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) 23/8

p/q is terminating if :
a) p and q are co-prime and
b)q is of the form , where n and m are non-negative integers.

Since 23 and 8 have no common factors, so 23 and 8 are co-prime.
For denominator 8 we have
So, denominator is of the form , where n=3 and m=0.
Thus , 23/8 is a terminating decimal.




ii)125/441

p/q is terminating if :
a) p and q are co-prime and
b)q is of the form , where n and m are non-negative integers.

Since 125 and 441 have no common factors, so 125 and 441 are co-prime.

For denominator 441 we have

So, denominator is not of the form .
Thus 125/441 has a non terminating repeating decimal expansion.



iii)35/50
35/50=7/10
p/q is terminating if :
a) p and q are co-prime and
b)q is of the form , where n and m are non-negative integers.
Since 7 and 10 have no common factors, so 7 and 10 are co-prime.
For denominator 10 we have

So, denominator is of the form , where n=1 and m=1.
Thus , 7/10 that is 35/50 is a terminating decimal.



iv)77/210

77/210=11/30

p/q is terminating if :
a) p and q are co-prime and
b)q is of the form , where n and m are non-negative integers.

Since 11 and 30 have no common factors, so 11 and 30 are co-prime.

For denominator 30 we have

So, denominator is not of the form .
Thus 11/30 that is 77/210 has a non terminating repeating decimal expansion.



v)

p/q is terminating if :
a) p and q are co-prime and
b)q is of the form , where n and m are non-negative integers.

Since 129 and have no common factors, so they are co-prime.

For denominator we have
So, denominator is not of the form .
Thus has a non terminating repeating decimal expansion.



vi)987/10500

987/10500= 47/500

p/q is terminating if :
a) p and q are co-prime and
b)q is of the form , where n and m are non-negative integers.

Since 47 and 500 have no common factors, so they are co-prime.

For denominator we have

So, denominator is of the form , where n=2 and m=3.
Thus , 47/500 that is 987/10500 is a terminating decimal.