"Alligation"? Is that what Alligators do?
"A vessel contains 70 lts of liquid which have 10% of water." Okay, 10% of 70 is 7 so we have 7 l of water in 70 l overall.
"How much water is added to the mixture so that the final mixture contains 12.5% of water?"
Let x be the amount of water added, in liters. Then we have x+ 7 liters of water, in x+ 70 liters overall. The concentration is now . Solve that for x.