Question

The smallest rational number by which $\dfrac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is
A. $\dfrac{1}{10}$
B. $\dfrac{3}{10}$
C. $3$
D. $30$

Answer 1

Hint: Basically, rational numbers are the numbers that are expressed in the form of fraction of two whole numbers. Rational numbers are of two types, one is terminating in which the division terminates after a certain number steps and has a finite number of decimal parts after the decimal point, and the second type is non - terminating in which the decimal part is infinite or recurring. We use a long division method to convert rational numbers into decimals. Decimal expansion of the number is the representation of the number in which the decimal point is used. Decimal numbers can be finite or repeating. Every decimal number can be expressed in the form of fraction.

Complete step by step solution :
The given rational number is $\dfrac{1}{3}$
The decimal expansion of $\dfrac{1}{3}$ is \[0\text{ }.333\ldots \] which is non - terminating.
Since this is a multiple choice question, so we can check if we get the required answer for any of the four options
Let us check for option a),
$\dfrac{1}{3}\times \dfrac{1}{10}=\dfrac{1}{30}=0.0333...$ which is again non - terminating. So , option a) is not correct.
Now for option b),
$\dfrac{1}{3}\times \dfrac{3}{10}=\dfrac{1}{10}=0.1$
which is terminating and the decimal point terminates after one place.
So this option may be correct but we need the smallest rational number which when multiplied to $\dfrac{1}{3}$ , terminates after one place of decimal.
Now let us also check for option c) and d),
$\dfrac{1}{3}\times 3=1$
which is also terminating but it is not true as $3$ is at least greater than $\dfrac{3}{10}$. So it is not the smallest.
Also, $30$ is greater than $\dfrac{3}{10}$. So, this can also not be true.

Hence option b) is the correct answer.

Note:
Since the rational number gives terminating or non - terminating recurring decimal . Thus it can be said that the number whose decimal expansion is terminating or non terminating is rational.