Question

The proportion of milk and water in 3 samples is \[2:1,3:2,5:3\]. A mixture consisting of equal quantities of all 3 samples is made. The proportion of milk and water in the mixture is
(a) \[227:133\]
(b) \[133:227\]
(c) \[99:61\]
(d) \[61:99\]

Answer 1

Hint: We solve this problem by considering the proportion of milk and water in each sample and then find the total proportion to find the ratio of milk and water in the mixture. The proportion of milk and water is calculated by using the definition of ratios. If ratio of milk and water is given as \[a:b\] then, the proportion of milk is given as \[\dfrac{a}{a+b}\] and proportion of water is \[\dfrac{b}{a+b}\].

Complete step by step answer:
Let us take each sample and calculate the proportions of milk and water.
Let us consider the first sample which has a proportion of milk and water as \[2:1\].
We know that if ratio of milk and water is given as \[a:b\] then, the proportion of milk is given as \[\dfrac{a}{a+b}\] and proportion of water is \[\dfrac{b}{a+b}\].
By using this definition, proportion of milk in first sample is
\[\Rightarrow {{M}_{1}}=\dfrac{2}{2+1}=\dfrac{2}{3}\]
Now, proportion of water in first sample is
\[\Rightarrow {{W}_{1}}=\dfrac{1}{2+1}=\dfrac{1}{3}\]
Now, let us consider the second sample having the proportion of milk and water as \[3:2\]
By using this definition, proportion of milk in second sample is
\[\Rightarrow {{M}_{2}}=\dfrac{3}{3+2}=\dfrac{3}{5}\]
Now, proportion of water in second sample is
\[\Rightarrow {{W}_{2}}=\dfrac{2}{3+2}=\dfrac{2}{5}\]
Now, let us consider the third sample having the proportion of milk and water as \[5:3\]
By using this definition, proportion of milk in third sample is
\[\Rightarrow {{M}_{3}}=\dfrac{5}{5+3}=\dfrac{5}{8}\]
Now, proportion of water in third sample is
\[\Rightarrow {{W}_{3}}=\dfrac{3}{5+3}=\dfrac{3}{8}\]
Now, let us find the total proportions of milk and water in the mixture by adding all the proportions in each sample.
By adding all the proportions of milk in each sample we get
\[\begin{align}
  & \Rightarrow M={{M}_{1}}+{{M}_{2}}+{{M}_{3}} \\
 & \Rightarrow M=\dfrac{2}{3}+\dfrac{3}{5}+\dfrac{5}{8} \\
 & \Rightarrow M=\dfrac{227}{120} \\
\end{align}\]
Now, similarly by adding all the proportions of water in each sample we get
\[\begin{align}
  & \Rightarrow W={{W}_{1}}+{{W}_{2}}+{{W}_{3}} \\
 & \Rightarrow W=\dfrac{1}{3}+\dfrac{2}{5}+\dfrac{3}{8} \\
 & \Rightarrow W=\dfrac{133}{120} \\
\end{align}\]
Now, the ratio of proportions of milk and water in mixture is given as
\[\begin{align}
  & \Rightarrow M:W=\dfrac{227}{120}:\dfrac{133}{120} \\
 & \Rightarrow M:W=227:133 \\
\end{align}\]
Therefore, the ratio of proportions of milk and water in the mixture is \[227:133\]

So, the correct answer is “Option a”.

Note: Students may make mistakes in considering the ratio order. They mentioned the order as milk to water so the first part in the ratios mentioned will be proportions of milk and second part is proportion of water. The ratio of milk and water is\[a:b\]then\['a'\]represents the proportion of milk and\['b'\]represents the proportion of water. This needs to be taken care of.