Three vessels whose volumes are in the ratio of 3:5:7 are full of lime and silica. The ratio of lime to silica in the 1st, 2nd, and 3rd vessels is 5:1, 7:3, and 3:4, respectively. If all three mixtures are mixed in another vessel, then find the ratio of lime and silica in the final mixture.

The correct option is 3:2.

Let us step-by-step calculate the total amount of lime and silica present in the final mixture after combining the three vessels.

We are given that the volumes of the three vessels are in the ratio 3 : 5 : 7.
Let the capacities (total volumes) of the three vessels be:

Volume of Vessel 1 = 3 units
Volume of Vessel 2 = 5 units
Volume of Vessel 3 = 7 units

Now, let us calculate the individual amounts of lime and silica in each vessel based on their respective ratios.

1. First Vessel:
The ratio of lime to silica is 5 : 1.
Total ratio parts = 5 + 1 = 6.
Amount of Lime in 1st vessel = $\frac{5}{6}\times 3=\frac{5}{2}$ units.
Amount of Silica in 1st vessel = $\frac{1}{6}\times 3=\frac{1}{2}$ units.

2. Second Vessel:
The ratio of lime to silica is 7 : 3.
Total ratio parts = 7 + 3 = 10.
Amount of Lime in 2nd vessel = $\frac{7}{10}\times 5=\frac{7}{2}$ units.
Amount of Silica in 2nd vessel = $\frac{3}{10}\times 5=\frac{3}{2}$ units.

3. Third Vessel:
The ratio of lime to silica is 3 : 4.
Total ratio parts = 3 + 4 = 7.
Amount of Lime in 3rd vessel = $\frac{3}{7}\times 7=3$ units.
Amount of Silica in 3rd vessel = $\frac{4}{7}\times 7=4$ units.

Now, we find the total amount of Lime and total amount of Silica in the combined mixture:

Total Lime = $\frac{5}{2}+\frac{7}{2}+3$

Total Lime = $\frac{5+7}{2}+3=6+3=9$ units.

Total Silica = $\frac{1}{2}+\frac{3}{2}+4$

Total Silica = $\frac{1+3}{2}+4=2+4=6$ units.

Finally, the ratio of lime to silica in the final mixture is:

Ratio = Total Lime : Total Silica = 9 : 6

Simplifying the ratio by dividing both terms by 3 gives:

Ratio = 3 : 2