Question

Without actually performing the long division, state whether 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) $\frac{35}{50}$ [NCERT]
(iv) $\frac{77}{210}$ [NCERT]
(v) $\frac{129}{2^2\times 5^7\times 7^{17}}$

(vi) $\frac{987}{10500}$

Answer 1

(i) The given number is $\frac{23}{8}$.

Here, $8=2^3$ and 2 is not a factor of 23.

So, the given number is in its simplest form.

Now, $8=2^3$ is of the form $2^m\times 5^n$, where m = 3 and n = 0.

So, the given number has a terminating decimal expansion.

(ii) The given number is $\frac{125}{441}$.

Here, $441=3^2\times 7^2$ and none of 3 and 7 is a factor of 125.

So, the given number is in its simplest form.

Now, $441=3^2\times 7^2$ is not of the form $2^m\times 5^n$.

So, the given number has a non-terminating repeating decimal expansion.

(iii) The given number is $\frac{35}{50}$ and HCF(35, 50) = 5.

∴ $\frac{35}{50}=\frac{35÷5}{50÷5}=\frac{7}{10}$

Here, $\frac{7}{10}$ is in its simplest form.

Now, $10=2\times 5$ is of the form $2^m\times 5^n$, where m = 1 and n = 1.

So, the given number has a terminating decimal expansion.

(iv) The given number is $\frac{77}{210}$ and HCF(77, 210) = 7.

∴ $\frac{77}{210}=\frac{77÷7}{210÷7}=\frac{11}{30}$

Here, $\frac{11}{30}$ is in its simplest form.

Now, $30=2\times 3\times 5$ is not of the form $2^m\times 5^n$.

So, the given number has a non-terminating repeating decimal expansion.

(v) The given number is $\frac{129}{2^2\times 5^7\times 7^{17}}$.

Clearly, none of 2, 5 and 7 is a factor of 129.

So, the given number is in its simplest form.

Now, $2^2\times 5^7\times 7^{17}$is not of the form $2^m\times 5^n$.

So, the given number has a non-terminating repeating decimal expansion.

(vi) The given number is $\frac{987}{10500}$.

$\frac{987}{10500}=\frac{987÷21}{10500÷21}=\frac{47}{500}$

Now, 500 = 2² × 5³

The denominator can be written in the form of 2^m × 5ⁿ.

So, the given number has a terminating decimal expansion.

$\frac{987}{10500}=\frac{47}{500}=\frac{47}{5^3\times 2^2}\times \frac{2}{2}=\frac{94}{5^3\times 2^3}=\frac{94}{1000}=0.094$