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Mathematics

Without actually performing the long division, find if 98710500\dfrac{987}{10500} will have terminating or non-terminating repeating decimal expansion. Give reasons for your answer.

Rational Irrational Nos

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Answer

GCD of numerator and denominator is 21. Reducing the number to its lowest form:

98710500=21×4721×500=47500\dfrac{987}{10500} = \dfrac{\cancel{21} \times 47}{\cancel{21} \times 500} \\[0.5em] = \dfrac{47}{500}

Prime factorization of denominator 500:

250022505125525551\begin{array}{l|l} 2 & 500 \ \hline 2 & 250 \ \hline 5 & 125 \ \hline 5 & 25 \ \hline 5 & 5 \ \hline & 1 \end{array}

500 = 2 x 2 x 5 x 5 x 5 = 22 x 53

Denominator is of the form 2m x 5n, where m, n are non-negative integers.

∴ The given number 98710500\dfrac{987}{10500} has a terminating decimal expansion.

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