Question

There are two containers of milk. There is twice as much milk in the first container as in the second. After using $2\mathrm{gal}$. of milk from the second container and $3\mathrm{gal}.$ of milk from the first container, the second container had $4.5$ times less milk than the first. How many gallons of milk were in both containers originally?

Answer 1

Step-1: Form linear equations using given data:

Given that,

• There are two containers of milk
• Initially, there is twice as much milk in the first container as in the second
• After using $2\mathrm{gal}$. of milk from the second container and $3\mathrm{gal}.$ of milk from the first container, the second container had $4.5$ times less milk than the first

Let the initial gallon of milk in first container be $x$
Let the initial gallon of milk in second container be $y$
According to question, there is twice as much milk in the first container as in the second
$\Rightarrow x=2y...\left(i\right)$

And, after using $2\mathrm{gal}$. of milk from the second container and $3\mathrm{gal}.$ of milk from the first container, the second container had $4.5$ times less milk than the first

$\Rightarrow \left(x-3\right)-\left(y-2\right)=4.5\left(y-2\right)$

Step-2: Solve for the quantity of milk in both containers originally:

$\left(x-3\right)-\left(y-2\right)=4.5\left(y-2\right)$

$\Rightarrow$ $x-3=5.5\left(y-2\right)$

$\Rightarrow$ $2y-3=5.5\left(y-2\right)\left[\because x=2y\right]$

$\Rightarrow$ $3.5y=8$

$\Rightarrow$ $y=\frac{8}{3.5}$

$\Rightarrow$ $y=2.2857\approx 2.29gal$
Substituting the value of $y=2.29$ in equation $\left(i\right)$
$$\begin{gathered} \Rightarrow x=2\times 2.29 \\ =4.58\mathrm{gal} \end{gathered}$$

Hence, the initial gallons of milk in first and second container is $4.58\&2.29$ respectively.