Question

How do you solve $3.2=-0.4n$?

Answer 1

Hint: We solve the given linear equation by simplifying the equation. We convert the decimal numbers into fractions. We cross-multiply the equations. The process of cross-multiplication is equivalent to taking the L.C.M of the denominators and multiplying with that. Then we apply the multiplication to get the value of n. We also take the L.C.M to multiply both sides and verify the answer.

Complete step by step answer:
The given equation $3.2=-0.4n$ is a linear equation of n. We first convert the decimal numbers into fractions.
So, \[3.2=-0.4n\Rightarrow \dfrac{32}{10}=\dfrac{-4n}{10}\]
We apply cross-multiplication to multiply -4n with 10 and 32 with 10.
\[\begin{align}
  & \Rightarrow \dfrac{32}{10}=\dfrac{-4n}{10} \\
 & \Rightarrow 10\left( -4n \right)=32\times 10 \\
\end{align}\]
The multiplication value of 10 with -4n gives us value \[10\left( -4n \right)=-40n\]. Similarly, the multiplication value of 32 with 10 gives us value \[32\times 10=320\].
The equation becomes \[-40n=320\]. Dividing with -40 we get
\[\begin{align}
  & -40n=320 \\
 & \Rightarrow \dfrac{-40n}{-40}=\dfrac{320}{-40} \\
 & \Rightarrow n=-8 \\
\end{align}\]
We complete the division to get \[n=-8\].
Therefore, the solution of the equation $3.2=-0.4n$ is \[n=-8\].
The L.C.M of 10 and 10 gives 10. So, multiplying both sides of $3.2=-0.4n$ with 10, we get
\[\begin{align}
  & \Rightarrow \dfrac{32}{10}\times 10=\dfrac{-4n}{10}\times 10 \\
 & \Rightarrow 32=-4n \\
 & \Rightarrow n=\dfrac{32}{-4}=-8 \\
\end{align}\].

So, the correct answer is “n=-8”.

Note: The process of cross-multiplication comes from the L.C.M of the denominators. The L.C.M gets multiplied with the both sides of the equation which in turn gives the cross multiplication. In the case of denominators being co-primes, both processes are the same.
For example, if we take $\dfrac{x}{36}=\dfrac{y}{27}$, we can cross multiply to get $27x=36y$ and then we divide both side with their G.C.D value 9.
That gives us
$\begin{align}
  & \dfrac{27x}{9}=\dfrac{36y}{9} \\
 & \Rightarrow 3x=4y \\
\end{align}$
We can also use the L.C.M of the denominators which is 108 and multiply both sides of $\dfrac{x}{36}=\dfrac{y}{27}$ to get the final answer directly.
\[\begin{align}
  & \dfrac{x}{36}\times 108=\dfrac{y}{27}\times 108 \\
 & \Rightarrow 3x=4y \\
\end{align}\].