Ratio and Proportion Concept clarity.

Question

The milk and water in two vessels $A$ and $B$ are in ratio of $4:3$ and $2:3 $ respectively. In what ratio should the liquids in both vessels be mixed to obtain a new mixture in vessel $C$ consisting of half milk and half water?

Solution: Let $X$ be the amount of mixture taken from $A$. Let $Y$ be the amount of mixture taken from $B$.

\begin{equation} \frac{\frac{4x}{7}+\frac{2y}{5}}{\frac{3x}{7}+\frac{3y}{5}} = \frac{1}{1} \tag{1} \end{equation}

Now solving for $\dfrac{x}{y}$ we will get the solution.

But I got confused when I thought why not I should do: Let $X$ be the total mixture from $A$. Let $Y$ be the total mixture from $B$. Then:

If total mixture of $A$ is $X$ then milk will be $4x$ and water will be $3x$. If total mixture of $B$ is $Y$ then milk will be $2y$ and water will be $3y$. So if we add both the mixture, the resultant mixture should be $1/1$.

\begin{equation} \frac{4x+2y}{3x+3y} = \frac{1}{1} \tag{2} \end{equation}

So I unable to understand what is the difference between both the two equations ( (1) and (2) ) in terms of meaning ( results of $\dfrac{x}{y}$ from both the equations is obviously different ).

As I am thinking that whether we take some part of both the mixture and mix it to get the $1/1$ resultant mixture or we take the both whole mixtures and mix it to get the $1/1$ resultant mixture, the resultant ratio (i.e $\dfrac{x}{y}$) of both the equations should be same. I know i am missing something but I am not getting it. I hope you understand my dilemma and help me through it.

Answer 1

If total mixture (taken from) of A is x then milk will be 4x and water will be 3x.

No, that would claim $x=4x+3x$, since the amount of fluid equals the amount of milk plus the amount of water.

The fluid in A contains a $4:3$ milk to water ratio, so you are taking $\tfrac 47x$ of milk, and $\tfrac 37x$ of water, for a total of $x$.

So, you mixing $\tfrac 47x$ and $\tfrac 25y$ milk, and $\tfrac 37x$ and $\tfrac 35y$ water, each from A and B respectively.

The ratio of milk to water in the resulting mixture is: $(\tfrac 47 x+\tfrac 25y)\div(\tfrac 37x+\tfrac 35y)$, or $(20x+14y)\div(15x+21y)$.

We would like this ratio to equal $1/1$ (1:1 water to milk), so , we obtain $5x=7y$ or $x/y=7/5$ .. a 7:5 ratio of fluid from A to fluid from B.

If the containers are the same size, this means taking $\tfrac 7{12}$'s of A and $\tfrac 5{12}$'s of B.

Answer 2

A ratio how many parts of $A$ to how many parts of $B$. Not how many parts of $A$ in the whole.

A mixture that is "half and half" means $1$ part (half of it) is milk and $1$ part (half of it) is water. So the ratio is $1:1$. Not $1:2$. ($1:2$ would mean $1$ part mild to $2$ parts water. So it would be $\frac 13$ mile and $\frac 23$ water.)

So if you have $x$ portions of $A$ and $y$ partions of $B$ then:

$A$ if $\frac 47$ milk so you have $\frac 47 y$ portions of milk from $A$. And $B$ is $\frac 25$ milk so you have $\frac 25y$ portions of milk from $B$. And in total you have $\frac 47x + \frac 25y$ portions of milk.

And the same for water: $A$ is $\frac 37$ water and $B$ is $\frac 35$ water so you have $\frac 37x + \frac 25y$ total portions of water.

And we want

$\frac {\frac 47x + \frac 25y}{\frac 37x + \frac 25y} = \frac 11$

Or $\frac 47x + \frac 25y = \frac 37x + \frac 25y$

.....

If total mixture of A is X then milk will be 4x and water will be 3x. If total mixture of B is Y then milk will be 2y and water will be 3y.

Well, no, $4x + 3x \ne X$. If the total mixture from $A$ was $7X$ then milk will be $4x$ and water will be $3x$.

Sin if the total mixture of $B$ was $5Y$ you'd get $2Y$ milk and $3Y$ water.

So you'd have $\frac {\frac 754x + 2y}{\frac 753x + 3y} = \frac 11$ or

$\frac {4x + \frac 57 2y}{ 3x + \frac 57 3y} =\frac 11$.

Better yet if you took $35X$ from $A$ and $35Y$ from $B$ you would have $\frac {4*5x + 3*7Y}{3*5x + 2*7y} = \frac 11$.

Those will work.

Answer 3

Another way to look at it.

$C$ will contain the total of $X+Y$ milk and water, a half of which must be milk.

If $X$ amount is taken from $A$, then the amount of milk taken will be $\frac47 X$.

If $Y$ amount is taken from $B$, then the amount of milk taken will be $\frac25 Y$.

Hence: $$\begin{align}\frac47 X+\frac25Y&=\frac12(X+Y) \Rightarrow \\ \frac47X-\frac12X&=\frac12Y-\frac25 Y\Rightarrow \\ \frac1{14}X&=\frac1{10}Y \Rightarrow \\ \frac{X}{Y}&=\frac75.\end{align}$$