QUESTION

# Find the number of digits after decimal in $\dfrac{18}{{{5}^{3}}}$.(a) 5(b) 4(c) 3(d) 2

Hint: In this question, we will use the rule that the integer number when divided by ${{\left( 5\times 2 \right)}^{n}}$, its digits after the decimal point will be $n$.

We know that, any number when divided to 10, its decimal point moves one digit to the left from where it was in a number.
For example, $\dfrac{3789}{10}=378.9$
Similarly, any number when divided to 100, its decimal point moves two digits to the left from where it was in a number.
For example, $\dfrac{3789}{100}=37.89$
And so on.
Hence, we can say that, for an integer number, the number of digits after zero will be equal to the number of zeros after 1 in the numbers 10, 100, 1000,…, by which it is divided. But if that integer number is multiple of 10, the number of decimal digits will decrease.
Also, we know that, prime factorisation of 10, 100, 1000, … is given by $5\times 2,\,{{\left( 5\times 2 \right)}^{2}},\,{{\left( 5\times 2 \right)}^{3}},\,...$ .
From this, we see that, number of zeroes in 10,100, 1000,... is equal to the exponential power of $5\times 2$ in its prime factorisation.
Now, to check the number of digits after decimal point in given expression $\dfrac{18}{{{5}^{3}}}$, we try to write it in a form of integer number divided by $5\times 2$ raised to some power.
Let us multiply and divide ${{2}^{3}}$ in numerator and denominator of given expression, so we get,
$\dfrac{18}{{{5}^{3}}}=\dfrac{18\times {{2}^{3}}}{{{5}^{3}}\times {{2}^{3}}}$
Taking power common in denominator, we get,
$\dfrac{18}{{{5}^{3}}}=\dfrac{18\times {{2}^{3}}}{{{\left( 5\times 2 \right)}^{3}}}$
Here, the number in the numerator is integer and not a multiple of 10. Also, it is divided by ${{\left( 5\times 2 \right)}^{3}}$, where exponential power of 10 is 3.
Hence, the numbers of digits after the decimal point in $\dfrac{18}{{{5}^{3}}}$ are 3.
Therefore, the correct answer is option (c).

Note: In this type of question, you can perform the long division to find the number after division and check the decimal points. But that will become difficult for bigger numbers. But by this method you can easily check the number of decimals digits for any number.