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# In what ratio must water be mixed with milk to gain 16% on selling the mixture at cost price?A. 1:6B. 4:25C. 2:3D. 4:3

Last updated date: 12th Aug 2024
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Hint: This is a mixture alligation problem. To solve this first we will obtain the gain amount and cost price of mixture. And then by using a mixture and alligation rule we will obtain the ratio or water and milk.

Let the cost price of 1 liter of milk = Rs.1
According to the question, cost price of milk is equal to the selling price of mixture (mixture of milk and water)
Hence the selling price of 1 liter of mixture = Rs.1
Gain = 16%
Hence the cost price of mixture = $\dfrac{{100}}{{100 + gain\% }} \times SP$ ( Here SP means selling price )
$= \dfrac{{100}}{{100 + 16}} \times 1$
$= \dfrac{{100}}{{116}} = \dfrac{{25}}{{29}}$rs.
By the rule of mixture and alligation,
Cost price of 1 liter of water, c = 0
Cost price of 1 liter of milk, d = 1
Mean price, m = cost price of mixture =$\dfrac{{25}}{{29}}$
Hence, d-m = $1 - \dfrac{{25}}{{29}} = \dfrac{{29 - 25}}{{29}} = \dfrac{4}{{29}}$
And m-c = $\dfrac{{25}}{{29}} - 0 = \dfrac{{25}}{{29}}$
Therefore the ratio of water and milk in the mixture is,
d-m : m-c
$= \dfrac{4}{{29}}:\dfrac{{25}}{{29}}$
As the denominator of both the terms are equal, hence the ratio is 4:25.
Therefore, in a 4:25 ratio water must be mixed with milk to gain 16% on selling the mixture at cost price.

So, the correct answer is “Option B”.

Note: You might get confused because the selling price of the mixture is equal to the cost price of the milk as it is trickily mentioned in the question.
We can also solve this by using alternative methods.
Alternative method- Let the amount of water mixed is x liter
CP of milk = SP of mixture = 1(let)
Actual CP of mixture is 100/116
By adding water we get profit 16/116
Therefore for every 100 liter of milk, 16 liter of water is added.
Hence the ratio of water and milk in the mixture is 16/100 = 4/25 = 4:25