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: We use the fact that a rational number when expressed in decimal expansion will terminate after one digit after decimal if the prime factorization of the denominator is in the form ${{2}^{m}}\times {{5}^{n}}$ where $m$and $n$are natural numbers. So we multiply each of the numbers in options to $\dfrac{1}{3}$ and then check the prime factorization of denominator if it is in the form of ${{2}^{m}}\times {{5}^{n}}$. \[\]

Complete step by step answer:
We know that any rational number can be expressed in the form of $\dfrac{p}{q}$ where $p$ is any integer and $q$ is a nonzero. Here $p$ is called numerator and $q$ is called denominator. The positive rational number of $\dfrac{p}{q}$ can be repressed in a the sequence of decimal digits with a separator called decimal point as
$\dfrac{p}{q}={{b}_{k}}{{b}_{k-1}}...{{b}_{0}}\cdot {{a}_{1}}{{a}_{2}}...$
Here ${{b}_{k}},{{b}_{k-1}},...,{{b}_{0}},{{a}_{1}},{{a}_{2}},...$ are decimal digits ranging from 0 to 9 and $k$ is non-negative number. Here ${{a}_{1}}$is the first digit after decimal place. \[\]
 We also know that any composite number $n$ can be expressed as the product of its prime factors say ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ with exponents say ${{e}_{1}},{{e}_{2}},...,{{e}_{n}}$as
\[n={{p}_{1}}^{{{e}_{1}}}{{p}_{2}}^{{{e}_{2}}}...{{p}_{n}}^{{{e}_{n}}}\]

We also know that a rational number terminates after one place the decimal point when expressed in decimals only if the denominator has 10 as a factor. If we take the prime factorization of 10 we shall have$10=2\times 5$. So if we find the prime factorization of denominator and express it in the form ${{2}^{m}}\times {{5}^{n}}$ where $m,n$ are natural numbers then we can divide the numerator by 10 to get a decimal number terminated after one digit in the decimal expansion.
Let us check option A and multiply $\dfrac{1}{10}$ to$\dfrac{1}{3}$. We have,
\[\dfrac{1}{3}\times \dfrac{1}{10}=\dfrac{1}{30}\]
Here the denominator is 30. Let us find the prime factorization of 30. We have,
\[30=2\times 3\times 5\]
We cannot express the denominator in the form of ${{2}^{m}}\times {{5}^{n}}$ because an additional prime factor 3 is involved in the prime factorization. So $\dfrac{1}{30}$ if expressed in decimal form will not terminate after one digit in the decimal expansion.
Let us check option A and multiply $\dfrac{3}{10}$ to$\dfrac{1}{3}$. We have,
\[\dfrac{1}{3}\times \dfrac{3}{10}=\dfrac{1}{10}\]
Here the denominator is 10. Let us find the prime factorization of 30. We have,
\[10=2\times 5\]
So we have the denominator 10 in the form ${{2}^{m}}\times {{5}^{n}}$ where$m=n=1$ . So $\dfrac{1}{10}$ if expressed in decimal form will not terminate after one digit in the decimal expansion. We can check it by expressing $\dfrac{1}{10}$ in decimal expansion as
\[\dfrac{1}{10}=0.1\]
The other options have numbers 3, 30 which are larger than $\dfrac{3}{10}$ and hence we reject them.

So, the correct answer is “Option B”.

Note: The decimal representation if terminated is called terminating decimal. If a certain number of digits are repeated infinitely and periodically in the decimal expansion then it is called recurring decimals. If it does not repeat or terminate then it is the decimal expansion of irrational numbers. If the decimal expansion gets terminated after two digits of decimal then the denominator is in the form of ${{2}^{m}}\times {{5}^{n}}$ where $m,n$ are even natural numbers.