Question

Without actual division, find which of the following rational numbers are terminating decimals. (i) $\frac{13}{80}$ (ii) $\frac{7}{24}$ (iii) $\frac{5}{12}$ (iv)$\frac{31}{375}$ (v) $\frac{16}{125}$

Answer 1

For terminating decimals, factors of denominator should be in the form of $2^m\times 5^n$.

(i) $\frac{13}{80}$

Here, denominator is $80$

$80=2\times 2\times 2\times 2\times 5=2^4\times 5$

hence denominator doesn't have any factor other than $2$ or $5.$

So, the number is terminating.

(ii) $\frac{7}{24}$

Denominator is $24.$

$24=2\times 2\times 2\times 3=2^3\times 3$

Hence, denominator has factor other than $2$ or $5.$

So, the number is non-terminating.

(iii) $\frac{5}{12}$

Denominator is $12.$

$12=2\times 2\times 3=2^2\times 3$

Hence, denominator has factor other than $2$ or $5.$

So, the number is non-terminating.

(iv) $\frac{31}{375}$

Denominator is $375.$

$375=5\times 5\times 5\times 3=5^3\times 3$

Hence, denominator has factor other than $2$ or $5.$

So, the number is non-terminating.

(v) $\frac{16}{125}$

denominator is $125.$

$125=5\times 5\times 5=5^3$

Hence, the denominator doesn't have any factor other than $2$ or $5.$

So, the number is terminating.