In 2 litres of first container,
Spirit = $$2 \times \frac{4}{5} = \frac{8}{5}$$
water = $$2 \times \frac{1}{5} = \frac{2}{5}$$

In 3 litres of first container,
Spirit =$$3 \times \frac{11}{15} = \frac{11}{5}$$
water = $$3 \times \frac{4}{15} = \frac{4}{5}$$

In 4 litres of first container,
Spirit =$$4 \times \frac{7}{10} = \frac{14}{5}$$
water = $$4 \times \frac{3}{10} = \frac{6}{5}$$

∴ Required ratio
$$ = \left( {\frac{8}{5} + \frac{{11}}{5} + \frac{{14}}{5}} \right) : \left( {\frac{2}{5} + \frac{4}{5} + \frac{6}{5}} \right) $$
$$ = \frac{{33}}{5} : \frac{{12}}{5} $$
= 33 : 12
= 11 : 4

Quantity of milk in 729 litre of mixture
= 7 × $$\frac{{729}}{{9}}$$ = 567 litre
Quantity of water
= 729 - 567
= 162 litres
Let x litre of water be added to become ratio 7 : 3

Or, $$\frac{{7}}{{3}}$$ = $$\frac{{567}}{{162 + {\text{x}}}}$$
Or, 162 × 7 + 7x = 567 × 3
Or, 7x = 1701 - 1134 = 567
Or, x = $$\frac{{567}}{{7}}$$ = 81 litres water is to be added.

In 300 kg of solution,
Percentage of sugar = 40%
∴ Sugar = $$\frac{{300 \times 40}}{{100}}$$   = 120 kg
Let x kg of sugar be added.
Now, according to question
$$\eqalign{ & \frac{{120 + x}}{{300 + x}} = \frac{1}{2} \cr & \Rightarrow 240 + 2x = 300 + x \cr & \Rightarrow 2x - x = 300 - 240 \cr & \Rightarrow x = 60\,{\text{kg}} \cr} $$

Cost of (18g of 1^st metal + 18g of 2^nd metal) = Rs. 165.60
Cost of (1g of 1^st metal + 1g metal of 2^nd metal) = Rs. 9.20
Hence cost of 1g of 2^nd metal,
= 9.20 - 6.70
= Rs. 2.5
Mean price = Rs. $$\frac{{87}}{{18}}$$

Now, $$\frac{{{\text{quantity}}\,{\text{of}}\,{1^{{\text{st}}}}\,{\text{metal}}}}{{{\text{quantity}}\,{\text{of}}\,{2^{{\text{nd}}}}\,{\text{metal}}}}$$     $$ = $$ $$\frac{{14}}{6}:\frac{{56}}{{30}} = 5:4$$
Quantity of 2^nd metal = $$\frac{{18 \times 4}}{9}$$  = 8g

Suppose vessel initially contains 8 litres of liquid.
Let x litres of this liquid be replaced with water.
Quantity of water in new mixture = $$\left( {3 - \frac{{3x}}{8} + x} \right)$$   litres
Quantity of syrup in new mixture = $$\left( {5 - \frac{{5x}}{8}} \right)$$   litres
$$\eqalign{ & \refore {3 - \frac{{3x}}{8} + x} = {5 - \frac{{5x}}{8}} \cr & \Rightarrow 5x + 24 = 40 - 5x \cr & \Rightarrow 10x = 16 \cr & \Rightarrow x = \frac{8}{5} \cr} $$
So, part of mixture replaced
$$\eqalign{ & = {\frac{8}{5} \times \frac{1}{8}} \cr & = \frac{1}{5} \cr} $$

$$\eqalign{ & {\text{According to question,}} \cr & {\text{Mixture }} - {\text{1 = 20 litres}} \cr & {\text{Mixture }} - {\text{2 = 36 litres}} \cr & {\text{In Mixture }} - {\text{1 ratio of }} \cr & {\text{ }}\frac{{{\text{Spirit}}}}{{{\text{Water}}}} = \left. {\frac{3}{7}} \right\rangle 10{\text{ units}} \cr & {\text{In Mixture }} - 2{\text{ ratio of}} \cr & {\text{ }}\frac{{{\text{Spirit}}}}{{{\text{Water}}}} = \left. {\frac{7}{5}} \right\rangle 12{\text{ units}} \cr & \cr & {\text{10 units }} \to {\text{20 litres}} \cr & \,\,\,{\text{1 unit }}\,\,\, \to {\text{2 litres}} \cr & {\text{12 units }} \to {\text{36 litres}} \cr & \,\,\,{\text{1 unit }}\,\,\, \to {\text{3 litres}} \cr & \cr & \refore {\text{In Mixture }} - {\text{1 ratio of }} \cr & \frac{{{\text{Spirit}}}}{{{\text{Water}}}} = \frac{{3 \times 2}}{{7 \times 2}} = \frac{6}{{14}} \cr & \cr & \refore {\text{In Mixture }} - 2{\text{ ratio of }} \cr & \frac{{{\text{Spirit}}}}{{{\text{Water}}}} = \frac{{7 \times 3}}{{5 \times 3}} = \frac{{21}}{{15}} \cr & \cr & {\text{Ratio of spirits and water}} \cr & {\text{ = }}\frac{{6 + 21}}{{14 + 15}} \cr & {\text{ = }}\frac{{27}}{{29}} \cr & {\text{ = 27:29}} \cr} $$

By rule of alligation, we have:

Ratio of 1^st and 2^nd parts
= 4 : 6
= 2 : 3
∴ Quantity of 2^nd kind
$$\eqalign{ & = \left( {\frac{3}{5} \times 1000} \right)\,{\text{kg}} \cr & = 600\,{\text{kg}} \cr} $$

Heads Count = 200
Legs count = 580
Average Legs count for per head = $$\frac{{580}}{{200}}$$ = $$\frac{{29}}{{10}}$$

Rabbits : Pigeons = $$\frac{{9}}{{10}}$$ : $$\frac{{11}}{{10}}$$ = 9 : 11
Number of Pigeons = $$200 \times \frac{{11}}{{20}}$$  = 110

Let price of water = Rs. 0
According to question,
Cost price of pure milk = Rs. 1.80
Cost price of mixture = Rs. 0.90
Now using Alligation method.

Therefore amount of water is equal to amount of Milk in mixture. i.e. 16 litter Milk in Given Mixture

Let cost of 1 litre milk be Rs. 1 Milk in 1 litre mixture in A = $$\frac{8}{13}$$ litre;
Cost price of 1 litre mixture in A = Rs. $$\frac{8}{13}$$
Milk in 1 litre mixture in B = $$\frac{5}{7}$$ litre;
Cost price of 1 litre mixture in B = Rs. $$\frac{5}{7}$$
Milk in 1 litre of final mixture
$$\eqalign{ & = {\frac{{900}}{{13}} \times \frac{1}{{100}} \times 1} \cr & = \frac{9}{{13}}{\text{ litre}}{\text{}} \cr} $$
Mean price = Rs. $$\frac{9}{{13}}$$
By rule of alligation, we have:

∴ Required ratio = $$\frac{2}{{91}}$$ : $$\frac{1}{{13}}$$ = 2 : 7