Question

Which of the following is a vulgar fraction?
A) $\dfrac{3}{{10}}$
B) $\dfrac{{13}}{{10}}$
C) $\dfrac{{10}}{3}$
D) None of these

Answer 1

Hint: Any number of the form $\dfrac{p}{q}$ is called a fraction if $p$ and $q$ are integers and $q \ne 0$. $p$ is the numerator and $q$ is the denominator of the given fraction. A fraction is called a vulgar fraction if it has a non-zero denominator and upon conversion into the decimal form it should be a non-terminating decimal number.

Complete step by step answer:
We are given three fractions, $\dfrac{3}{{10}}$, $\dfrac{{13}}{{10}}$ and $\dfrac{{10}}{3}$. The objective is to determine which of these fractions is a vulgar fraction. If none of these is a vulgar fraction, then we have to select the (D) option.

A) Now, we will first check for the term $\dfrac{3}{{10}}$
It is clear that the given term has an integer $3$ above the line and an integer $10$ below the line, and both the numbers are separated by ‘/ ’. So, it is a fraction term with numerator $3$ and denominator $10$.
Now by the definition of a vulgar fraction, the number should have a non-zero denominator and upon conversion in the decimal form, it should be a non-terminating decimal number.
The fraction $\dfrac{3}{{10}}$ has a denominator $10$ which is non-zero integers.
Now, convert the fraction $\dfrac{3}{{10}}$ into a decimal number. It gives
$\dfrac{3}{{10}} = 0.3$
Since the decimal form is a terminating decimal form, hence this is not a vulgar fraction.

B) Now, we will check for the term $\dfrac{{13}}{{10}}$
It is clear that the given term has an integer $13$ above the line and an integer $10$ below the line, and both the numbers are separated by ‘/ ’. So, it is a fraction term with numerator $13$ and denominator $10$.
The fraction $\dfrac{{13}}{{10}}$ has a denominator $10$ which is non-zero integers.
Now, convert the fraction $\dfrac{{13}}{{10}}$ into a decimal number. It gives
$\dfrac{{13}}{{10}} = 1.3$
Since the decimal form is a terminating decimal form, hence this is not a vulgar fraction.

C) Now, we will check for the term $\dfrac{{10}}{3}$
It is clear that the given term has an integer $10$ above the line and an integer $3$ below the line, and both the numbers are separated by ‘/ ’. So, it is a fraction term with numerator $10$ and denominator $3$.
The fraction $\dfrac{{10}}{3}$ has a denominator $3$ which is non-zero integers.
Now, convert the fraction $\dfrac{{10}}{3}$ into a decimal number. It gives
$\dfrac{{10}}{3} = 3.3333... = 3.\overline 3 $
Since the decimal form is non-terminating, hence this is a vulgar fraction.
The vulgar fraction is (C) $\dfrac{{10}}{3}$.

Therefore, option (C) is correct.

Note:
When a fraction term is converted into the decimal form, then the decimal form can be of three types. Terminating decimals are those decimals that have a finite number of digits, for example $2.56$. Non-terminating non-repeating decimals are those decimals that do not have a finite number of digits after the decimal, for example $1.01001000100001....$. Non-terminating repeating decimals are those decimals that have a finite group of numbers that repeat themselves after the decimal, for example $1.02020202....$, it can be written as $1.\overline {02} $.