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• RD Sharma Class 10 Solutions

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

### (i)23/8

Solution:

Denominator = 8

⇒ 8 = 23 x 5

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

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

### (i) 3/8

Solution:

Rational number is 3/8.

We can see that 8 = 23 is of the form 2m 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 2m 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 2m 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 2m x 5n where m = 0 and n = 4.

So, the given number has terminating decimal expansion.

∴ 14588/ 625 = (14588 x 24)/ (24 x 54 ) = 233408/10

= 233408/10000 = 23.3408

### (v) 129/(22 x 57)

Solution:

We can see, 22 x 57 is of the form 2m 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 2m 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.

### (i) 43.123456789

Solution:

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

### (ii) [Tex] [/Tex]

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}*** QuickLaTeX cannot compile formula: *** Error message: Error: Nothing to show, formula is empty 

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.

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