# Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion

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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) $\frac{987}{10500}$

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(i) $\frac{23}{8}$

We have, $\frac{23}{8}$ and here the denominator is 8.

⇒ 8 = 23 x 5

We see that the denominator 8 of $\frac{23}{8}$ is of the form 2m x 5n, where m, n are non-negative integers.

Hence, $\frac{23}{8}$ has terminating decimal expansion. And, the decimal expansion of $\frac{23}{8}$ terminates after three places of decimal.

(ii) $\frac{125}{441}$

We have, $\frac{125}{441}$ and here the denominator is 441.

⇒ 441 = 32 x 72

We see that the denominator 441 of $\frac{125}{441}$ is not of the form 2m x 5n, where m, n are non-negative integers.

Hence, $\frac{125}{441}$ has non-terminating repeating decimal expansion.

(iii) $\frac{35}{50}$

We have, $\frac{35}{50}$ and here the denominator is 50.

⇒ 50 = 2 x 52

We see that the denominator 50 of $\frac{35}{50}$ is of the form 2m x 5n, where m, n are non-negative integers.

Hence, $\frac{35}{50}$ has terminating decimal expansion. And, the decimal expansion of $\frac{35}{50}$ terminates after two places of decimal.

(iv) $\frac{77}{210}$

We have, $\frac{77}{210}$ and here the denominator is 210.

⇒ 210 = 2 x 3 x 5 x 7

We see that the denominator 210 of $\frac{77}{210}$ is not of the form 2m x 5n, where m, n are non-negative integers.

Hence, $\frac{77}{210}$ has non-terminating repeating decimal expansion.

(v) $\frac{129}{(2^2 \times 5^7 \times 7^{17})}$

We have, $\frac{129}{(2^2 \times 5^7 \times 7^{17})}$ and here the denominator is 22 x 57 x 717

Clearly,

We see that the denominator is not of the form 2m x 5n, where m, n are non-negative integers.

And hence, $\frac{129}{(2^2 \times 5^7 \times 7^{17})}$ has non-terminating repeating decimal expansion.

(vi) $\frac{987}{10500}$

We have, $\frac{987}{10500}$

But, $\frac{987}{10500}$ = $\frac{47}{500}$ (reduced form)

And now the denominator is 500.

⇒ 500 = 22 x 53

We see that the denominator 500 of $\frac{47}{500}$ is of the form 2m x 5n, where m, n are non-negative integers.

Hence, $\frac{987}{10500}$ has terminating decimal expansion. And, the decimal expansion of $\frac{987}{10500}$ terminates after three places of decimal.