Question

How do you simplify $6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}$?

Answer 1

Hint: For this problem they have asked to simplify the given value. We can observe that we need to sum the three fractions out of them two are mixed fractions. So, we need to first convert the mixed fraction into normal fractions. We know that the normal fractional form of mixed fraction $a\dfrac{b}{c}$ will be $\dfrac{a\times c+b}{c}$. So, we will consider each mixed fraction individually and use the above rule to convert the mixed fraction into normal fraction. After that we will add all the three-fractions based on LCM of the denominators and simplify the equation to get the required result.

Complete step by step solution:
Given that, $6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}$.
In the above value, we can observe the mixed fractions $6\dfrac{2}{9}$, $7\dfrac{2}{5}$.
Considering the mixed fraction $6\dfrac{2}{9}$.
We know that the normal fractional form of the mixed fraction which is form of $a\dfrac{b}{c}$ will be $\dfrac{a\times c+b}{c}$. From this the normal fractional form of $6\dfrac{2}{9}$ will be
$\begin{align}
  & \Rightarrow 6\dfrac{2}{9}=\dfrac{6\times 9+2}{9} \\
 & \Rightarrow 6\dfrac{2}{9}=\dfrac{54+2}{9} \\
 & \Rightarrow 6\dfrac{2}{9}=\dfrac{56}{9} \\
\end{align}$
Considering the mixed fraction $7\dfrac{2}{5}$. Now the normal fraction form of the mixed fraction $7\dfrac{2}{5}$ will be
$\begin{align}
  & \Rightarrow 7\dfrac{2}{5}=\dfrac{7\times 5+2}{5} \\
 & \Rightarrow 7\dfrac{2}{5}=\dfrac{35+2}{5} \\
 & \Rightarrow 7\dfrac{2}{5}=\dfrac{37}{5} \\
\end{align}$
Substituting the values, in the given equation, then we will get
$\Rightarrow 6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}=\dfrac{56}{9}+\dfrac{37}{5}+\dfrac{5}{9}$
Considering the LCM of the denominators $9$, $5$ which is $45$, performing addition in the above equation, then we will get
$\begin{align}
  & \Rightarrow 6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}=\dfrac{56\times 5+37\times 9+5\times 5}{45} \\
 & \Rightarrow 6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}=\dfrac{280+333+25}{45} \\
 & \Rightarrow 6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}=\dfrac{638}{45} \\
\end{align}$

Hence the simplified value of $6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}$ will be $\dfrac{638}{45}$.

Note: We can also write the obtained fraction in mixed fraction by dividing the numerator $638$ with $45$. When we divide $638$ with $45$, then we will get $14$ as quotient and $8$ as reminder. So, we can write the above value as
$\Rightarrow 6\dfrac{2}{9}+7\dfrac{2}{5}+\dfrac{5}{9}=14\dfrac{8}{45}$.