Real-Life Applications of Real Numbers

Summary

Exercise 1.6 of RD Sharma's Class 10 Mathematics textbook focuses on the concept of Euclid's division algorithm and its applications. This section covers finding the HCF (Highest Common Factor) of two or more numbers using the division algorithm, understanding the relationship between HCF and LCM (Least Common Multiple), and solving problems related to divisibility. Students learn to apply Euclid's algorithm systematically to find the HCF of large numbers, use the HCF to simplify fractions, and solve word problems involving HCF and LCM concepts.