Question

How do you write the fraction $ \dfrac{10}{30} $ in simplest form?

Answer 1

Hint: In this question, we have to solve a fraction and get its simplest form. A fraction is a number that contains the numerator and the denominator, the numerator says “a” is written above the horizontal line and the denominator says “b” is written below the horizontal line, it is used to represent a small portion of the whole thing, that is $ \dfrac{a}{b} $. Therefore, to solve this problem we will use the factorization method. We will find the factors of the numerator and the denominator with the help of LCM and then make the necessary calculations, to get the required solution to the problem.

Complete step by step answer:
According to the question, we have to simplify a fraction into its simplest form
Thus, we will use the factorization method to get the solution.
The fraction given to us is $ \dfrac{10}{30} $ ----------- (1)
Therefore, we will rewrite the equation (1) in the form of $ \dfrac{m}{n} $ , such that both m and n have no common divisors, that is
 $ \dfrac{10}{30}=~\dfrac{m}{n} $
Now, we first take the factors of the numerator which is number 10, so we will find the least common multiple of 10, that is
 $ \begin{align}
  & \text{ 2}\left| \!{\underline {\,
  10 \,}} \right. \\
 & \text{ 5}\left| \!{\underline {\,
  5 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align} $
Therefore, LCM(10)= $ 2\times 5\times \text{1} $ ----------- (2)
Now, we first take the factors of the denominator which is number 30, so we will find the least common multiple of 30, which is
 $ \begin{align}
  & \text{ 2}\left| \!{\underline {\,
  30 \,}} \right. \\
 & \text{ 5}\left| \!{\underline {\,
  15 \,}} \right. \\
 & \text{ 3}\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align} $
Therefore, LCM(30)= $ 2\times \text{5}\times \text{3}\times \text{1} $ ----------- (3)
Thus, we will substitute the value of equation (2) and (3) in equation (1), we get
 $ \dfrac{2\times 5\times 1}{2\times 5\times 3\times 1} $
Now, we know that the same terms in the numerator and the denominator cancel out each other, and we get the quotient as 1 that is we will cancel 2, 5, and 1 in the numerator and the denominator, therefore we get
 $ \dfrac{1}{3} $
Thus, we cannot further simplify the above fraction as both the numerator which is 1, and the denominator which is 3 have no common divisor, thus we get
$\Rightarrow$ $ \dfrac{10}{30}=~\dfrac{m}{n}=\dfrac{1}{3} $
Therefore, for the fraction $ \dfrac{10}{30} $ , its simplest form is $ \dfrac{1}{3} $ , which is our required solution.

Note:
 While solving this problem, always find the factors of the numerator and the denominator carefully to avoid confusion and mistakes in the answer. One of the alternative methods to solve this problem is taking common 10 in both the numerator and the denominator and then cancel out, to get the required solution to the problem.