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**Hint:**Use rule of the allegation and before that find the individual ration of each is water and milk complete step by step solution. Multiplying or dividing each term by the same nonzero number will give an equal ratio. For example, the ratio 2:4 is equal to the ratio 1:2. To tell if two ratios are equal, use a calculator and divide. If the division gives the same answer for both ratios, then they are equal.

**Complete step by step solution:**In the first mixture of water and milk, we have

Total part in the first mixture $=5$

Milk is the first mixture $=\,\dfrac{2}{5}$ part

Water in the first mixture $=\,\dfrac{3}{5}$ part

In the second mixture of water and milk, we have

Total part in second mixture $=\,10$

Milk in the second mixture $=\,\dfrac{3}{10}$ part

Water in the second mixture $=\,\dfrac{7}{10}$ part

Now, in the final part, it is written that the ratio of milk to water $=\,4:7$

So, milk in the final part $=\,\dfrac{4}{11}$

As the total in the final part is 11

And water in the final part $=\,\dfrac{7}{11}$

Therefore if we apply the rule of the allegation which is, when different quantities of different ingredients are mixed together to produce a mixture of a mean value, the ratio of their quantities is inversely proportional to the difference on the cost from the mean value

So, using the rule of the allegation, we get

Milk: water $=\,\dfrac{7}{11}:\dfrac{55}{2}=\,7:4$

**Therefore the answer is 7:4, it this is the ratio of milk and water is the new mixture is which ratio of milk to water is $4:7$.**

**Note:**In this type of question if we follow the information given and adding a few rules like the rule of allegation in this will help to solve his type of question very easily. Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. Eg. $4:5\,=\,8:10\,=\,12:15$. Also, 4: 6 = 2: 3.

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