**Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a nonterminating repeating decimal expansion:**

**Solution:**

**(i) 13/3125**

13/3125 = 13/5^{5}

The denominator is of the form 5^{5}.

Hence, the decimal expansion of is 13/3125 terminating.

**(ii) 17/8**

17/8 = 17/2^{3}

The denominator is of the form 2^{3}.

Hence, the decimal expansion of is 17/8 terminating.

**(iii) 64/455**

64/455 = 64/(5 × 7 × 13)

Since the denominator is not in the form 2^{m} × 5^{n}, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

**(iv) 15/1600**

15/1600 = 15/(2^{6} × 5^{2)}

The denominator is of the form 2^{m} × 5^{n}.

Hence, the decimal expansion of 15/1600 is terminating.

**(v) 29/343**

29/343 = 29/7^{3}

Since the denominator is not in the form 2^{m} × 5^{n}, and it contains 7as its factors, its decimal expansion will be non-terminating repeating.

**(v) 23/(2 ^{3} x 5^{2})**

The denominator is of the form 2^{m} × 5^{n}. Hence, the decimal expansion of 23/(2^{3} x 5^{2}) is terminating

**(vi) 129/(2 ^{2}x5^{7}x7^{5})**

Since the denominator is not of the form 2^{m} × 5^{n}, and it also has 7 as its factor, the decimal expansion of 129/(2^{2}x5^{7}x7^{5}) is non-terminating repeating.

**(vii)6/15 = (2 x 3)/(3 x 5)**

The denominator is of the form 5^{n}. Hence, the decimal expansion of 6/15 is terminating.

**(viii) 35/50 = (5 x 7)/(5 x 10) = 7/(2 x 5)**

The denominator is of the form 2^{m} × 5^{n}. Hence, the decimal expansion of 35/50 is terminating.

**(ix) 77/210 = (11 x 7)/(30 x 7) = 11/30 = 11/(3 x 5 x 2)**

Since the denominator is not of the form 2^{m} × 5^{n}, and it also has 3 as its factors, the decimal expansion of 77/210 is non-terminating repeating.

**Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.**

**Solution:**

**(i) 13/3125**

Therefore, 13/3125 = 0.00416

**(ii) 17/8**

Therefore, 17/8 = 2.125

**(iii) 15/1600**

Therefore, 15/1600 = 0.009375

**(iv) 23/200**

Therefore, 23/200 = 0.115

**(v) 2/5**

Therefore, 2/5 = 0.4

**(vi) 35/50**

Therefore, 35/50 = 0.7

**3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form 𝑝/𝑞 , what can you say about the prime factor of q? **

**(i) 43.123456789 **

**(ii) 0.120120012000120000… **

**(iii)**

**Solution:**

(i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form 𝑝/𝑞 and q is of the form 2^{m} × 5^{n} i.e., the prime factors of q will be either 2 or 5 or both.

(ii) 0.120120012000120000 …

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii)

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form 𝑝/𝑞 and q is not of the form 2^{m} × 5^{n} i.e., the prime factors of q will also have a factor other than 2 or 5.

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