Question

Without performing long division, determine which of the following rational numbers has a terminating decimal expansion: 7/15, -12/840, 3/80, 15/220. Also write its decimal expansion.

Answer 1

Answer:

Answer: 0.0375 terminating decimal expansion (3/80).

Answer 2

For teminating decimal expansion, the denominator should be in the form $2^{m} 5^{n}$

1) $\frac{7}{15} = \frac{7}{3^{1}5^{1}  }$, as it does not satisfy the above rule, so it doesn't have a terminating decimal expansion.

2) $\frac{-12}{840} = \frac{-3^{1}2^{2}  }{2^{3}3^{1}5^{1} 7^{1} } = -\frac{1}{2^{1}5^{1} 7^{1}  }$, the denominator has a 7, as yet still does not satisfy the above statement, so it doesn't have a terminating decimal expansion.

3) $\frac{3}{80} = \frac{3}{2^{4} 5^{1}  }$, it satisfies the above statement, so it has a terminating decimal expansion.

4) $\frac{15}{220} = \frac{3^{1} 5^{1} }{2^{2} 5^{1} 11^{1}  }  = \frac{3^{1} }{2^{2} 11^{1} }$, it does not satisfy the above statement, so it does not have a terminating decimal expansion.

Thanks and have a great day!