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A mixture contains milk and water in the ratio $5:1$. On adding \[5{\text{ }}litres\] of water, the ratio of milk to water becomes $5:2$. The quantity of the milk in the original mixture is:
A) 16 Lts
B) 22.75 Lts
C) 25 Lts D) 32.5 Lts

Last updated date: 16th Jun 2024
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Hint: We need to have the concept of Ratio & Proportion very well. Here, the mixture is of two liquids (water and milk). This type of problem is solved mostly in a similar way considering some value of both liquids. Take two unknown values in ratio according to question i.e., two different conditions. Then form a linear equation & solve it to get the quantity of milk in the original mixture.

Complete step-by-step answer:
Let the amount of milk and water in the mixture be \[X\;and{\text{ }}Y{\text{ }}litres\] respectively.
According to the problem,$5:1$ ratio of milk and water in the mixture is $5:1$
$x:y = 5:1 \Rightarrow \dfrac{x}{y} = \dfrac{5}{1}$ [ by ratio & proportion, we can represent proportional ratio in fraction form]
   \Rightarrow 5y = x \\
   \Rightarrow y = \dfrac{x}{5}.........(1) \\
} $ [ by cross multiplication]
As per given information in the question, \[5{\text{ }}litres\] of water is added in the original mixture then we know milk & water proportion in the mixture will be changed & a new mixture will be formed.
Thus, the quantity of milk will remain the same i.e., $x$ litres but quantity of water will be $y + 5\,$litres.
As per problem, Ratio of milk & water in new mixture is $5:2$,
$\therefore \dfrac{x}{{y + 5}} = \dfrac{5}{2}$
As, quantity of milk is to be determined, y is written in terms of x or we can say putting value of $y = \dfrac{x}{5}$,
$ \Rightarrow \dfrac{x}{{\dfrac{x}{5} + 5}} = \dfrac{5}{2}$ [From eq. (1)]
$ \Rightarrow \dfrac{{5x}}{{x + 25}} = \dfrac{5}{2}$ [ solving the denominator by addition]
$ \Rightarrow \dfrac{{5x}}{{x + 25}} = \dfrac{5}{2}$ [ simplifying the above equation]
$ \Rightarrow 10x = 5x + 125$ [ By cross multiplication ]
\[ \Rightarrow 5x = 125\] [ Taking all x term values in one side]
$ \Rightarrow x = 25$
Hence as assumed, Quantity of the milk in the original mixture = $25$ litres (c).
Hence, the correct option is (C)

Note: In this type of problem, what is to be determined should be checked in the question carefully because there are chances of ignorance towards what has been asked for in this question that can lead you to the wrong ultimate answer instead of knowing all the concepts & procedures to be applied . In this problem, the quantity of milk in the original mixture is asked. Sometimes, the addition of a liquid is done in the wrong place (like quantity of water), this can be avoided by reading thoroughly. Finally put the value of x in any of the equations to check whether it is satisfying the equation or not. Concept of Ratio & Proportion is important for it to be solved.