In a mixture, water and milk are in the ratio of 3:5. In the second mixture, the ratio of water and milk is 9:11. If these two mixtures are mixed to form a new mixture in which water and milk are in the ratio 5:7, then what is the ratio of these two mixtures in the new mixture?

Let's solve this using the method of Alligation by focusing on the proportion of water in each mixture.
Proportion of water in the 1st mixture (m
1
​
) =
3+5
3
​
=
8
3
​


Proportion of water in the 2nd mixture (m
2
​
) =
9+11
9
​
=
20
9
​


Proportion of water in the final mixture (m
f
​
) =
5+7
5
​
=
12
5
​


To simplify calculations, find the LCM of the denominators (8,20,12), which is 120.
Multiply each proportion by 120:

Value for 1st mixture =
8
3
​
×120=45

Value for 2nd mixture =
20
9
​
×120=54

Mean value =
12
5
​
×120=50

Now apply alligation rule:

Mixture 1 (45) Mixture 2 (54)
\ /
(50)
/ \
(54 - 50) (50 - 45)
= 4 = 5
Thus, the required ratio is 4:5.