Write actual division, find which of the following rational numbers are terminating decimals.
Question:

Write 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) $\frac{16}{125}$

Solution:

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

Denominator of $\frac{13}{80}$ is 80 .

And,

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

Therefore, 80 has no other factors than 2 and 5.

Thus, $\frac{13}{80}$ is a terminating decimal.

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

Denominator of $\frac{7}{24}$ is 24 .

And,

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

So, 24 has a prime factor 3, which is other than 2 and 5.

Thus, $\frac{7}{24}$ is not a terminating decimal.

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

Denominator of $\frac{5}{12}$ is 12 .

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

So, 12 has a prime factor 3, which is other than 2 and 5.

Thus, $\frac{5}{12}$ is not a terminating decimal.

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

Denominator of $\frac{31}{375}$ is 375

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

So, the prime factors of 375 are 5 and 3.

Thus, $\frac{31}{375}$ is not a terminating decimal.

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

Denominator of $\frac{16}{125}$ is 125 .

And,

$125=5^{3}$

Therefore, 125 has no other factors than 2 and $5 .$

Thus, $\frac{16}{125}$ is a terminating decimal