**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 = 2

^{3}x 5The denominator 8 of the fraction 23/8 is of the form 2

^{m}x 5^{n}, 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 = 3

^{2}x 7^{2}The denominator 441 of 125/441 is not of the form 2

^{m}x 5^{n}, 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 5

^{2}The denominator 50 of the fraction 35/50 is of the form 2

^{m}x 5^{n}, 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 5^{n}, where m, n are non-negative integers.Therefore, 77/210 has non-terminating repeating decimal expansion.

**(v) 129/(2**^{2} x 5^{7} x 7^{17})

^{2}x 5

^{7}x 7

^{17})

**Solution:**

The denominator = 2

^{2}x 5^{7}x 7^{17}.The denominator of the fraction cannot be expressed in the form 2

^{m}x 5^{n}, 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 = 2

^{2}x 5^{3}The denominator 500 of 47/500 can be expressed in the form 2

^{m }x 5^{n}, 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 = 2

^{3}is of the form 2^{m}x 5^{n}, where m = 3 and n = 0.Therefore, the given number has terminating decimal expansion.

**(ii) 13/125**

**Solution:**

We can see that 125 = 5

^{3 }is of the form 2^{m}x 5^{n}, where m = 0 and n = 3.Therefore, the given rational number has terminating decimal expansion.

∴ 13/ 125 = (13 x 2

^{3})/(125 x 2^{3})= 104/1000

= 0.104

**(iii) 7/80**

**Solution:**

We can see, 80 = 2

^{4}x 5 is of the form 2^{m}x 5^{n}, where m = 4 and n = 1.Therefore, the given number has terminating decimal expansion.

∴ 7/ 80 = (7 x 5

^{3})/ (2^{4}x 5 x 5^{3})= 7 x 125 / (5 x 2)

^{4}= 875/10000

= 0.0875

**(iv) 14588/625**

**Solution:**

We can see, 625 = 5

^{4}is of the form 2^{m}x 5^{n}where m = 0 and n = 4.So, the given number has terminating decimal expansion.

∴ 14588/ 625 = (14588 x 2

^{4})/ (2^{4}x 5^{4}) = 233408/10^{4 }= 233408/10000 = 23.3408

**(v) 129/(2**^{2 }x 5^{7})

^{2 }x 5

^{7})

**Solution:**

We can see, 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 terminating decimal expansion.

∴ 129/ 2

^{2}x 5^{7}= 129 x 2^{5}/ 2^{2}x 5^{7}x 2^{5}= 4182/10

^{7}=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 = 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 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 5^{n}, 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.