We can create the following ratios:

Vessel 1: water : milk = x : 2x

Vessel 2: water : milk = 2y : 5y

We can create the equation:

(x + 2x)/(2y + 5y) = 1/4

3x/7y = 1/4

12x = 7y

We see that we can let x = 7 and y = 12. So we have:

Vessel 1: water = 7 and milk = 14

Vessel 2: water = 24 and milk = 60

Therefore, in the final mixture, the ratio of water to milk is:

(7 + 24)/(14 + 60) = 31/74

Answer: A
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Quantity of water in the resultant mixture
1*1/3 + 4*2/7 = 1/3 + 8/7 = 31/21

Quantity of milk in the resultant mixture
1*2/3 + 4*5/7 = 2/3 + 20/7 = 74/21

Required ratio
31/21 : 74/21
= 31 : 74
IMO A

I need details,, why cant sole by using allegation method.

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Hi,

Here is the alligation diagram that can be used while solving questions like this. You need to keep two things in mind while using this:

1. The higher value (HV), the lower value (LV) and the mean value (MV) have to be all values that are associated to the word per (for example: percents, averages, costs)
2. You select a quantity that overlaps between the higher, lower and mean values.



Now here the question gives us two solutions of milk and water,

Solution 1 - W : M -----> 1 : 2 -----> W = 1/3 and M = 2/3

Solution 2 - W : M -----> 2 : 5 -----> W = 2/7 and M = 5/7

Now we need to mix these two in the ratio of 1 : 4, so we have been given HV - MV : MV - LV = 1 : 4

Drawing the mixture diagram for solution 1



Drawing the mixture diagram for solution 2



Combining these two ratios we get


_________________

w/m=1/2; w=1/3 parts
w/m=2/5; w=2/7 parts

1/3____m_____2/7
_____1_____4_____

1/4=2/7-m/m-1/3, m-1/3=4(2-7m/7), 3m-1=12(2-7m/7),
21m-7=24-84m, 105m=31, m=31/105=w/total,
milk=105-31=74, w/m=31/74

Ans (A)

fraction of water in mixture = (1/5)*(1/3)+(4/5)*(2/7)=155/(15*35)
Fraction of milk in mixture = (1/5)*(2/3)+(4/5)*(5/7) = 370/(15*35)
Ratio of Water to Milk = 155/370 = 31:74.

We can let the first vessel contain 1 liter of water and 2 liters of milk. So the first vessel contains 3 liters of content. Since the content of the first vessel to that of the second vessel is 1:4, the second vessel contains 3(4) = 12 liters of content. Since the ratio of water and milk in the second vessel is 2:5, we can create the equation:

2x + 5x = 12

7x = 12

x = 12/7

Therefore, the second vessel contains 2(12/7) = 24/7 liters of water and 5(12/7) = 60/7 liters of milk.

Finally, the ratio of water and milk in the resulting mixture is:

(1 + 24/7)/(2 + 60/7) = (7 + 24)/(14 + 60) = 31/74

Answer: A
_________________

Assumption 1:
Assume that Vessel 1 contains 30 litres (or whichever units you commonly use) of the mixture of water + milk and Vessel 2 contains 70 litres of the mixture.

Assumption 2:
Combined this gives us 100 litres. For simplicity, we further assume that all contents of both vessels have been mixed (this would be nice to mention maybe ). I am going to define this as Final Mixture or FM.

Now we are told that in the FM the ratio of milk to water is 1:4. We then transform to the quantity of FM defined in Assumption 2 above i.e. 20 litres of milk and 80 litres of water are there in FM.

This where it gets a bit tricky.
In Vessel 1 we had 10 litres of milk but we have 30 litres in the FM so we need to reduce the 30 litres by the appropriate scale factor which is 10/3. doing the same for Vessel 2 gives us (8/7)*20
So the amount of milk in FM is equal to 621/21

Next, we calculate the amount of water in the FM which comes down to (8/7)*50 + (2/3)*20 which is equal to 1480/21

After simplifying the resulting fractions we are now left with 620:1480
Dividing this by 20 yields 31:74

Option A

Alligation is used when we are given the final weighted average value and are asked to find out the weights in the original value. We use weighted average when we are provided with the initial weights and we are asked to find out the final weighted average value.

In this question, we are given the latter information, so we use weighted average. Remember, alligation is just another way to represent weighted averages to make solving questions more convenient and intuitive, so there's no such big difference between the two. It's a matter of convenience.

Water to Total = (1/3 + 4*2/7)/5 = 31/105

Water to Milk = 31:74

A