Question

How do you simplify $\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}$ ?

Answer 1

Hint: In this problem we need to calculate the sum of the three given fractions. We know that the addition of fraction uses the concept of LCM. Generally, we have the addition of the two fractions $\dfrac{1}{a}$, $\dfrac{1}{b}$ as $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{b+a}{ab}$. Now we will use the above formula to calculate the sum of the first two fractions. Now we will simplify the obtained result and convert it to another fraction. Now we will again add the obtained fraction with another remaining fraction by using the above method. Now we will simplify the obtained equation to get the result.

Formulas Used:
1. $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{b+a}{ab}$

Complete step by step solution:
Given that, simplify the given expression is $\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}$
Considering the first two fractions $\dfrac{1}{3}$, $\dfrac{1}{2}$.
Now the sum of the above two fractions is
$\dfrac{1}{3}+\dfrac{1}{2}$
Using the known formula $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{b+a}{ab}$ in the above equation, then we will get
$\dfrac{1}{3}+\dfrac{1}{2}=\dfrac{2+3}{2\times 3}$
Simplifying the above equation, then we will get
$\Rightarrow \dfrac{1}{3}+\dfrac{1}{2}=\dfrac{5}{6}$
Now we can write the given expression $\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}$ as
$\Rightarrow \dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}=\dfrac{5}{6}+\dfrac{1}{5}$
Now we will consider the fractions $\dfrac{5}{6}$, $\dfrac{1}{5}$.
Sum of the fractions $\dfrac{5}{6}$, $\dfrac{1}{5}$ is
$\dfrac{5}{6}+\dfrac{1}{5}$
Again, using the rule $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{b+a}{ab}$ to calculate the above sum, then we will get
$\Rightarrow \dfrac{5}{6}+\dfrac{1}{5}=\dfrac{5\times 5+6}{6\times 5}$
Simplifying the above equation, then we will get
$\begin{align}
  & \Rightarrow \dfrac{5}{6}+\dfrac{1}{5}=\dfrac{25+6}{30} \\
 & \Rightarrow \dfrac{5}{6}+\dfrac{1}{5}=\dfrac{31}{30} \\
\end{align}$

Hence the value of the given equation $\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}$ is $\dfrac{31}{30}$.

Note:
We can also directly calculate the value of the given expression by using the well know formula $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{bc+ac+ab}{abc}$. From the above formula the value of $\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}$ will be
$\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}=\dfrac{2\times 5+3\times 5+3\times 2}{3\times 2\times 5}$
Simplifying the above equation by substituting the known values, then we will get
$\begin{align}
  & \Rightarrow \dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}=\dfrac{10+15+6}{30} \\
 & \Rightarrow \dfrac{1}{3}+\dfrac{1}{2}+\dfrac{1}{5}=\dfrac{31}{30} \\
\end{align}$
From both the methods we got the same result.