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) 13/3125 (ii) 17/8 (iii) 23/75 (iv) 6/15 (v) 1258/625 (vi) 77/210

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The given number is in its lowest form. Prime factorization of denominator 3125: 3125 = 5 x 5 x 5 x 5 x 5 x 1 = 5⁵ x 1 = 1 x 5⁵ = 2⁰ x 5⁵ [ 2⁰ = 1] Denominator is of the form 2^m x 5ⁿ, where m, n are non-negative integers. ∴ The given number has a terminating decimal expansion. The given number is in its lowest form. Prime factorization of denominator 8: 8 = 2 x 2 x 2 x 1 = 2³ x 1 = 2³ x 5⁰ [ 5⁰ = 1] Denominator is of the form 2^m x 5ⁿ, where m, n are non-negative integers. ∴ The given number has a terminating decimal expansion. The given number is in its lowest form. Prime factorization of denominator 75: 75 = 3 x 5 x 5 x 1 = 3 x 5² x 1 = 3 x 5² x 2⁰ [ 2⁰ = 1] Denominator has a prime factor 3 other than 2 or 5. ∴ The given number has a non-terminating repeating decimal expansion. Both numerator and denominator contain common factor 3. Reducing the number to its lowest form: The Denominator 5 = 2⁰ x 5¹ Denominator is of the form 2^m x 5ⁿ, where m, n are non-negative integers. ∴ The given number has a terminating decimal expansion. The given number is in its lowest form. Prime factorization of denominator 625: 625 = 5 x 5 x 5 x 5 x 1 = 5⁴ x 1 = 1 x 5⁴ = 2⁰ x 5⁴ [ 2⁰ = 1] Denominator is of the form 2^m x 5ⁿ, where m, n are non-negative integers. ∴ The given number has a terminating decimal expansion. Both numerator and denominator contain common factor 7. Reducing the number to its lowest form: Prime factorization of denominator 30: 30 = 2 x 3 x 5 Denominator has a prime factor 3 other than 2 or 5. ∴ The given number has a non-terminating repeating decimal expansion.

Mathematics

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) 13/3125 (ii) 17/8 (iii) 23/75 (iv) 6/15 (v) 1258/625 (vi) 77/210

Rational Irrational Nos

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