Question

In one glass, milk and water are mixed in the ratio of \[3:5\] and in another glass they are mixed in the ratio \[6:1\]. In what ratio should the content of the two glasses be mixed together so that the next mixture contains milk and water in the ratio of \[1:1\].
A. \[20:7\]
B. \[8:3\]
C. \[27:4\]
D. \[25:9\]

Answer 1

Hint: We are given two solutions and the ratios of milk and water and we have to make a third solution which we will get by mixing some particular quantities of the earlier two given solutions. Now we are going to form equations by these ratios which are as follows, \[3:5=3x:5x\] and \[6:1=6y:y\] it is because ratios are simplified version of large or actual numbers having same factors, so we are assuming that \[8x\] is the quantity of solution 1 and \[7y\] is the quantity of solution 2. In the end we will compare milk and water quantity, separately, of two given solutions for both components with the final solution, which is for milk \[3x+6y=1\] and for water it is \[5x+y=1\]. In the end we will get \[8x:7y\].

Complete step-by-step answer:
Now in this question we are given with the ratios of milk and water quantity for three solutions but the third one has to be created by us and the quantity of components is given in it but we do not have the components in their original separated form or pure form so we have to make use of the solutions given and by altering their quantity we can form our final solution of the given ratio.
First we need to form equations from the ratios of the two given solutions of the components separately which means we will find the total milk quantity by adding the milk part of the ratios of the two given solutions and then equate it to the milk part of the final solution to form an equation, similarly we will do for water by adding the water part of the ratios of the two given solutions and then equate it to the water part of the final solution to form it’s equation too.
But first let us assume that the quantities of milk and water in solution 1 are, \[3:5=3x:5x\], \[3x\] is milk and \[5x\] is water. Similarly the quantities of milk and water in solution 2 are, \[6:1=6y:y\], \[6y\] is milk and \[y\] is water. In the final solution the quantities are 1 for milk and 1 for water and we will let it be like this because then our question will be easy to solve.
Consider \[8x\] be the amount of solution 1 and \[7y\] be the amount of solution 2 in the third solution, we get,
Now forming equations, first the equation for the quantity of milk is,
\[3x+6y=1\], where we add the quantities of milk in both the given solutions and we want it to be 1
Now, the equation for the quantity of water is,
\[5x+y=1\], where we add the quantities of water in both the given solutions and we want it to be 1
Now solving the two formed equations of milk and water quantity, we get,
\[3x+6y=1\], milk equation
\[5x+y=1\], water equation
Now multiplying the water equation by 6, we get,
\[6\left( 5x+y=1 \right)\]
\[30x+6y=6\]
Now subtracting the milk equation from water equation, we get,
\[30x+6y=6\]
\[-3x-6y=-1\]
\[27x=5\]
\[x=\dfrac{5}{27}\]
Now calculating the value of \[y\] by putting the value of \[x\] in any of the equations, that might be either water or milk. To get the amount of solution 2 we multiply the value of \[x\] by 8.
Putting the value of amount of solution 1 in original water equation, we get,
\[5x+y=1\]
\[\begin{align}
  & 5\left( \dfrac{5}{27} \right)+y=1 \\
 & \dfrac{25}{27}+y=1 \\
 & y=1-\dfrac{25}{27} \\
 & y=\dfrac{27-25}{27}=\dfrac{2}{27} \\
\end{align}\]
Now we end up with the value of \[y\] as well, to get the amount of solution 2 we will multiply it by 7.
Now we need to calculate the ratio of amount of solution 1 to the amount of solution 2 which is,
\[x:y=\dfrac{5}{27}:\dfrac{2}{27}\]
\[\begin{align}
  & x:y=\dfrac{5}{27}\times \dfrac{27}{2} \\
 & x:y=5:2 \\
\end{align}\]
But now in the end we need to find the ratio of amount of solution 1 to the amount of solution 2 as by adding the amount of components we get the amount of solution, so we need to multiply our ratio by \[\dfrac{8}{7}\], we get,
\[8x:7y=20:7\], which means 20 parts of solution 1 and 7 parts of solution 2 are mixed to form 27 parts of solution 3.
Hence the correct answer is,
Option. A. \[20:7\]

Note: The possible mistake in these questions is to not multiply the ratio of variables by the amount of solutions in this case we need to multiply our ratio of variables with \[\dfrac{8}{7}\] because the amount of solution is calculated by the amount of its components added, in solution 1 it is \[3x\left( milk \right)+5x\left( water \right)=8x\left( solution \right)\] and in solution 2 it is \[6y\left( milk \right)+y\left( water \right)=7y\left( solution \right)\]. We could have assigned a variable to the third solution and it could act as a third solution which is also a shortcut, we could have equated both the equations as their third variable was same like, \[\begin{align}
  & 3x+6y=z \\
 & 5x+y=z \\
 & 3x+6y=5x+y \\
\end{align}\]
\[\begin{align}
  & 5y=2x \\
 & \dfrac{x}{y}=\dfrac{5}{2} \\
\end{align}\]
At the end multiply by \[\dfrac{8}{7}\] to get the ratio of solution 1 amount to solution 2 amount.