Question

Find the value of $x$ if \[870\times 5\dfrac{2}{3}+4\dfrac{4}{5}\div 1\dfrac{3}{5}=x\]

Answer 1

Hint: We first convert the given improper fractions into proper fractions. We change the division into multiplication. We complete the solution to find the answer.

Complete step-by-step answer:
Improper fractions are those fractions who have greater value in the numerator than the denominator.
We need to convert the mixed fraction which is representation in the form sum of an integer and a proper fraction. We express the process in the form of variables.
Then the equational condition will be $\dfrac{a}{b}=x+\dfrac{c}{b}$. The representation of the mixed fraction will be $x\dfrac{c}{b}$. The solution of the equation $x+\dfrac{c}{b}=\dfrac{bx+c}{b}$ and $a=bx+c$.
We first convert the given improper fractions in \[870\times 5\dfrac{2}{3}+4\dfrac{4}{5}\div 1\dfrac{3}{5}=x\].
So, \[5\dfrac{2}{3}=\dfrac{5\times 3+2}{3}=\dfrac{17}{3},4\dfrac{4}{5}=\dfrac{5\times 4+4}{5}=\dfrac{24}{5},1\dfrac{3}{5}=\dfrac{5\times 1+3}{5}=\dfrac{8}{5}\].
We now put those values in the equation and get
\[\begin{align}
  & 870\times 5\dfrac{2}{3}+4\dfrac{4}{5}\div 1\dfrac{3}{5}=x \\
 & \Rightarrow 870\times \dfrac{17}{3}+\dfrac{24}{5}\div \dfrac{8}{5}=x \\
 & \Rightarrow \dfrac{870\times 17}{3}+\dfrac{24}{5}\times \dfrac{5}{8}=x \\
\end{align}\]
We now simplify the equation
\[\begin{align}
  & \dfrac{870\times 17}{3}+\dfrac{24}{5}\times \dfrac{5}{8}=x \\
 & \Rightarrow x=4930+3=4933 \\
\end{align}\]
Therefore, the value of $x$ is 4933.
So, the correct answer is “4933.”.

Note: We need to be careful about the change of operation from division of multiplication. This process also changes the fraction to its inverse form. We need to remember that the denominator in both cases of improper fraction and the mixed fraction will be the same. The only change happens in the numerator. The relation being the equational representation of $a=bx+c$.