Question

Without performing division, state whether the following rational numbers will have a terminating decimal form or a non-terminating, repeating decimal form.

i) \(\frac{13}{3125}\)

ii) \(\frac{11}{12}\)

iii) \(\frac{64}{455}\)

iv) \(\frac{15}{1600}\)

v) \(\frac{29}{343}\)

Answer 1

i) \(\frac{13}{3125}\)

Note: We check whether the denominator is of the form 2ⁿ. 5^m or not? If yes, the rational number can be expressed as a terminating decimal. If not, it can’t be expressed as a terminating decimal.

[!! Denominator is of the form 2^m × 5ⁿ . Hence a terminating decimal.] 

3125 = 5 

∴ \(\frac{13}{3125}\) is a terminating decimal.

ii) \(\frac{11}{12}\)

The denominator 12 is not a factor of 11. Moreover 12 = 2² × 3.

[!! Denominator is not of the form 2^m × 5ⁿ .] 

∴ \(\frac{11}{12}\) is a non terminating, repeating decimal.

iii) \(\frac{64}{455}\)

[!! Denominator is not of the form 2^m × 5ⁿ . Hence a non terminating decimal.] 

∴ 455 = 5 × 7 × 13 

Hence \(\frac{64}{455}\) is a non terminating, repeating decimal

iv) \(\frac{15}{1600}\)

∴ 1600 = 2⁶ × 5² [∵ The denominator is of the form 2 . 5 ] 

Hence \(\frac{15}{1600}\) is a terminating decimal.

v) \(\frac{29}{343}\)

343 = 7³ [Not of the form 2ⁿ. 5^m] ∴ \(\frac{29}{343}\) is a non terminating, repeating decimal.