Question

Number $\dfrac{{\text{3}}}{{{\text{625}}}}$is a terminating or non-terminating repeating decimal expansion. Write in a decimal form

Answer 1

Hint- The problems focus on making students understand exactly when the decimal expansion of a rational number is terminating and when it is non-terminating repeating. terminating means that a number stops after a few digits of the decimal while non-terminating the value of the number goes to infinity digits after the decimal point.
Complete step-by-step solution -
For example, if denominator of a number is rewrite in the form of ${{\text{2}}^{\text{m}}}\times{{\text{5}}^{\text{n}}}$, where m, n are non-negative integer. Then we say it is a terminating repeating decimal expansion.
According to question,
We have, $\dfrac{{\text{3}}}{{{\text{625}}}}$ is a rational number.
and here the denominator is 625.
Now, we rewrite denominator as
625 = ${5^4}$
As we see the denominator of a given number.
Now, we will see that the denominator 625 of $\dfrac{{\text{3}}}{{{\text{625}}}}$ is of the form ${{\text{2}}^{\text{m}}}\times{{\text{5}}^{\text{n}}}$,
where m, n are non-negative integers.
$\therefore $ We can write the given number as $\dfrac{3}{{625}} = \dfrac{3}{{{5^4}}}$.
Multiplying the numerator and denominator by ${2^4}$, we get:
$\dfrac{3}{{625}} = \dfrac{{3 \times {2^4}}}{{{5^4} \times {2^4}}} = \dfrac{{48}}{{{{(10)}^4}}} = \dfrac{{48}}{{10000}} = 0.0048$.
Hence, $\dfrac{{\text{3}}}{{{\text{625}}}}$ has terminating decimal expansion

Note- In this question, we have learnt about the decimal expansion of a rational number that is either terminating or non-terminating repeating without the actual knowledge of when it’s terminating or non-terminating. And also, if the given rational number has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.