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Which of the following rational numbers $\frac{16}{225}, \frac{5}{18}, \frac{2}{21} and \frac{7}{250}$ have a terminating decimal?

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\frac{16}{225}

only

\frac{5}{18}

and

\frac{2}{21}

\frac{2}{21}

only

\frac{16}{225}

and

\frac{7}{250}

answer is D.

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Detailed Solution

We have to find which of the following rational numbers

\frac{16}{225}, \frac{5}{18}, \frac{2}{21} and \frac{7}{250}

have a terminating decimal.
Let

\frac{p}{q}

can be a rational number.
Then, prime factorisation of q is of the form

2^{m} \times 5^{n}

where m and n are non negative integers.
Let us consider

\frac{16}{225}

.
Here,

q = 225

.
225 can be written as

3^{2} \times 5^{2}

\Rightarrow

It has a terminating decimal.
Let us now consider

\frac{5}{18}

.
Here,

q = 18

.
18 can be written as

2 \times 3^{2}

\Rightarrow

It does not have a terminating decimal.
Now let us consider

\frac{2}{21}

.
Here,

q = 21

.
21 can be written as

3 \times 7

\Rightarrow

It does not have a terminating decimal.
Let us now consider

\frac{7}{250}

.
Here,

q = 250

.
250 can be written as

2^{1} \times 5^{3}

\Rightarrow

It has a terminating decimal.
Therefore, the rational numbers

\frac{16}{225}

and

\frac{7}{250}

has a terminating decimal.
Hence, option 4 is correct.

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Which of the following rational numbers 16225, 518, 221 and 7250 have a terminating decimal?

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Detailed Solution

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Which of the following rational numbers $\frac{16}{225}, \frac{5}{18}, \frac{2}{21} and \frac{7}{250}$ have a terminating decimal?