Question

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) \(\frac{23}{8}\)

(ii) \(\frac{125}{441}\)

(iii) \(\frac{35}{50}\)

(iv) \(\frac{77}{210}\)

(v) \(\frac{129}{(2^2 \times 5^7 \times 7^{17})}\)

(vi) \(\frac{987}{10500}\)

Answer 1

(i) \(\frac{23}{8}\)

We have, \(\frac{23}{8}\) and here the denominator is 8. 

⇒ 8 = 2³ x 5 

We see that the denominator 8 of \(\frac{23}{8}\) is of the form 2^m x 5ⁿ, where m, n are non-negative integers. 

Hence, \(\frac{23}{8}\) has terminating decimal expansion. And, the decimal expansion of \(\frac{23}{8}\) terminates after three places of decimal.

(ii) \(\frac{125}{441}\)

We have, \(\frac{125}{441}\) and here the denominator is 441. 

⇒ 441 = 3² x 7² 

We see that the denominator 441 of \(\frac{125}{441}\) is not of the form 2^m x 5ⁿ, where m, n are non-negative integers. 

Hence, \(\frac{125}{441}\) has non-terminating repeating decimal expansion.

(iii) \(\frac{35}{50}\)

We have, \(\frac{35}{50}\) and here the denominator is 50. 

⇒ 50 = 2 x 5² 

We see that the denominator 50 of \(\frac{35}{50}\) is of the form 2^m x 5ⁿ, where m, n are non-negative integers. 

Hence, \(\frac{35}{50}\) has terminating decimal expansion. And, the decimal expansion of \(\frac{35}{50}\) terminates after two places of decimal.

(iv) \(\frac{77}{210}\)

We have, \(\frac{77}{210}\) and here the denominator is 210. 

⇒ 210 = 2 x 3 x 5 x 7 

We see that the denominator 210 of \(\frac{77}{210}\) is not of the form 2^m x 5ⁿ, where m, n are non-negative integers. 

Hence, \(\frac{77}{210}\) has non-terminating repeating decimal expansion.

(v) \(\frac{129}{(2^2 \times 5^7 \times 7^{17})}\)

We have, \(\frac{129}{(2^2 \times 5^7 \times 7^{17})}\) 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, n are non-negative integers. 

And hence, \(\frac{129}{(2^2 \times 5^7 \times 7^{17})}\) has non-terminating repeating decimal expansion.

(vi) \(\frac{987}{10500}\)

We have, \(\frac{987}{10500}\) 

But, \(\frac{987}{10500}\) = \(\frac{47}{500}\) (reduced form) 

And now the denominator is 500. 

⇒ 500 = 2² x 5³ 

We see that the denominator 500 of \(\frac{47}{500}\) is of the form 2^m x 5ⁿ, where m, n are non-negative integers. 

Hence, \(\frac{987}{10500}\) has terminating decimal expansion. And, the decimal expansion of \(\frac{987}{10500}\) terminates after three places of decimal.